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PARA- AND ORTHOHYDROGEN
L. Farkas, Cambridge*
Contents
I. Introduction.
II. The theory of the para- and ortho-states of H₂ and its consequences.
1. Quantum-mechanical explanation of the para- and ortho-states. 2. Thermodynamic consequences of the splitting of terms.
III. Experimental data on the properties of para- and orthohydrogen: 1. Preparation of pure parahydrogen. 2. Measurement of the concentration in a mixture of pH₂—oH₂. 3. Specific heat. 4. Heats of transition. 5. Entropy, free energy, chemical constant, and vapor pressure. 6. Optical properties. 7. Magnetic properties of pH₂ and oH₂. 8. Chemical properties, gas-kinetic quantities.
IV. Mutual conversion of pH₂ and oH₂. 1. Homogeneous conversion: a) Irradiation. b) Thermal conversion. c) Transformations with the aid of paramagnetic bodies. d) Conversion of liquid and solid hydrogen. 2. Heterogeneous catalysis of conversion. a) Low-temperature mechanism. b) High-temperature mechanism.
V. Applications. 1. Measurement of the concentration of H-atoms. 2. Energy exchange at the boundary surface metal—H₂. 3. Determination of the coefficients of self-diffusion of H₂.
Literature
I. Introduction
The discovery of para- and orthohydrogen was one of the best experimental confirmations of the conclusions of quantum mechanics, which had predicted in advance the existence of these two kinds of hydrogen. Further investigations of their properties revealed a whole series of possibilities for application and results important for physical chemistry. The historical development that led to the discovery of both hydrogens was, in brief, as follows.
It had long been known (Mecke⁵⁶, ⁵⁷) that there is an alternation of intensities or the disappearance of lines in the band spectra of molecules with identical nuclei. Heisenberg⁴⁶, ⁴⁷, and later Hund⁵⁰, connected this with specific quantum-mechanical symmetry conditions, which, when applied to the case of H₂, lead to the conclusion that its molecular states must be divided into two types that practically do not combine with one another. The properties of the spectra can be explained by this circumstance. An important consequence of this splitting of terms was discovered by Dennison, who showed that at low—
* Ergebn. d. exakt. Naturwiss., 12, 163, 1933. Translation by M. Gogoberidze.
at what temperatures hydrogen is in many respects similar to a mixture of gases that are not in a state of thermal equilibrium with one another. This supposition enabled him to explain the peculiar course, discovered as early as 1912 by Eucken¹⁸, of the curve of the rotational heat capacity of H₂ at low temperatures. Thus it could be expected that, as a result of the spontaneous establishment of equilibrium at low temperatures, certain thermal properties of H₂ would change with time. Bonhoeffer and Harteck³, ⁴, ⁵, ⁶, ⁷ deserve the credit for discovering the possibility of catalyzing the transition to equilibrium, and also for developing a simple and sensitive method for detecting both kinds of hydrogen. In this way they not only gave the first confirmation of Dennison’s theory, but were also able to discover a whole series of characteristic differences between the two modifications. Already after Bonhoeffer and Harteck, Giok and Johnston³⁷ carried out “transition experiments” (Umwandlungsversuche), in which they observed the change with time of the vapor pressure at the triple point; however, because of the smallness of the observed effect, they could draw no definite conclusion about the existence of the modifications, and only the measurements of Bonhoeffer and Harteck confirmed the reality of their observations. Almost simultaneously with their measurements, Eucken and Hiller¹⁹, ²⁰ discovered the existence of two modifications of H₂ by measuring the specific heats of hydrogen at low temperatures.
The experimental method of Bonhoeffer and Harteck lay at the basis of most of the subsequent experimental investigations of para- and orthohydrogen. In our subsequent exposition we shall not follow the historical sequence of the works, but shall pay attention only to the logical connection.
II. Theory of the para- and ortho-states of H₂ and its consequences
1. Quantum-mechanical explanation of the para- and ortho-state
By para- (pH₂) and ortho- (oH₂) hydrogen we shall mean two different kinds of ordinary hydrogen molecules, differing from one another in the orientation of the nuclear magnets and in their rotational quantum numbers. In pH₂, in the normal state, the nuclear magnets are antiparallel (↑↓), and the molecule can then have only even rotational quantum numbers. In oH₂, the nuclear magnets are parallel (↑↑), and the rotational quantum numbers can assume only odd values. Whereas an asymmetrically constructed molecule, for example HCl, can freely pass from a rotational state with an even quantum number to a rotation with an odd one, in a molecule built of identical nuclei, such as H₂, such a transition does not occur spontaneously. Thus the difference between pH₂ and oH₂ in ordinary hydrogen is not reduced merely to the designation of a certain internal structure, but as a result
the prohibition of the transition \(p \rightleftarrows o\) indicates the existence of two gases, distinct in many respects and capable of being separated. The mutual relation between the orientation of the nuclear spins and the rotational quantum numbers, as well as the prohibition of the transition \(p \rightleftarrows o\), follow from quantum mechanics and will be examined below.
The energy states of an atomic system can be determined with the aid of the Schrödinger equation, which for the internal degrees of freedom of the molecule \(\mathrm{H}_2\) may be written as follows (the electrons and nuclei are here treated as point charges, i.e. without taking spin into account):
\[ \left\{ \frac{1}{m}(\Delta_1+\Delta_2)+\frac{1}{M}\Delta_{\mathrm{I,II}}+ \frac{8\pi^2}{h^2}(E_n-V) \right\}\Psi_n=0 . \tag{1} \]
The potential energy is then expressed as follows:
\[ V=\frac{e^2}{r_{\mathrm{I,II}}} +\frac{e^2}{r_{1,2}} -\frac{e^2}{r_{\mathrm{I},1}} -\frac{e^2}{r_{\mathrm{I},2}} -\frac{e^2}{r_{\mathrm{II},1}} -\frac{e^2}{r_{\mathrm{II},2}}, \]
where 1 and 2 denote the coordinates of the electron, and I and II the nuclei.
For the ground state of the molecule \(\mathrm{H}_2\), and also for all \(\Sigma\)-states, the solution of the above equation may be written approximately in the form:
\[ \Psi_{nj}=\chi_n(1,2,\mathrm{I},\mathrm{II})\,P_j(\vartheta,\varphi). \tag{2} \]
Here \(\chi\) denotes a function which is unchanged for all rotational states belonging to one and the same electronic-vibrational state \((n)\) and depends only on the configuration of the molecule (on the mutual distances of the nuclei and electrons). \(\chi\) also does not depend on the position of the molecule in space, whereas \(P_j(\vartheta,\varphi)\) depends only on the direction of the line joining the nuclei: \(\vartheta\) and \(\varphi\) are the polar angle and the azimuth of the line joining the nuclei in a coordinate system fixed in space.
For \(P_j(\vartheta,\varphi)\), on the basis of the above Schrödinger equation, one may write the equation:
\[ \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left(\sin\vartheta\frac{\partial P_j}{\partial\vartheta}\right) +\frac{1}{\sin^2\vartheta}\frac{\partial^2 P_j}{\partial\varphi^2} +\frac{4\pi^2 E_j}{h^2} R^2 M P_j=0, \tag{3} \]
where \(MR^2=2J\) is the moment of inertia of the molecule \(\mathrm{H}_2\).* The solutions of this equation have the form:
\[ P_j=P_{jm}(\cos\vartheta)e^{\pm im\varphi}\qquad (m\leq j), \tag{4} \]
* The \(R\) introduced here is the mean internuclear distance \(r_{\mathrm{I,II}}\) for the electronic-vibrational state under consideration. For the ground term of hydrogen the moment of inertia, according to Takeo Hori’s investigations\({}^{49}\) on band spectra in the lowest vibrational state \((\tfrac12\)-quantum), is \(J=4.66\cdot10^{-41}\,\mathrm{g\,cm^2}\).
and the eigenvalues
\[ E_j=\frac{j(j+1)h^2}{8\pi^2 I}. \tag{5} \]
The eigenvalue \(E_j\) represents the energy of the rotational state with quantum number \(j\) and with angular momentum
\[ \frac{h}{2\pi}\sqrt{j(j+1)}. \]
The functions \(P_{jm}(\cos\vartheta)\) are spherical surface functions,* and we see that, since \(E_j\) does not depend on \(m\), each eigenvalue \(E_j\) corresponds to \(2j+1\) eigenfunctions:
\[ P_{j0},\quad P_{j1}e^{+i\varphi},\quad P_{j1}e^{-i\varphi},\quad P_{j2}e^{+2i\varphi}\ldots P_{jj}e^{+ji\varphi},\quad P_{jj}e^{-ji\varphi}. \]
This \((2j+1)\)-fold degeneracy of the eigenvalues \(E_j\) means that their statistical weight is
\[ g_j=2j+1. \]
In the old quantum theory, instead of \(2j+1\) independent eigenfunctions, we had \(2j+1\) integral spatial quantizations of the angular momentum \(\frac{h}{2\pi}j\) in a magnetic field (\(m\) is therefore also called the magnetic quantum number).
Until now we have considered nuclei without taking account of the nuclear magnetic moment. The mutual relation between the rotational quantum number and the direction of the nuclear spin follows from the fundamental principle discovered by Heisenberg \(^{46,47}\).
The indistinguishability of electrons and nuclei is taken into account by the eigenfunctions of the Schrödinger equation (neglecting electronic and nuclear spins) in that, upon interchange of the Cartesian coordinates of particles, these functions change only their sign. Thus there exist two types of wave functions (one with the sign \(+\), and the other with the sign
\[ {}^{*}\ \text{The functions } P_{jm}(x) \text{ are defined by the equation} \]
\[ P_{jm}(x)=(1-x^2)^{\frac{m}{2}}\frac{d^m P_j(x)}{dx^m}, \qquad m\leq j, \]
and
\[ P_j=\frac{1}{2^j j!}\frac{d^j(x^2-1)^j}{dx^j}. \]
The first spherical surface functions have the form:
\[ P_{00}=1,\quad P_{10}=\cos\vartheta,\quad P_{11}=\sin\vartheta, \]
\[ P_{20}=\frac{1}{2}(3\cos^2\vartheta-1),\quad P_{21}=3\sin\vartheta\cos\vartheta,\quad P_{22}=3\sin^2\vartheta, \]
\[ P_{30}=\frac{1}{2}(5\cos^3\vartheta-3\cos\vartheta),\quad P_{31}=\frac{3}{2}\sin\vartheta(5\cos^2\vartheta-1), \]
\[ P_{32}=15\sin^2\vartheta\cos\vartheta,\quad P_{33}=15\sin^2\vartheta. \]
...from which, in nature, for atomic spectra only those are realized which, when the electronic spin is taken into account, upon interchange of some pair of electrons change their sign. Thus, for example, one must have
\[
\Psi(1\ldots k\ l\ldots n\ s_1\ldots s_k\ s_l\ldots s_n)=
\]
\[
=-\Psi(1\ldots l\ k\ldots n\ s_1\ldots s_l\ s_k\ldots s_n),
\tag{6}
\]
where \(k\) and \(l\) are Cartesian coordinates, and \(s_k\) and \(s_l\) are two values of the spin coordinates of the \(k\)-th and \(l\)-th electrons.
This “antisymmetry” of the wave functions with respect to all electrons is a consequence of the Pauli exclusion principle; on this basis, wave functions having electron pairs coinciding in all coordinates must be equal to zero.
Since, in the case of the \(\mathrm{H}_2\) molecule, the eigenfunction of the rotational states in the ground state is known, it is possible to determine in which state replacement of the Cartesian coordinates of the nuclei will cause a change of sign of the function, and in which state it will not. In Figs. 1a and 1b the positions of the electrons and nuclei before and after interchange are shown schematically. In order to make it more convenient to compare the wave functions belonging to these two configurations, we introduce still another, intermediate configuration 1c, which differs from 1a by the opposite direction of the line connecting the nuclei; in other words, the internal structure of 1a and 1c is identical, and only \(\vartheta\) has been replaced by \(\pi-\vartheta\) and \(\varphi\) by \(\varphi+\pi\). If this substitution is introduced into \(P_{jm}\), then it is multiplied by \((-1)^j\). Now, in order to pass from 1c to 1b, it is necessary to reflect both electrons in mirror image with respect to the line connecting the nuclei and with respect to the perpendicular restored to its midpoint. In the ground state, under this transformation \(\chi_0\) does not change, so that the eigenfunction \(\chi_0 P_{jm}\) for even rotational terms is symmetric \(\{(-1)^{2j}=+1\}\), and for odd rotational terms is antisymmetric \(\{(-1)^{2j+1}=-1\}\), with respect to the Cartesian coordinates of the nuclei*.
Fig. 1 a–c: scheme of exchange of nuclei in the \(\mathrm{H}_2\) molecule.
* Similar electronic terms we shall henceforth call even and denote by the index \(g\).
** In the first excited state \(\chi_1\) nevertheless changes its sign (denoting the odd term by \(u\)), so that in this state the even rotational terms are antisymmetric with respect to the Cartesian coordinates of the nuclei, and the odd rotational terms are symmetric (see Wigner and Witmer \(^{74}\), where the symmetry properties \(g\) and \(u\) are denoted by \(+\) and \(-\)).
If one also takes into account the nuclear spins of the protons and regards them as equal to \(\dfrac{1}{2}\dfrac{h}{2\pi}\), then they, like two electrons in the helium atom, may be parallel or antiparallel. The antiparallel orientation of the nuclear spins corresponds to a singlet nuclear state with statistical weight 1, while the parallel orientation of the spins corresponds to a triplet nuclear state with statistical weight 3. It is impossible to say in advance which of the two possible orientations of the nuclear spins will occur in reality, for example in even rotational states. Analysis of the spectra of \(\mathrm{H}_2\) molecules shows that in the lower electronic state of the molecule each rotational state is possible only with one of the two orientations of the nuclear spins. If these are the even rotational terms associated with the antiparallel orientation of the nuclear spins, then the series of statistical weights \(g_j\) has the form:
\[ 1,\ 3\cdot 3,\ 5,\ 3\cdot 7,\ 9,\ 3\cdot 11, \tag{7} \]
in the opposite case we obtain
\[ 3\cdot 1,\ 3,\ 3\cdot 5,\ 7,\ 3\cdot 9,\ 11. \tag{7a} \]
The possibility of choosing between these two cases was given by Dennison\(^{16}\), who was able to establish the temperature dependence of the rotational heats of \(\mathrm{H}_2\) solely from the statistical weight according to (7)*. With respect to the symmetry properties of the wave functions this means that upon exchange of the nuclei and their spins the sign of the function must change, that in this case the Pauli prohibition is also valid for protons**. States that are antisymmetric in the Cartesian coordinates of the nucleus will be denoted by us as ortho-(o)-states, and co-
* For more detail see Chapter II, 2 and Chapter III, 3.
** Although for \(\mathrm{H}_2\) the proper functions with respect to all particles, taking into account their spins, are antisymmetric, this does not, however, in any way necessarily belong to the proper functions of a molecule composed of identical nuclei. The rules established by Heitler and Herzberg\(^{48}\), as well as by Wigner\(^{46}\), for the symmetry properties of the wave functions of diatomic molecules composed of identical nuclei state that nuclei with odd atomic weight obey Fermi statistics (the wave function is antisymmetric), while nuclei with even atomic weight obey Bose statistics (the wave function is symmetric). Protons furnish an example of the former, and nitrogen nuclei an example of the latter. (The deuteron \(\mathrm{H}^2\) likewise obeys Bose statistics; see Lewis and Ashley, Phys. Rev. 43, 837, 1933.) The relation between rotational quantum numbers and the orientation of nuclear spins is established for diatomic molecules in exactly the same way as for \(\mathrm{H}_2\); in the case of \(\Sigma_g^-\)-terms and Fermi statistics it is necessary to multiply the statistical weights of the even rotational terms by \(i(2i+1)\), and of the odd rotational terms by \((i+1)(2i+1)\), where \(i\) is the nuclear spin. In the case of \(\Sigma_u\)-terms or Bose statistics for the nuclei one must proceed in the opposite way. Molecules whose nuclei have no nuclear spin (\(i=1\)) form a special case, in which half of the rotational terms [namely those whose statistical weight would have had to be multiplied by \(i(2i+1)\)] completely disappear.
states are symmetric—like the para-\((p)\)-states.
Between the \(p\)- and \(o\)-states there is a very strict prohibition of transitions. If the nuclei had no magnetic moment, then the probability of transition due to spontaneous radiation would be exactly equal to zero, since the integral determining the transition probability (the \(x\)-component) vanishes in this case*:
\[ \int (x_1+x_2-X_{\mathrm I}-X_{\mathrm{II}})\Psi_{0j}^{*}\Psi_{0j+1}\,d\tau=0 . \tag{8} \]
As a result of the interaction of the nuclear magnets, according to Wigner \(^{75}\), small terms are added to the proper functions \(\Psi_{0j}\) and \(\Psi_{0j+1}\), violating their mutual symmetry or antisymmetry in the Cartesian coordinates of the nuclei. Instead of \(\Psi_{0j}\) one may put
\[ \Psi'_{0j}=\Psi_{0j}+\sum_{j'} a_{j'}\Psi_{0j'}+\sum_{nj'} b_{nj'}\Psi_{nj'}, \]
where \(j\) denotes summation over rotational states, and \(n\)—over electronic states. If, instead of \(\Psi_{0j+1}\), the corresponding expression is substituted into integral (8), we obtain terms of various kinds: first, the integral (8) itself, which, however, is equal to zero; second, terms
\[ a_{j'}\int \Psi_{0j'}(x_1+x_2-X_{\mathrm I}-X_{\mathrm{II}})\Psi_{0j+1}^{*}\,d\tau, \]
but these also vanish even when \(j'\) and \(j+1\) denote two \(p\)- or two \(o\)-states (coincidence of terms of the rotational spectra). The remaining terms, of order
\[ b_{nj'}\int \Psi_{nj'}(x_1+x_2-X_{\mathrm I}-X_{\mathrm{II}})\Psi_{nj'+1}^{*}\,d\tau \]
may be neglected; because of the large distance from the ground state of \(\mathrm{H}_2\) to the next electronic state \((E\sim 10\ \mathrm{V})\) and the smallness of the energy of interaction of the nuclei \((\sim 3\cdot 10^{-4}\ \mathrm{V})\), these terms are so small that they lead to a transition probability of \(10^{-10}\ \mathrm{sec}\).
In addition to these extraordinarily slow molecular transitions, there also exist bimolecular transitions between \(p\)- and \(o\)-states. For example, one should consider the nonmechanical exchange of protons in the collision of two \(\mathrm{H}_2\) molecules, during which disorientation of the nuclear spins may occur. The rate of this reaction was calculated by Halle and Oppenheimer \(^{43}\), who found that its half-period is equal (at atmospheric pressure) to approximately—
* In the first excited electronic state \({}^{1}\Sigma_u\) of the \(\mathrm{H}_2\) molecule, according to the note on the preceding page, the even rotational terms form \(o\)-states, and the odd ones \(p\)-states. In general, states with a larger statistical weight will be denoted as ortho states.
** This is easy to verify if one recalls that the electric moment cannot change upon interchange of the nuclei, whereas the sign of (8) must change, which is possible only when (8) is identically equal to zero.
considerably three years*). In a collision of a \(p\mathrm{H}_2\)- or \(o\mathrm{H}_2\)-molecule with some foreign molecule, a transition of the two forms of hydrogen into one another is also possible. The transition \(p \rightleftarrows o\) in a collision with some foreign paramagnetic molecule will be discussed in Chapter IV, 10. As for the transition in a collision with a diamagnetic molecule, it is still insufficiently investigated. However, a rough estimate shows that such a transition \(p\mathrm{H}_2\) to \(o\mathrm{H}_2\) (or conversely) at room temperature occurs on the average once in \(10^{18}\) collisions.
2. Thermodynamic consequences of the decomposition of the terms
The decomposition of the hydrogen terms into two mutually non-combining, or only very weakly combining, systems has made it possible to derive a number of consequences, especially with respect to the thermal properties of \(\mathrm{H}_2\). Starting from the differences of the energies of the rotational terms and from their statistical weights, one can calculate the thermal properties of both kinds of \(\mathrm{H}_2\).
The ratio \(\beta = \dfrac{p\mathrm{H}_2}{o\mathrm{H}_2}\), corresponding to thermodynamic equilibrium, is determined by Boltzmann’s law:
\[ \beta(T)=\frac{Z_{p\mathrm{H}_2}}{Z_{o\mathrm{H}_2}}, \tag{9} \]
where
\[ Z_{p\mathrm{H}_2}=\sum_{j\;(\mathrm{even})}(2j+1)e^{-\frac{E_j}{kT}} =1+5e^{-6\sigma}+ \]
\[ +9e^{-20\sigma}+13e^{-42\sigma}+\ldots \tag{10} \]
and
\[ Z_{o\mathrm{H}_2}=3\sum_{j\;(\mathrm{odd})}(2j+1)e^{-\frac{E_j}{kT}} =3\left(3e^{-2\sigma}+7e^{-12\sigma}+\right. \]
\[ \left.+11e^{-30\sigma}+\ldots\right) \tag{11} \]
denote sums over states. Here
\[ \sigma=\frac{h^2}{8\pi^2 J kT}=\frac{84.997}{T}. \tag{12} \]
Expression (9) gives, for high temperatures \(\left(kT>\dfrac{h^2}{4\pi^2 J}\right)\), asympto-
* The yield of the reaction (collision) is approximately equal to \(e^{-\left(\frac{M}{m}\right)^{\frac{1}{2}}}\).
PARA- AND ORTHOHYDROGEN
statistical value \(\frac{1}{3}\), while for low temperatures \(\left(kT \ll \frac{h^2}{4\pi^2 J}\right)^*\) it becomes infinite. The limiting value \(\beta=\frac{1}{3}\) depends on the ratio of the nuclear multiplicities; in practice it is established already at room temperature, and a further rise in temperature no longer changes the concentration of the mixture. Thus ordinary hydrogen consists of \(\frac{1}{4}\,p\mathrm{H}_2\) and \(\frac{3}{4}\,o\mathrm{H}_2^{**}\).
In Table 1, after Harkness and Deming\(^{44}\), values of \(\beta\) and \(\%\,p\mathrm{H}_2=\frac{100\beta}{1+\beta}\) are given for various temperatures (see also Fig. 2).
TABLE 1
Thermal equilibrium ratio and percentage content of \(p\mathrm{H}_2\) as a function of temperature
| \(T\) | \(\beta\) | \(\%\,p\mathrm{H}_2\) | \(T\) | \(\beta\) | \(\%\,p\mathrm{H}_2\) | \(T\) | \(\beta\) | \(\%\,p\mathrm{H}_2\) |
|---|---|---|---|---|---|---|---|---|
| 20 | 544,8 | 99,82 | 76 | 1,046 | 51,13 | 95 | 0,8701 | 40,48 |
| 21 | 363,5 | 99,73 | 77 | 1,017 | 50,41 | 100 | 0,6262 | 38,51 |
| 22 | 251,6 | 99,60 | 78 | 0,9894 | 49,73 | 105 | 0,5829 | 36,82 |
| 23 | 179,8 | 99,45 | 79 | 0,9626 | 49,05 | 110 | 0,5456 | 35,30 |
| 24 | 132,2 | 99,25 | 80 | 0,9377 | 48,39 | 115 | 0,5152 | 34,00 |
| 25 | 99,57 | 99,01 | 81 | 0,9140 | 47,75 | 120 | 0,4897 | 32,87 |
| 30 | 32,07 | 96,98 | 82 | 0,8916 | 47,13 | 130 | 0,4498 | 31,03 |
| 35 | 14,28 | 93,45 | 83 | 0,8702 | 46,53 | 140 | 0,4208 | 29,62 |
| 40 | 7,780 | 88,61 | 84 | 0,8500 | 45,95 | 150 | 0,3994 | 28,54 |
| 45 | 4,853 | 82,91 | 85 | 0,8307 | 45,37 | 160 | 0,3835 | 27,72 |
| 50 | 3,327 | 76,89 | 86 | 0,8123 | 44,82 | 170 | 0,3715 | 27,09 |
| 55 | 2,443 | 70,96 | 87 | 0,7981 | 44,39 | 180 | 0,3555 | 26,23 |
| 60 | 1,890 | 65,39 | 88 | 0,7781 | 43,76 | 210 | 0,3463 | 25,72 |
| 65 | 1,521 | 60,33 | 89 | 0,7621 | 43,25 | 230 | 0,3409 | 25,42 |
| 70 | 1,264 | 55,83 | 90 | 0,7469 | 42,75 | 250 | 0,3277 | 25,34 |
| 75 | 1,077 | 51,86 | 91 | 0,7323 | 42,27 | 270 | 0,3337 | 25,13 |
* At the temperature \(T=\frac{h^2}{4\pi^2 kJ}=105^\circ\mathrm{K}\) the ratio \(\frac{p\mathrm{H}_2}{o\mathrm{H}_2}\) already increases twofold in comparison with the limiting value \(\frac{1}{3}\). We see that this “characteristic” temperature lies so high in the case of hydrogen because the first rotational quantum \(\frac{h^2}{4\pi^2 J}\) is especially large here. For other molecules this temperature is smaller in the ratio \(\frac{M}{M_{\mathrm{H}_2}}\), so that, if \(p\)- and \(o\)-states exist, deviations from the normal ratio \(\frac{p}{o}\) should be expected only at very low temperatures.
** In what follows this mixture will be denoted as normal \(\mathrm{H}_2\) \((n\mathrm{H}_2)\), regardless of the temperature at which it is considered. Conversely, by equilibrium hydrogen \((g\mathrm{H}_2)\) we shall mean hydrogen which, at the given temperature, is in thermodynamic equilibrium with respect to \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\). At high temperatures \(g\mathrm{H}_2=n\mathrm{H}_2\).
Upon lowering the temperature, owing to the already mentioned prohibition of the transition \(p \rightleftarrows o\), the equilibrium corresponding to formula (9) is not established; rather, as Denisson1 first pointed out, the ratio \(\frac{1}{4}p\mathrm{H}_2 + \frac{3}{4}o\mathrm{H}_2\) is preserved, so that ordinary hydrogen, cooled to low temperatures, is not in a state of thermodynamic equilibrium. The \(p\mathrm{H}_2\) molecules collect in the zero, and the \(o\mathrm{H}_2\) molecules in the first rotational quantum state*. Assuming that the concentration of the mixture \(\left(\beta=\frac{1}{3}\right)\) does not change with changing temperature,
Fig. 2. Dependence of the concentration of \(p\mathrm{H}_2\) on temperature.
Denisson1 calculated the curves of rotational heats and found that they agree quantitatively with the experimental data of Eucken2. In Chapter III, 3, the experimental data are given.
The heat capacities may be calculated on the basis of the general formula
\[ C=\frac{dE}{dT}, \tag{13} \]
where
\[ E=\frac{3}{2}RT+RT\frac{d\ln Z}{dT}. \tag{14} \]
TABLE 2
Rotational energy of hydrogen in calories per mole
| \(T\) | \(E^{\mathrm{rot}}_{p\mathrm{H}_2}\) | \(E^{\mathrm{rot}}_{o\mathrm{H}_2}\) | \(E^{\mathrm{rot}}_{g\mathrm{H}_2}\) | \(E^{\mathrm{rot}}_{n\mathrm{H}_2}\) |
|---|---|---|---|---|
| 0 | 0.00 | 337.17 | 0.00 | 252.88 |
| 15 | 0.00 | 337.17 | 0.04 | 252.88 |
| 20 | 0.00 | 337.17 | 0.63 | 252.88 |
| 25 | 0.00 | 337.17 | 3.39 | 252.88 |
| 30 | 0.00 | 337.17 | 10.28 | 252.88 |
| 40 | 0.05 | 337.17 | 38.63 | 252.89 |
| 50 | 0.20 | 337.18 | 78.38 | 252.94 |
| 75 | 5.77 | 337.22 | 165.61 | 254.36 |
| 100 | 30.56 | 338.59 | 219.78 | 262.17 |
| 125 | 80.09 | 341.83 | 258.41 | 276.39 |
| 150 | 146.61 | 351.40 | 292.94 | 300.20 |
| 175 | 219.06 | 368.54 | 328.44 | 331.17 |
| 200 | 290.22 | 393.59 | 366.76 | 367.75 |
| 225 | 357.04 | 425.69 | 408.19 | 408.53 |
| 250 | 419.27 | 463.46 | 452.30 | 452.41 |
| 275.1 | 473.34 | 532.16 | 494.84 | 494.91 |
| 298.1 | 529.12 | 546.92 | 542.46 | 542.47 |
and after transformation, for the purely rotational heat capacity we obtain:
\[ C^{\mathrm{rot}}=-R\frac{d}{dT}\frac{d\ln Z}{d\frac{1}{T}} =R\sigma^{2}\frac{d^{2}\ln Z}{d\sigma^{2}} . \tag{15} \]
In Table 2 (Giauque\({}^{40}\)) the purely rotational energies are given
\[ E^{\mathrm{rot}}=E-\frac{3}{2}RT \]
according to the formulas
\[ E^{\mathrm{rot}}_{pH_2} = N\frac{\sum'(2j+1)E_j e^{-\frac{E_j}{kT}}}{Z_{pH_2}}, \qquad (j\text{ even}) \tag{16a} \]
\[ E^{\mathrm{rot}}_{oH_2} = N\frac{3\sum'(2j+1)E_j e^{-\frac{E_j}{kT}}}{Z_{oH_2}}, \qquad (j\text{ odd}) \tag{16b} \]
\[ E^{\mathrm{rot}}_{gH_2} = N\frac{\sum(2j+1)E_j e^{-\frac{E_j}{kT}} +3\sum'(2j+1)E_j e^{-\frac{E_j}{kT}}} {Z_{oH_2}+Z_{pH_2}}, \tag{16c} \]
\[ E^{\mathrm{rot}}_{nH_2} = \frac{1}{4}E^{\mathrm{rot}}_{pH_2} +\frac{3}{4}E^{\mathrm{rot}}_{oH_2}. \tag{16d} \]
For the rotational heat capacities the following expressions are used:
\[ C^{\mathrm{rot}}_{pH_2} = \frac{N}{kT^{2}} \left[ \frac{\sum(2j+1)E_j^{2}e^{-\frac{E_j}{kT}}}{Z_{pH_2}} - \left( \frac{\sum(2j+1)E_j e^{-\frac{E_j}{kT}}}{Z_{pH_2}} \right)^{2} \right], \tag{17a} \]
\[ C^{\mathrm{rot}}_{oH_2} = \frac{N}{kT^{2}} \left[ \frac{3\sum(2j+1)E_j^{2}e^{-\frac{E_j}{kT}}}{Z_{oH_2}} - \left( \frac{3\sum'(2j+1)E_j e^{-\frac{E_j}{kT}}}{Z_{oH_2}} \right)^{2} \right]; \tag{17b} \]
\[ C^{\mathrm{rot}}_{gH_2} = \frac{N}{kT^{2}} \left[ \frac{\sum(2j+1)E_j^{2}e^{-\frac{E_j}{kT}}+3\sum'\ldots} {Z_{pH_2}+Z_{oH_2}} - \right. \]
\[ \left. \left( \frac{\sum(2j+1)E_j e^{-\frac{E_j}{kT}}+3\sum'\ldots} {Z_{pH_2}+Z_{oH_2}} \right)^{2} \right], \tag{17c} \]
\[ C^{\mathrm{rot}}_{nH_2} = \frac{1}{4}C^{\mathrm{rot}}_{pH_2} +\frac{3}{4}C^{\mathrm{rot}}_{oH_2}. \tag{17d} \]
TABLE 3*
Rotational heat capacity of hydrogen in calories per degree per mole
| \(T\) | \(C_{pH_2}^{\mathrm{rot}}\) | \(C_{oH_2}^{\mathrm{rot}}\) | \(C_{gH_2}^{\mathrm{rot}}\) | \(C_{nH_2}^{\mathrm{rot}}\) |
|---|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 15 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 20 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 25 | 0.0000 | 0.0000 | 0.9196 | 0.0000 |
| 30 | 0.0001 | 0.0000 | 1.8795 | 0.0000 |
| 40 | 0.0049 | 0.0000 | 3.4465 | 0.0012 |
| 50 | 0.0349 | 0.0000 | 4.1042 | 0.0100 |
| 75 | 0.5177 | 0.0079 | 2.7263 | 0.1353 |
| 100 | 1.5041 | 0.0731 | 1.7498 | 0.4309 |
| 125 | 2.3981 | 0.3131 | 1.4138 | 0.8343 |
| 150 | 2.8451 | 0.5271 | 1.3801 | 1.1066 |
| 175 | 2.9046 | 0.8464 | 1.4708 | 1.3610 |
| 200 | 2.7674 | 1.1512 | 1.5965 | 1.5553 |
| 225 | 2.5777 | 1.3023 | 1.7148 | 1.6211 |
| 250 | 2.4056 | 1.6049 | 1.8101 | 1.8051 |
| 273.1 | 2.2819 | 1.7378 | 1.8756 | 1.8738 |
| 298.1 | 2.1862 | 1.8377 | 1.9254 | 1.9248 |
Table 3 contains the rotational heat capacities up to room temperatures (see also Fig. 3). In all four cases the asymptotic value is \(k\), and for \(gH_2\) and \(nH_2\) this value has already almost been reached at room temperature. The values of \(C_{nH_2}^{\mathrm{rot}}\) given in Table 3, as we shall see in Chapter III, 3, are in complete agreement with the experimentally determined rotational heat capacities for the various kinds of hydrogen. From this follow two very important consequences: first, that the adoption of series (7) for the statistical weights of the rotational states of \(H_2\) is correct, i.e. that the Pauli exclusion principle is applicable to protons, and, second, that the splitting of terms predicted by quantum mechanics and the prohibition of transitions in \(H_2\) do indeed exist.
Fig. 3. Rotational heat capacity of the various kinds of \(H_2\) at low temperatures.
* The rotational heat capacities were calculated by Dennison\(^{15}\) and by Dieke\(^{40}\), and corrected by Dennison\(^{16}\) and Beutler\(^{1}\).
III. Experimental Data on the Properties of Para- and Orthohydrogen
1. Preparation of Pure Parahydrogen
From Table 1 we see that at low temperatures the equilibrium shifts toward \(p\mathrm{H}_2\). The establishment of equilibrium, which under ordinary conditions proceeds very slowly, can be extraordinarily accelerated with the aid of heterogeneous catalysts, first discovered by Bonhoeffer and Harteck \(^{4,5}\). One of such catalysts is, for example, activated charcoal (Chapter IV, 2a). It was established that \(\mathrm{H}_2\), adsorbed on charcoal, at low temperatures passes into equilibrium hydrogen. To obtain pure \(p\mathrm{H}_2\) or mixtures rich in \(p\mathrm{H}_2\), the following apparatus is used: a vessel of quartz or of Jena glass, to which it is advisable to give the shape shown in Fig. 4, is filled with technical activated charcoal (see also Chapter IV, 2a, where the question of the activity of various kinds of charcoal will be considered) and at a temperature of several hundred degrees is degassed in vacuum for approximately 1 hour. After cooling to a low temperature, the charcoal is saturated with hydrogen. In this case a vessel containing about \(100\ \mathrm{cm}^3\) of activated charcoal, when hydrogen is passed through it, gives per minute about \(200\ \mathrm{cm}^3\) of gas enriched in \(p\mathrm{H}_2\) (at atmospheric pressure)*.
Fig. 4. Apparatus for obtaining \(p\mathrm{H}_2\).
If the activated charcoal is cooled to the temperature of liquid hydrogen, practically pure \(p\mathrm{H}_2\) is obtained. Sometimes liquid air, liquid \(\mathrm{O}_2\), and \(\mathrm{N}_2\) are used as the temperature bath. Assuming that complete equilibrium has been established, the percentage content of \(p\mathrm{H}_2\) can be determined from the temperature of the bath with the aid of Table 1.
In clean glass vessels \(p\mathrm{H}_2\), at ordinary temperature and in the absence of \(\mathrm{O}_2\), is a practically stable gas. It can be passed through rubber tubes and lubricated stopcocks and can be collected over mercury (but not over water; see Chapter IV, 1c).
* According to Taylor and Sherman \(^{72,73}\), instead of activated charcoal one may also use a nickel–kieselguhr preparation obtained by reducing \(\mathrm{NiO}+\) kieselguhr in hydrogen (with a metallic nickel content of 15%).
** Naturally, gas rich in \(p\mathrm{H}_2\) is also obtained when hydrogen adsorbed approximately at atmospheric pressure is desorbed after some time by connecting the vessel to an evacuated flask.
2. Measurement of concentration in a \(pH_2—oH_2\) mixture
The determination of the concentration of parahydrogen is based on the difference in the heat capacities of the two modifications. However, direct measurement of the heat capacities is extremely difficult, and therefore it is usually replaced by measurements of thermal conductivities, whose relation to the heat capacities is known in advance. The advantage of this method, first applied by Bonhoeffer and Harteck, in addition to its unusual simplicity, is also the circumstance that concentration measurements require a very small amount of gas.
TABLE 4
Calculation of the concentration of \(pH_2\). Pressure of \(H_2\) 58 mm
| Wire resistance | Excess concentration of \(pH_2\) (in percent) relative to the original hydrogen |
|---|---|
| 121.40 | 100 \((pH_2)\) |
| 122.18 | 84 |
| 123.94 | 69.2 |
| 126.12 | 29.8 |
| 127.40 | 11.0 |
| 128.12 | 0.0 \((nH_2)\) |
The thermal conductivity \(\lambda\) of both modifications is obtained from the following general equation*:
\[ \lambda=\left(\frac{2.25\,R}{C_v}+1\right)\eta C_v, \tag{18} \]
where \(\eta\) is the coefficient of internal friction, and \(C_v\) is the heat capacity. The factor in parentheses takes into account the circumstance that the greater part of the translational energy will be carried precisely by the fast particles, so that they will play a greater role in the thermal conductivity than those molecules whose energy is found mainly in the form of rotational energy. Since \(\eta\) is the same for both modifications of \(H_2\) (see Chapter III, 8), we obtain:
\[ \frac{\lambda_{pH_2}}{\lambda_{nH_2}} = \frac{C_{pH_2}+4.48}{C_{nH_2}+4.48} = \alpha(T), \tag{19} \]
\[ \lambda_{xH_2} = \lambda_{nH_2} \left( \frac{4}{3}(1-x)+\frac{4x-1}{3}\alpha \right), \tag{20} \]
where \(x\) denotes the content of \(pH_2\). Thus, within the range \(80—250^\circ\) K the thermal conductivities of the two modifications differ substantially from one another.
For measuring thermal conductivities, Schleiermacher’s method is used. Concentration measurements of the \(H_2\) modifications are best carried out on the apparatus shown in Fig. 5. The vessel in which the thermal-conductivity measurement is made is placed in a bath with liquid air and, immediately before each measurement, is filled with the \(H_2\) under investigation to a pressure of \(40—60\) mm. Along the axis of the vessel a platinum wire of thickness \(5—10\,\mu\) is stretched, its length being selected so that the total resistance at \(20^\circ\)C is about \(300—400\,\Omega\). After the vessel has been placed in liquid air and filled with the \(H_2\) under investigation, the wire
* See Herzfeld, Kinetic Theory of Gases, p. 92.
is heated by the current passed through it to a temperature of \(160^\circ\mathrm{C}\)*. The resulting magnitude of the overheating temperature, under otherwise constant conditions, is the smaller, the higher the percentage content of \(p\mathrm{H}_2\) in the gas under study, since the thermal conductivity of \(p\mathrm{H}_2\) is greater than that of \(o\mathrm{H}_2\).
Labels in Fig. 5: test hydrogen; to pump; hydrogen of unknown composition; valve for fine regulation; microscope for reading the manometer; precision resistance box; galvanometer; \(10\Omega\); \(1000\Omega\); 6–10 V.
Fig. 5. Apparatus for measuring the concentration of \(p\mathrm{H}_2\).
The heating temperature of the wire is determined by measuring its resistance with a Wheatstone bridge. The calculation of the concentration from the resistance value is facilitated by the fact that, from the difference between the resistance of the wire in \(n\mathrm{H}_2\) and in \(p\mathrm{H}_2\), the intermediate concentrations can be calculated by a simple proportion. The linearity of the relation between the resistance and the \(p\mathrm{H}_2\) content was found experimentally and is justified to an accuracy of up to \(1\%\), although the difference between the resistances of the filament in \(n\mathrm{H}_2\) and in \(p\mathrm{H}_2\) amounts to no more than \(5\%\) of the total resistance (see Fig. 6). Theoretically this relation is not self-evident. The values of the resistance of the wire are obtained from the equation
\[ \frac{i^2 R_{x\mathrm{H}_2}}{i^2 R_{p\mathrm{H}_2}} = \frac{T_{x\mathrm{H}_2}}{T_{p\mathrm{H}_2}} = \frac{\displaystyle \int_{T_0}^{T_{x\mathrm{H}_2}} \lambda_{x\mathrm{H}_2}\,dT} {\displaystyle \int_{T_0}^{T_{p\mathrm{H}_2}} \lambda_{p\mathrm{H}_2}\,dT}, \tag{21} \]
Fig. 6. Experimental proof of the dependence of the resistance of the wire on the concentration of \(p\mathrm{H}_2\).
* At this temperature the difference between the thermal conductivities of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) reaches its greatest value (see the curves of rotational heats in Fig. 3).
where, instead of \(\lambda_{\mathrm{H}_2}\), it is necessary to substitute the thermal conductivity according to formula (20). The resistance of the platinum wire \(R\) may be regarded as proportional to the absolute temperature, and here we neglect the very small differences in the currents at \(T_{\chi\mathrm{H}_2}\) and \(T_{p\mathrm{H}_2}\). Table 4 gives the values of the resistances for a wire \(6.5\ \mathrm{cm}\) long and \(6\ \mu\) thick for various samples of \(\mathrm{H}_2\), and the concentrations of \(p\mathrm{H}_2\) calculated from them. The pressure was \(58\ \mathrm{mm}\), and the heating voltage \(6\ \mathrm{V}\); the value of the filament resistance corresponds to a temperature of about \(180^\circ\mathrm{K}\).
For concentration measurements at a pressure of \(0.5\ \mathrm{mm}\), Geib and Harteck \(^{35}\) modified the above-described method of measuring thermal conductivities in such a way that, instead of one of the bridge resistances (in Fig. 5), a second element is introduced, the temperature of which is maintained equal to \(0^\circ\mathrm{C}\) and by means of which small pressure fluctuations are compensated.
For concentration measurements at very low pressures, or in cases where only very small quantities of gas are available, Farkas \(^{30}\) developed still another method, also based on the measurement of thermal conductivity.
Since heat transfer at low pressures is proportional to the pressure, very precise and readily reproducible pressure values in individual measurements can be obtained by means of the following artificial device. The measuring vessel, placed in liquid air, is filled with hydrogen to a pressure of \(0.05\ \mathrm{mm}\), so that, at a current \(i_1\) (several milliamperes), the measuring wire will have a resistance \(R_1\), corresponding to a temperature \(T_1\).* For different hydrogen mixtures the pressure \(p\) in the vessel is different; for example, for \(n\mathrm{H}_2\) and \(p\mathrm{H}_2\), they are related as the areas \(T_{\text{liquid air}}T_1BA\) and \(T_{\text{liquid air}}T_1B'A'\) in Fig. 7, which depicts the relation under consideration,
\[ \frac{p_{p\mathrm{H}_2}}{p_{n\mathrm{H}_2}} = \frac{\displaystyle \int_{T_1}^{T_1} C_v^{n\mathrm{H}_2}\,dT} {\displaystyle \int_{T_1}^{T_1} C_v^{p\mathrm{H}_2}\,dT} = \frac{T_{\text{liquid air}}\,T_1BA}{T_{\text{liquid air}}\,T_1B'A'}. \tag{22} \]
If the current is increased to \(i_2\), the wire will assume a temperature \(T_2\), which is the higher the richer the gas under study is in \(p\mathrm{H}_2\). In the cases \(p\mathrm{H}_2\) and \(n\mathrm{H}_2\) we have, approximately,
\[ \frac{T_{\text{liquid air}}\,T_1BA}{T_{\text{liquid air}}\,T_1B'A'} = \frac{T_{\text{liquid air}}\,T_2''CA}{T_{\text{liquid air}}\,T_2' C'A'}. \]
* Practically it is most convenient to proceed in such a way that somewhat more gas than is necessary is admitted into the measuring vessel, and then it is pumped out until the wire assumes the desired temperature. The resistance \(R_1\), corresponding to the temperature \(T_1\), is selected beforehand, even before the measurements, and, observing the passage of the galvanometer pointer through the zero point at the moment when the wire has resistance \(R_1\), one can very accurately establish reproducible pressures of \(\mathrm{H}_2\).
The greatest differences will be achieved when \(T_1 \sim 160^\circ\text{C}\) and \(T_2 \sim 0^\circ\text{C}\). Empirically, in this case as well there exists a linear dependence of the temperature established at a current \(i_2\) on the concentration of \(p\text{H}_2\).
The measuring vessel, which is an apparatus for measuring thermal conductivity and is shown in Fig. 8, is immersed as deeply as possible in liquid air. The measuring wire is a platinum wire \(0.01\) mm thick and about \(5\) cm long.* Before carrying out the actual concentration measurements using this method, it is advisable to make preliminary (calibration) measurements with \(p\text{H}_2\) and \(n\text{H}_2\). It is also necessary to ensure that the surface of the wire has as far as possible constant properties; this is achieved by cleaning off contaminants and by heating in vacuum.
Fig. 7. Diagram of excess pressure and superheating temperature in measurements of \(p\text{H}_2\) concentration according to A. Farkas.
Fig. 8. Vessel for measuring thermal conductivity at low pressures.
3. Specific heat
In Chapter II, 2, theoretical formulas were given for the temperature dependence of the rotational heats for the various kinds of hydrogen. The older measurements of Eucken ¹⁸, and later those of Scheel and Heuse, Brinkworth, Giacomini, and also Partington and Howe**, carried out on \(n\text{H}_2\), agree within the errors of experiment with the course of the rotational heats for a mixture consisting of \(\frac{1}{4}\,p\text{H}_2\) and \(\frac{3}{4}\,o\text{H}_2\).
* Platinum wire coated with quartz (“Taylor process wire” from Backer & Co., Newark, N.Y., U.S.A.) can also be used with good results.
* Ann. Phys. 40, 473, 1913.
* Phil. Mag. 50, 146, 1925.
* Proc. Roy. Soc., Lond. 107, 510, 1925.
* Proc. Roy. Soc., Lond. 109**, 286, 1925.
The data of these measurements are given in Fig. 9, curve I. Since this mixture, according to Dennison’s theory, is not in a state of thermal equilibrium, Eucken and Hiller\(^{19,20}\) again investigated the course of the rotational heats of \( \mathrm{H}_2 \), paying special attention to the change of the heat capacity with time. The hydrogen was placed in a steel vessel of small heat capacity at a density 150 times greater than normal, and was kept for a long time at the temperature of liquid air. Immediately after the vessel was filled, measurement of the rotational heats gives the temperature course of curve I, already established by the old measurements; after 1–2 weeks curves II, III, and IV are obtained, which, according to the formula \(xC_{p\mathrm{H}_2} + (1-x)C_{o\mathrm{H}_2}\), correspond
Fig. 9. Experimentally determined decrease in the rotational heat of hydrogen according to Eucken, Clusius, and Hiller.
to a content of \(p\mathrm{H}_2\) of 31.1, 36.4, and 43.1%. Curve IV was obtained for a hydrogen sample kept at 150 atm at \(70^\circ\mathrm{K}\) in contact with platinized asbestos. Thus all the curves indicate the transition of \(n\mathrm{H}_2\) into equilibrium hydrogen.
Dennison’s theory received especially good confirmation as a result of measurements on almost pure \(p\mathrm{H}_2\), obtained at \(20^\circ\mathrm{K}\) with the aid of activated charcoal, in the work of Clusius and Hiller\(^{12}\) (Fig. 9, curve V). The most essential point here is that the maximum of \(2.7\ \mathrm{cal/grad}\) at \(160^\circ\mathrm{K}\) exceeds the heat capacity of the classical rotator by almost \(0.7\ \mathrm{cal}\).
The temperature course of the specific heat of liquid \(p\mathrm{H}_2\) was investigated by Clusius and Hiller\(^{12}\) and compared with the corresponding data for \(n\mathrm{H}_2\) (measurements of Simon and Lange\(^{67}\)); no difference was observed beyond the limits of experimental error (see Fig. 10). For solid \(p\mathrm{H}_2\), the temperature course of the specific heats in the range from \(13.95^\circ\mathrm{K}\) (melting point) to approximat-
approximately down to \(10^\circ\mathrm{K}\) coincides with the course established by Simon and Lange^67^ for mixed crystals consisting of \(\frac{1}{4}\, \(p\mathrm{H}_2\) and \(\frac{3}{4}\, \(o\mathrm{H}_2\). The characteristic temperature for \(p\mathrm{H}_2\) is approximately \(91^\circ\).
The change in the specific heats of solid hydrogen with different contents of \(p\mathrm{H}_2\) was traced down to rather low temperatures (\(2^\circ\mathrm{K}\)) by Simon, Mendelssohn, and Ruhemann.^68,69^ Since the specific heats of solid \(p\mathrm{H}_2\) obey the Debye third-power law in temperature and at \(2^\circ\mathrm{K}\) should be immeasurably small, an anomaly in the specific heats is apparently observed in mixed crystals of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) at temperatures below \(10^\circ\mathrm{K}\). The course of the specific heats is shown in Fig. 11. This anomaly of the specific heats is explained by the fact that the threefold degeneracy of the lower orthorotational states in the condensed system disappears and, as the temperature is lowered, the molecules in the crystal pass into the lowest of the three states. That this is indeed explained by the splitting of the threefold orthorotational state is evident from the fact that the specific heats are proportional to the content of \(o\mathrm{H}_2\), and that their maximum is reached not at \(\frac{1}{2}\,p\mathrm{H}_2 + \frac{1}{2}\,o\mathrm{H}_2\), as would be expected for an ordered mixed phase. On the assumption that the threefold
Fig. 10. Specific heat of liquid and solid \(p\mathrm{H}_2\) and \(n\mathrm{H}_2\).
Fig. 11. Specific heat of solid \(n\mathrm{H}_2\) and \(p\mathrm{H}_2\) at low temperatures.
the rotational state is split in such a way that equal distances are established between the split components; the magnitude of the splitting is found to be 7.5 cal (communication from Simon in a private letter). This number is not entirely exact; it must be checked by measurements at still lower temperatures.
The specific heats of mixed crystals with \(x\) parts \(p\mathrm{H}_2\) are obtained from the equation
\[ \Delta C_x=-xR\frac{d}{dT}\cdot\frac{d\ln Z}{d\frac{1}{T}} =\frac{xN\varepsilon^2}{kT^2}\, \frac{e^{-\frac{\varepsilon}{kT}}+4e^{-\frac{2\varepsilon}{kT}}+e^{-\frac{3\varepsilon}{kT}}}{Z^2}, \tag{23} \]
where \(Z=1+e^{-\frac{\varepsilon}{kT}}+e^{-\frac{2\varepsilon}{kT}}\) denotes the sum over states for the three ortho-states. With the above value \(\varepsilon=7.5\) cal, it gives rather well the course of the anomaly in the temperature region from 2 to 7°.
The additional entropy arising as a result of the splitting of the molecules into three ortho-states is obtained from the expression
\[ \Delta s_x=\int_0^T \Delta C_x\,d\ln T =-x\cdot 4.575\left[ \lg Z+\frac{1}{kT}\cdot \frac{\varepsilon e^{-\frac{\varepsilon}{kT}}+2\varepsilon e^{-\frac{2\varepsilon}{kT}}}{Z} \right]. \tag{24} \]
For temperatures above \(10^\circ\mathrm{K}\) this expression practically reduces to \(-x\cdot 4.575\lg 3\).
4. Heats of transition
Since the terms \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) do not optically combine, it is of special interest to determine calorimetrically the energy difference between the lowest \(o\mathrm{H}_2\)-term and the zero quantum para-state. Theoretically this energy difference is equal to 337.17 cal. By converting the gaseous mixture \(\frac{1}{4}p\mathrm{H}_2+\frac{3}{4}o\mathrm{H}_2\), by means of adsorption on active charcoal and subsequent evaporation, into \(p\mathrm{H}_2\), Elbe and Simon\({}^{17}\) found calorimetrically at \(77.5^\circ\mathrm{K}\) a heat of transition equal to \(74\pm 7\) cal, and at \(20.39^\circ\mathrm{K}\) equal to \(241\pm 10\) cal, which agrees well with the theoretical energy differences, equal to 83 and 252 cal respectively (see Table 2).
5. Entropy, free energy, chemical constant* and vapor pressure
The general expressions for the entropy and free energy have the form
\[ S=2.3R\left[ \frac{5}{2}\lg T-\lg P+\lg\left(\frac{2\pi M}{N}\right)^{\frac{3}{2}} \frac{ek^{\frac{5}{2}}}{h^3} +\lg Z+T\frac{d\lg Z}{dT} \right]; \tag{25} \]
\[ F=\varepsilon_0-2.3RT\left[ \frac{2}{5}\lg T-\lg P+\lg\left(\frac{2\pi M}{N}\right)^{\frac{3}{2}} \frac{k^{\frac{5}{2}}}{h^3} +\lg Z \right]. \tag{26} \]
* Gibson and Heitler\({}^{41}\), A. Eucken\({}^{21,22}\), Ludloff\({}^{55}\), F. Simon\({}^{70}\), Giauque and Johnston\({}^{37,38}\), Giauque\({}^{39,40}\).
From the last equation one obtains the chemical constant* \((P=1\ \mathrm{atm})\) of \(g\mathrm{H}_2\) at low temperatures (near \(20^\circ\mathrm{K}\)), when \(Z_{g\mathrm{H}_2}\sim 1\).
\[ i_{t,g\mathrm{H}_2}=\lg\left(\frac{2\pi M}{N}\right)^{\frac{3}{2}}\frac{k^{\frac{5}{2}}}{h^3} =-1.5885+\lg 2.0152=-1.132 \tag{27a} \]
\[ i_{t,g\mathrm{H}_2}=i_{t,p\mathrm{H}_2} \tag{27b} \]
and since \(Z_{o\mathrm{H}_2}\sim 9\), then
\[ i_{t,o\mathrm{H}_2}=i_{t,g\mathrm{H}_2}+\lg 9=0.178. \tag{27c} \]
At high temperatures, when
\[ Z_{g\mathrm{H}_2}=Z_{p\mathrm{H}_2}+Z_{o\mathrm{H}_2} =\sum (2j+1)e^{-j(j-1)\sigma} + \]
\[ +\,3\sum (2j+1)e^{-j(j+1)\sigma} =\frac{2}{\sigma}\lg\frac{2}{\sigma} =i_{\mathrm{rot}}+\lg T,^{**} \]
we obtain
\[ i_{h,g\mathrm{H}_2}=-1.132+\lg\frac{16\pi^2 kJ}{h^2}=-2.76. \tag{28a} \]
Further, the chemical constants for \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) are equal to:
\[ i_{h,p\mathrm{H}_2}=-1.132+\lg\frac{4\pi^2 kJ}{h^2}=-3.36, \tag{28b} \]
\[ i_{h,o\mathrm{H}_2}=-1.132+\lg\frac{12\pi^2 kJ}{h^2}=-2.885. \tag{28c} \]
In the intermediate region \((30^\circ—300^\circ\mathrm{K})\) the sum includes the term
\[ \lg Z_{g\mathrm{H}_2}=\frac{1}{R}\int_0^T \frac{dT}{T^2}\int_0^T C_v^{\mathrm{rot}}\,dT. \tag{29}^{***} \]
and, thus, the specific heat of rotation for \(g\mathrm{H}_2\) in this region is not equal to \(R\).
* The “chemical constant” will hereafter be called the part of the free energy proportional to \(T\).
** At high temperatures
\[ \sum_{\text{even}}(2j+1)e^{-j(j+1)\sigma} = \sum_{j\ \text{odd}}(2j+1)e^{-j(j+1)\sigma} \sim \]
\[ \sim \frac{1}{2}\int_0^\infty (2j+1)e^{-j(j+1)\sigma}\,dj =\frac{1}{2\sigma}. \]
*** By substituting
\[ C_v^{\mathrm{rot}} = -R\frac{d}{dT} \frac{d\ln Z}{d\frac{1}{T}} \]
it is easy to verify the formulas given above.
At low temperatures, thus, \(g\mathrm{H}_2\) and \(p\mathrm{H}_2\) behave as a monatomic gas (molecular weight 2.015); at high temperatures the chemical constant given here for \(g\mathrm{H}_2\) is greater by \(\lg 4\) than that calculated for the case of chemical equilibrium from the Stern–Sackur–Tetrode formula (\(-3.36\)). This is explained by the fact that in the chemical constants given here the multiplicity of the nucleus is taken into account. Accordingly, if the chemical constant of the H atom, instead of the normal value 1.282, is reckoned with account of the nuclear spin, equal to:
\[ i_{\mathrm H}=-1.5885+\frac{3}{2}\lg 1.0076+\lg 4=-0.981, \]
then the statistical weight of the H atom in the ground state \(({}^2S)\) is equal to \(2\cdot 2\). In calculating equilibrium states, for high temperatures, the assumption of nuclear multiplicity falls away (thus, for example, in the dissociation equilibrium \(\mathrm{H}_2 \rightleftarrows 2\mathrm{H}\) it reaches \(\lg 4\), Gibson and Heitler \(^{41}\)). For unexcited rotation or for other mixtures \(\left(\text{besides } \frac{1}{4}p\mathrm{H}_2+\frac{3}{4}o\mathrm{H}_2\right)\) this is inapplicable, and in such cases the multiplicity of the nucleus must be taken into account.
The chemical constants of any mixtures of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) at low temperatures can be calculated from the equation:
\[ i_{l,x}=i_{l,p\mathrm H_2}-[x\lg x+(1-x)\lg(1-x)]+(1-x)\lg 9, \tag{30} \]
and at high temperatures from the following:
\[ i_{h,x}=i_{h,p\mathrm H_2}-[x-\lg x+(1-x)\lg(1-x)]+(1-x)\lg 3. \tag{31} \]
The expression in brackets gives the entropy of mixing, which is here considered as that of a mixture consisting of different gases; \((1-x)\lg 9\) and \((1-x)\lg 3\) take into account the higher statistical weight of \(o\mathrm{H}_2\).
For the mixture \(\frac{1}{4}p\mathrm{H}_2+\frac{3}{4}o\mathrm{H}_2\) we obtain from (30), at low temperatures, the value \(\lg 4+\frac{3}{4}\lg 3\) relative to the normal chemical constant.
The difference of the chemical constants of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) in the case of dissociation equilibrium causes a different equilibrium concentration of H atoms. (In reality there is no strict equilibrium, since the H atoms, formed partly by recombination and partly by the reaction \(\mathrm H+p\mathrm H_2 \rightleftarrows o\mathrm H_2+\mathrm H\), rapidly destroy \(p\mathrm H_2\); see Chapter IV, 1b.)
The entropy values for \(g\mathrm H_2\) and \(p\mathrm H_2\) are obtained from the above formulas, for \(T=298.1^\circ\mathrm K\), \(P=1\ \mathrm{atm}\), as equal to \(33.98\ \frac{\mathrm{cal}}{\mathrm{deg}}\) and \(31.23\ \frac{\mathrm{cal}}{\mathrm{deg}}\). The free energy at \(T=298.1\mathrm K\) for \(p\mathrm H_2\) is greater than for \(g\mathrm H_2\) by the amount:
\[ F_{p\mathrm H_2}-F_{g\mathrm H_2}=529.12-542.46+4.575\cdot 298.1\cdot \lg 4 =807.66\ \mathrm{cal/mol}. \tag{32} \]
The entropy of a mixture consisting of \(x\,p\mathrm{H}_2\) and \((1-x)\,o\mathrm{H}_2\), at high temperatures, is greater by
\[ -2.3\,R[x\lg x+(1-x)\lg(1-x)+(1-x)\lg 3] \tag{33} \]
than for pure \(p\mathrm{H}_2\).
In the theoretical expression for the vapor pressure
\[ \lg P=-\frac{\lambda_0}{4.575T}+2.5\lg T-\frac{1}{4.575}\int_0^T \frac{dT}{T^2}\int D\left(\frac{91}{T}\right)dT+j \tag{34} \]
the vapor-pressure constant \(j\) must be replaced by the chemical constant for \(g\mathrm{H}_2\), which at low temperatures is equal to \(-1.132\).
From the old measurements of vapor pressure for the mixture \(\frac{1}{4}p\mathrm{H}_2=\frac{3}{4}o\mathrm{H}_2\), the same value \(j=-1.13\) was obtained for the vapor-pressure constant of this mixture, provided that the specific heats of these mixed crystals are extrapolated to absolute zero by means of one and the same Debye function \((\theta=91)\). The “correct” value of the vapor-pressure constant should have been equal to
\[ -1.132-\frac{1}{4}\lg\frac{1}{4}-\frac{3}{4}\lg\frac{3}{4}+\frac{3}{4}\lg 9 = -1.132+\lg 4+\frac{3}{4}\lg 3, \]
but, owing to the additional term \(-1.132\), it is reduced to the corresponding expression for the condensate—
\[ \frac{1}{4.575}\int_0^T \frac{dT}{T^2}\int_0^T C\,dT. \]
The ratio of the constituent parts of the mixture in the condensate, because of complete disorder, has the same value as in the gas:
\[ -\frac{1}{4}\lg\frac{1}{4}-\frac{3}{4}\lg\frac{3}{4}; \]
\(4.575\left(+\frac{3}{4}\lg 3\right)\) enters into the integral which expresses the anomaly of the specific heats obtained as a result of the splitting of the ortho-rotational state. The remaining part, \(\frac{3}{4}\lg 3\), disappears as a consequence of the splitting, arising at extremely low temperatures, of the degeneracy of the nuclear spins of \(o\mathrm{H}_2\) in the crystal,* which, like the splitting of the rotational degeneracy, creates an additional specific heat.
The entropy of \(n\mathrm{H}_2\), which can be calculated from experimental data, differs by the same amount
\[ +4.575\left(\lg 4+\frac{3}{4}\lg 3\right)=4.39\ \frac{\mathrm{cal}}{\mathrm{mol}} \]
from the theoretical value given above, \(33.98\ \frac{\mathrm{cal}}{\mathrm{mol}}\), which is obtained when the specific heat of the condensate \(n\mathrm{H}_2\) is extra—
* As a result of this splitting of the nuclear multiplet, the entropy of \(o\mathrm{H}_2\) at absolute zero is equal to zero, as should also have been expected on the basis of Nernst’s heat theorem.
extrapolate by means of the normal Debye function to absolute zero and do not take into account the entropy of the mixture \(pH_2 + oH_2\). The results of calculations of the entropy of \(nH_2\) from experimental data (according to Giauque \(^{40}\)) are given in Table 5.
TABLE 5
Calculation of the entropy of \(nH_2\)
| Contribution | Value |
|---|---|
| \(4{,}575 \displaystyle\int_{0}^{14}\frac{dT}{T^2}\int_{0}^{14}D\!\left(\frac{91}{T}\right)\,dT\) | 0,52 |
| Change of entropy on melting: \(\dfrac{23}{13{,}95}\) | 2,01 |
| Change of entropy on evaporation: \(\dfrac{217{,}8}{13{,}95}\) | 15,61 |
| Van der Waals correction, or | 0,03 |
| Berthelot correction | 0,13 |
| Work of compression — \(4{,}575\lg \dfrac{5{,}38}{76{,}0}\) | −5,26 |
| \(\displaystyle\int_{14}^{298{,}1} C_p\,d\ln T\) | 16,73 |
| Instead of the expected value | \(29{,}64—29{,}74\) |
| Instead of the expected value | \(33{,}98 - 4{,}39 = 29{,}59\) |
The vapor pressure, melting point, and triple point of \(pH_2\) were determined by Bonhoeffer and Harteck \(^{5}\), later by Hennig, Geiser, Otto, and Justi \(^{58}\), and, finally, by Keesom, Bijl, and Van der Horst \(^{52}\), and were compared with the corresponding values for \(nH_2\). The results are given in Table 6.
TABLE 6
Melting points and triple points of \(pH_2\) and \(nH_2\)
| °C | Vapor pressure \(nH_2\) | Vapor pressure \(pH_2\) | Note |
|---|---|---|---|
| 252,754 | 760,0 | 787,10 | Boiling point of \(pH_2\) |
| 252,871 | — | 760,00 | ″ ″ \(pH_2\) |
| 259,15 | 51,4 | 58,8 | Triple point of \(nH_2\) |
| 259,03 | — | 53,0 | ″ ″ \(pH_2\) |
Since \(pH_2\) has a higher vapor pressure than \(nH_2\), it follows from this that the heat of vaporization of \(pH_2\) is approximately 0.65% smaller than that of \(nH_2\) (the sums of the corresponding terms in the vapor-pressure formula for \(nH_2\) and \(pH_2\) are equal with accuracy up to terms of the second-
of that order). This is somewhat unexpected, since precisely in rotating molecules of \(o\mathrm{H}_2\) one would expect a more energetic transition into the disordered state and, consequently, a smaller heat of vaporization.
The decrease in vapor density for a mixture richer in \(p\mathrm{H}_2\) than \(n\mathrm{H}_2\), observed at the triple point, was, according to the experiments of Dziok and Johnston \(^{37}\), so small that on this basis they could not confidently draw a conclusion concerning the transformation of hydrogen cooled to a low temperature.
From the vapor-density curves for \(n\mathrm{H}_2\) one obtains a heat of vaporization, at absolute zero, equal to
\[ \lambda_0 = 181.9\,\frac{\mathrm{cal}}{\mathrm{mol}}, \]
and at the boiling temperature—about \(216\) cal. Since the heat of vaporization of pure \(p\mathrm{H}_2\) approximately coincides with this value, it follows from this that the rotation of ortho-molecules in the crystal lattice also occurs quite freely. As Pauling \(^{59}\) has shown, such behavior of the \(\mathrm{H}_2\) molecules represents an exceptional case, caused by the fact that in \(\mathrm{H}_2\) the rotational energy is especially large, while the heat of fusion is especially small.
Fig. 12. Scheme of terms of the Lyman bands.
Fig. 13. Scheme of terms of the Werner bands.
6. Optical properties*
The term schemes of the three lower singlet terms of hydrogen, shown in Figs. 12 and 13, make it possible to note characteristic properties of the spectra of \(\mathrm{H}_2\), caused by the splitting of the terms into \(p\)- and \(o\)-states. For simplicity, only those rotational terms which belong to states without vibrations are shown here (without preserving the energy scale).
* Cf. Weizel, Bandenspektren.
In the combination \(^{1}\Sigma_g^{+} \longleftrightarrow {}^{1}\Sigma_u^{+*}\) (Lyman bands, the longest-wave part of the absorption spectrum of \(\mathrm{H}_2\), Fig. 12), only the \(R\)- and \(P\)-branches take part. In the Descartes coordinates of the nuclei the antisymmetric rotational states are different; those states whose eigenfunctions, upon their mirror reflection from the center of gravity of the molecule, do not change sign we shall denote by \(\times\), and those that change sign by the sign \(|\). The combination rule reads: \(\times \longleftrightarrow |\) and \(\times \longleftrightarrow\); for the rotational quantum number we have \(\Delta j=\pm 1\). This is obtained because the electric moment of the molecule upon interchange of the nuclei, i.e. upon mirror reflection of the particles from the center of gravity of the molecule, must not change sign. In all cases where the matrix element of the electric moment under these operations changes its sign, the corresponding combinations disappear; thus, for example, for the matrix element of the \(x\)-component (and likewise for all the others) corresponding to the transition \(\times \longleftrightarrow |\):
\[ \begin{aligned} M_{01} &=\int \Psi_{0\times}(1,2,\mathrm{I},\mathrm{II}) (x_1+x_2-X_{\mathrm I}-X_{\mathrm{II}}) \Psi_{1|}(1,2,\mathrm{I},\mathrm{II})\,d\tau \\ &=\int -\Psi_{0\times}(1,2,\mathrm{II},\mathrm{I}) (x_1+x_2-X_{\mathrm{II}}-X_{\mathrm I}) \Psi_{0\times}(1,2,\mathrm{II},\mathrm{I})\,d\tau ; \end{aligned} \tag{35} \]
this equality can be satisfied only if the integral is equal to zero**.
In electronic transitions \(o\) combines only with \(o\) and \(p\) only with \(p\). Since the \(o\)-state, both in the ground and in the excited states, has three times the statistical weight of the \(p\)-state, consequently the number of transitions is also three times larger, and the \(o\)-lines must be three times more intense than the \(p\)-lines. The lines denoted by heavy strokes (Fig. 12) belong to the ortho system, and those by thin strokes to the para system.
In the combination \(^{1}\Pi_g \longleftrightarrow {}^{1}\Sigma_u^{+}\) (Werner bands, Fig. 13) \(R\)-, \(P\)- and \(Q\)-branches appear. The combination rule here is likewise \(\times \longleftrightarrow \perp\) and \(\times \longleftrightarrow\); \(\Delta j=\pm 1\) or \(0\). In the upper \(^{1}\Pi_g\)-state each rotational state in Descartes coordinates of the nuclei is split into one symmetric and one antisymmetric state*, and these have the sequence indicated in Fig. 13.
* The index \(+\) or \(-\) denotes further symmetry properties of the eigenfunctions of the state. For details see E. Wigner and Witmer, Z. Phys. 51, 859, 1928, where, however, they are denoted by the signs \('\) and \(''\); see Weizel, “Polyatomic Spectra,” Handbuch der Experimentalphysik, supplementary volume, 1931. The symmetry properties denoted by the letters \(g\) and \(u\) (in Wigner and Witmer \(+\) and \(-\)) are explained on p. 357.
** Here we neglect the exceedingly small perturbations of the symmetry properties of the eigenfunctions caused by the interaction of the nuclei with magnets; theoretically they create very small transition probabilities, for example \(\times \longleftrightarrow\), as was already discussed in Chapter II.
*** See footnote * on this page.
**** See Wigner and Witmer\(^{74}\).
The difference entering into the expression
\[ R(j)-Q(j)=Q(j+1)-P(j+1)+\varepsilon', \tag{36} \]
is called the combination defect. It is given by the expression:
\[ \varepsilon'=\left[F'_{\times}(j+1)-F'_{\times}(j)\right]-\left[F'_1(j+1)-F'_1(j)\right] \tag{37} \]
where \(F'\) is the value of the upper term.
In those cases when, for the difference of terms \(F''(j+1)-F''(j)\) in the lower state, one uses the value following from formula (5), \(\dfrac{(j+1)h}{4\pi^2 J}\), \(\varepsilon\) can be calculated from expression (36). Since there is no method independent of (5) for determining the magnitude \(\varepsilon'\), the experimental confirmation of the theoretical formulas remains uncertain to within the magnitude \(\varepsilon'\) (see Chapter III, 4).
The theoretically predicted alternation of intensities was observed experimentally, with greater or lesser clarity, in all bands.
Fig. 14. Above: a mixture rich in parahydrogen.
Below—ordinary hydrogen.
Measurements of the intensities by Kapustinsky and Eymers \(^{5}\) on the many-line spectrum of \(\mathrm{H}_2\) gave the ratio of intensities \(p:o\) very close to the theoretical value, equal to \(1:3\). In Fig. 14 a photograph by Bonhoeffer and Harteck \(^{5}\) is given, which makes clear the relation between the intensity of the lines of hydrogen rich in \(p\mathrm{H}_2\), and the lines of \(n\mathrm{H}_2\) (the \(p\)-lines are marked by dots).
The prohibition of mutual combinations \(p \longleftrightarrow o\) was also checked by Beutler \(^{1}\) for impact excitation in \(\mathrm{H}_2\) of electronic and vibrational quantum transitions. Selective excitation of Lyman bands with the aid of argon atoms of suitable energies (4 terms with energy \(94\,000\ \mathrm{cm}^{-1}\)) leads to such states in which only those \(\mathrm{H}_2\) terms are excited for which the energy received by the \(\mathrm{H}_2\) molecule is approximately equal to the excitation energy of the argon atom (resonance impact) and, at the same time, the combination rule \(p \longleftrightarrow p\) and \(o \longleftrightarrow o\) is fulfilled. Transitions for which resonance with argon is good, but for which the combination rule is not fulfilled, disappear. Such, for example, are the transitions:
\[ {}^{1}\Sigma_g^{0}(2)\to{}^{1}\Sigma_u^{3}(6) \quad\text{and}\quad {}^{1}\Sigma_g^{0}(4)\to{}^{1}\Sigma_u^{3}(6). \]
(The numbers in parentheses denote rotational quantum numbers, and the numbers at the upper right—the vibrational level.)
Intensity alternation was also observed in the Raman spectra of liquid and gaseous H\(_2\). The combination rule for Raman spectra has the form: \(\Delta j = \pm 2\) (for all diatomic molecules), so that liquid hydrogen exhibits two Raman frequencies, \(0 \to 2\) (354 cm\(^{-1}\)) and \(1 \to 3\) (588 cm\(^{-1}\)) (MacLennan and McLeod\(^{53}\)). These frequencies do not change at all upon condensation of H\(_2\), contrary to theoretical expectations. The transition \(1 \to 3\) is more intense than the transition \(0 \to 2\), since the concentration of \(o\)H\(_2\) in fresh liquid hydrogen is three times greater than the concentration of \(p\)H\(_2\). The same result was obtained by Rasetti\(^{62}\) for the Raman spectra of gaseous H\(_2\); here too the transitions between odd rotational quantum numbers, i.e. the \(o\)-transitions, are more intense than the \(p\)-transitions. The observed Raman frequencies of gaseous H\(_2\) agree very well with the theoretical values*.
7. Magnetic properties of \(p\)H\(_2\) and \(o\)H\(_2\)
The magnetic properties of both kinds of hydrogen were investigated chiefly in order to determine the magnetic moment of the proton**: by Frisch and Stern\(^{34}\), and later by Estermann, Frisch, and Stern\(^{17a}\), using the Stern–Gerlach method. For the deflection they used a field 10 cm long with an inhomogeneity of \(2 \cdot 10^5\ \frac{\text{gauss}}{\text{cm}}\). The deflection of the molecular beam of hydrogen was of the order of 0.05 mm.
The experiments showed that \(p\)H\(_2\) in the state without rotation (a molecular beam of pure \(p\)H\(_2\) at a temperature of about 85° K) has no magnetic moment. This was to be expected, since in this case the magnetic moments of the proton are antiparallel. In the two-quantum rotational state (a molecular beam of \(p\)H\(_2\) at 200 and 300° K), \(p\)H\(_2\) proves to be magnetic, and its rotational magnetic moment in this state is approximately equal to
\[ 1.8\frac{he}{4\pi Me} = 1.8\ \text{nuclear magnetons} = 1.8 \cdot 0.5 \cdot 10^{-23}\ \text{CGS units}. \]
In \(o\)H\(_2\) in the ground state the observed magnetic moment consists of two parts: of the magnetic rotational moment of the one-quantum state and of the magnetic moment of the parallel-oriented protons. Assuming that the magnetic rotational moment in the one-quantum \(o\)-state is equal to half the rotational moment of \(p\)H\(_2\) with \(j = 2\), we find that the nuclear moment of \(o\)H\(_2\) is approximately 5 nuclear magnetons. Hence for the moment of the proton there follows a value equal to approximately 2.5 nuclear magnetons. The ratio of the magnetic moment to the mechanical moment thus has, for the pro-
* The refractive indices of \(p\)H\(_2\) and \(n\)H\(_2\) are equal to one another within the limits of experimental error (C. and M. Cuthbertson\(^{14}\)).
** See, for example, “Advances in the Physical Sciences,” vol. XIV, issue 1, p. 99, the article by Frisch and Stern on the magnetic deflection of hydrogen molecules and the magnetic moment of the proton (Translator’s note).
instead of the expected value \(\dfrac{2e}{2Mc}\), a value approximately equal to \(2.5\dfrac{2e}{2Mc}\) (for the “rotating electron” this ratio has the value \(\dfrac{e}{Mc}\)).
Why the magnetic moment of the proton is anomalously large is still unclear. However, the observed rotational moment can be explained with the aid of Wiek’s hypothesis,* which assumes that the electron shell lags somewhat behind during rotation of the nucleus, leading to a fundamental rotational moment lying within the limits \(0.36—0.92\) nuclear magneton.
8. Chemical properties, gas-kinetic quantities
For a systematic investigation of possible differences in the chemical behavior of the two forms of \(\mathrm{H}_2\), almost no experiments have so far been undertaken.** The free energy of \(p\mathrm{H}_2\) at \(300^\circ\mathrm{K}\) is higher than that for \(n\mathrm{H}_2\) by \(807.66\ \dfrac{\mathrm{cal}}{\mathrm{mol}}\). Thus a \(p\mathrm{H}_2\)-electrode should have, relative to an \(n\mathrm{H}_2\)-electrode, an electromotive force equal to \(-0.035\ \mathrm{V}\). According to the experiments of Bonhoeffer and Harteck,^5 the corresponding case cannot occur on platinum electrodes, since under these conditions the platinum electrode causes the rapid catalytic decomposition of \(p\mathrm{H}_2\).
The gas-kinetic cross sections of the two forms of \(\mathrm{H}_2\), according to Harteck and Schmidt,^45 are equal to one another within the errors of experiment. Evidence for this is the complete identity of the temperature dependence of the internal friction of \(p\mathrm{H}_2\) and \(n\mathrm{H}_2\), from which follow the same Sutherland constant, the same force laws, and the same effective cross sections of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\).***
IV. Mutual conversion of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\)
1. Homogeneous conversion
The establishment of equilibrium with respect to the concentrations of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) for homogeneous systems has been investigated in the gaseous, dissolved, liquid, and solid states. It was established that the mutual conversion occurs either by the type of chemical reactions (exchange reactions), or under the influence of magnetic forces. In what follows, the various possible modes of conversion will be considered.
a) Radiation. Experimental data (in the absence of catalysts) give a value for the lower limit for conversion with radiation—
* Z. phys. 85, 25, 1933.
* Goldman’s experiments (Z. Phys. Chem. 305*, 1929), who found identical upper and lower limits for the explosions of normal gas and gas containing \(p\mathrm{H}_2\) rattlesnake gas, do not permit any further conclusions, since it has recently been shown that in the presence of \(\mathrm{O}_2\), \(p\mathrm{H}_2\) very rapidly converts into \(n\mathrm{H}_2\).
* According to the experiments of Ramsauer and Kollath (Ann. Phys. 7**, 176, 1930), the effective transverse cross sections of \(n\mathrm{H}_2\) and \(p\mathrm{H}_2\) are equal (at electron velocities below \(1\ \mathrm{V}\)).
with which the theoretical value of the half-conversion time turns out to be 300 years (see Chapter II, 1). Thus, the experiments of Bonhoeffer and Harteck show that the spontaneous transition \(nH_2 \to gH_2\) at \(85^\circ K\) and a pressure of \(60\) mm, since it proceeds monomolecularly, has a half-conversion period of more than 1 year. Likewise the reverse transition of \(pH_2\) into \(nH_2\) at room temperature (pressure about \(400\) mm) proceeds with a half-period probably greater than 2 years.
b) Thermal transition. \(pH_2\), which at room temperature is a stable gas, at a temperature of several hundred degrees already shows a noticeable tendency toward transition. The mechanism of this thermal transition was investigated by Farkas\(^{26,27}\) for temperatures from 600 to \(900^\circ C\) and for pressures from 50 to 700 mm. A quartz vessel placed in an electric furnace, whose temperature was kept constant, was used as the reaction vessel. Earlier, Bonhoeffer and Harteck\(^{5}\), in quartz and porcelain vessels, had established that the thermal transition is homogeneous in character, and showed that the reaction rate is entirely independent of the ratio of the surface of the vessel to its volume. The main experiments were carried out in flasks of capacity 1 liter by the static method, in such a way that the vessel was filled to the desired pressure with a gaseous mixture rich in \(pH_2\) (for the most part a mixture with \(46\%\ pH_2\) was used) and heated to the temperature of the experiment, and then, after a certain time, the transition was determined by concentration measurements.
Fig. 15. Time course of the thermal transition.
The course of the reaction in time is expressed by the exponential formula
\[ u_t = u_0 e^{-k(T,P)t}, \tag{38} \]
where \(u_t\) and \(u_0\) are the excess concentrations (equal to the concentrations of \(p\mathrm{H}_2\) minus the concentration of the equilibrium mixture) corresponding to the times \(t\) and \(t=0\), and \(k\) is a constant depending on the temperature \(T\) and pressure \(P\). In Fig. 15 the course of this reaction in time is shown on a logarithmic scale. At constant temperature \(k\) is proportional to the square root of the pressure (see Table 6a). Column 2 gives the half-conversion time, i.e., the intervals of time in which \(u_0\) decreases to \(\dfrac{u_0}{2}\), related to \(k\) by the expression
\[ k=\frac{\ln 2}{\tau_{\frac12}} . \]
TABLE 6a
Rate of reaction in the temperature conversion of \(p\mathrm{H}_2\)
| Pressure \(P\), in mm Hg | Temperature 923° K | Temperature 923° K | Temperature 923° K | \(k^*=\dfrac{k(P,T)}{\sqrt{[\mathrm{H}_2]}}\) |
|---|---|---|---|---|
| Pressure \(P\), in mm Hg | Half-conversion time \(\tau_{\frac12}\), in sec. | \(k(P,T)\) | \(\dfrac{k(P,T)}{\sqrt{P}}\) | \(k^*=\dfrac{k(P,T)}{\sqrt{[\mathrm{H}_2]}}\) |
| 50 | 648 | 0.00106 | 0.000150 | 0.0358 |
| 100 | 450 | 0.00153 | 0.000153 | 0.0366 |
| 200 | 318 | 0.00216 | 0.000154 | 0.0368 |
| 400 | 222 | 0.00310 | 0.000155 | 0.0370 |
If we replace the pressure, expressed in millimeters of mercury, by the concentration of \(\mathrm{H}_2\) in moles per liter, we obtain the constant \(k^*\) of the fifth column.
From the dependence of the reaction rate on the pressure it follows that the order of the reaction is \(3/2\),* and the mechanism that satisfies this order and the observed time course is the exchange reaction:
\[ \mathrm{H} + p\mathrm{H}_2 \rightleftarrows o\mathrm{H}_2 + \mathrm{H} \tag{39} \]
\[ \boxed{\uparrow\quad \uparrow}\quad \downarrow \qquad \uparrow\quad \uparrow \qquad \downarrow \]
The transition in this case occurs in such a way that, in collisions between the \(\mathrm{H}_2\) molecule and H atoms obtained as a result of thermal dissociation of \(\mathrm{H}_2\), exchange takes place between bound and free H atoms, owing to which the quan-
* Between the half-conversion time \(\tau_{\frac12}\), the pressure \(P\), and the order of the reaction \(n\), there exists, as is known, the following relation:
\[ \tau_{\frac12} P^{(n-1)}=\mathrm{const}. \]
the amounts of \(o\)- and \(p\)-molecules prove to be in the ratio corresponding to equilibrium \((3:1)\). In this mechanism the overall symmetry of the system \(\mathrm{H} + \mathrm{H}_2\) is preserved, as was theoretically shown by Wigner \(^{77}\). This circumstance is illustrated by the arrows which show [formula (39)] the orientation of the nuclear spins.
The mechanism described above corresponds to the reaction equation:
\[ -\frac{dx[\mathrm{H}_2]}{dt} = k_1[\mathrm{H}]x[\mathrm{H}_2] - k_2[\mathrm{H}](1-x)[\mathrm{H}_2], \tag{40} \]
where \(x\) denotes the content of \(p\mathrm{H}_2\). Integration gives:
\[ \left(x_t-\frac{k_2}{k_1+k_2}\right) = \left(x_0-\frac{k_2}{k_1+k_2}\right) e^{-t[\mathrm{H}](k_1+k_2)}. \tag{41} \]
Here \(\frac{k_1}{k_2}\) denotes the ratio of the concentrations of \(o\mathrm{H}_2\) and \(p\mathrm{H}_2\) in the state of equilibrium and, consequently, \(\frac{k_2}{k_1+k_2}\) denotes the equilibrium concentration of \(p\mathrm{H}_2\); replacing \(\left(x_t-\frac{k_2}{k_1+k_2}\right)\) by \(u_t\) and correspondingly \(\left(x_0-\frac{k_2}{k_1+k_2}\right)\) by \(u_0\), we obtain the time course shown in Fig. 15. The constant entering into the exponent with \(e\) is the sum of the rate constants of the reactions \(p \to o\) and \(o \to p\).
The concentration of H atoms entering into the reaction equation can be calculated from the dissociation equilibrium equation
\[ \mathrm{H}_2 \rightleftarrows 2\mathrm{H} - 102.8 \ \text{cal}. \]
If we denote the dissociation constant by
\[ K_c=\frac{[\mathrm{H}]^2}{[\mathrm{H}_2]} \]
(concentration in moles per liter), then the quantity \(\frac{k^*}{\sqrt{K_c}}\) gives the constant \((k_1+k_2)\) of equation (41).
Whereas the \(k^*\)-constants increase rapidly as a result of the increase of the H concentration with rising temperature, \((k_1+k_2)\) increase comparatively slowly (the value at \(923^\circ\mathrm{K}\) is somewhat out of line). The heats of activation of the reaction \(\mathrm{H}+p\mathrm{H}_2 \rightleftarrows o\mathrm{H}_2+\mathrm{H}\) calculated from the temperature dependence of these latter prove to be approximately equal to \(6000\) cal. The collision yields calculated by the formulae of kinetic theory are given in Table 7, column 5**. If in the formula
\[ Z=Se^{-\frac{Q}{RT}} \]
one substitutes the activation heat \(6000\) cal, then for the magnitude of the factor \(S\) we obtain
\[ \text{* In Farkas }^{27}\text{, Table 2, the values of }k^*\text{ (his }k_2\text{) were inadvertently given four times smaller.} \]
\[ \text{** These values differ somewhat from the figures obtained by Farkas }^{27}\text{ (Table 3), since the dissociation equilibrium was calculated on the basis of new data (Jijok }^{40}\text{).} \]
\[ \text{*** They are obtained from }(k_1+k_2)\text{ by division by} \]
\[ 2\sqrt{\pi}\left(\frac{d_{\mathrm{H}}+d_{\mathrm{H}_2}}{2}\right)\cdot\sqrt{3RT}\cdot 6.06\cdot10^{20} = 1.6\cdot10^{10}\sqrt{T}. \]
about \(1/10\). Of course, the activation heat and the steric factor given above are determined with some inaccuracy, because in determining them from the observed rates of transition the limits of error of thermal [H] have a strong effect. The complete reaction equation has the form:
\[ 2.3 \lg \frac{u_0}{u_t} = 1.6 \cdot 10^{10}\sqrt{T}\sqrt{[\mathrm{H}_2]}K_c \frac{1}{10} e^{-\frac{6000}{RT}}. \tag{42} \]
The reduced yield of collisions \(Z_{k_1+k_2}\) gives the sum of the probabilities of the reaction of spin reorientation in a collision of H and \(\mathrm{H}_2\). The probability of the reaction \(p \to o\) is equal to \(\frac{3}{4} Z_{k_1+k_2}\), and that of the reaction \(o \to p\): \(\frac{1}{4} Z_{k_1+k_2}\). Thus, in the case of equilibrium at high temperatures it should be
\[ Z_{p=o}\cdot \frac{1}{4}[\mathrm{H}_2] Z_{o\to p}\frac{3}{4}[\mathrm{H}_2]. \]
TABLE 7
Rate constants of the thermal reaction \(p\mathrm{H}_2 \rightleftarrows o\mathrm{H}_2\)
| \(T\) | \(k^*=\dfrac{k(P,T)}{\sqrt{\mathrm{H}_2}}\) | \(\lg K_c\) | \(k_1+k_2=\dfrac{k^*}{\sqrt{K_c}}\) | \(Z_{k_1+k_2}\) = collision yield |
|---|---|---|---|---|
| 873 | 0.0083 | \(-22.43\) | \(1.73\cdot 10^9\) | 0.00287 |
| 923 | 0.0373 | \(-21.03\) | \(1.22\cdot 10^9\) | 0.00251 |
| 973 | 0.263 | \(-19.75\) | \(2.00\cdot 10^9\) | 0.00400 |
| 1023 | 1.188 | \(-18.61\) | \(2.39\cdot 10^9\) | 0.00463 |
TABLE 8
Collision yield for the reaction \(\mathrm{H}+p\mathrm{H}_2 \rightleftarrows o\mathrm{H}+\mathrm{H}\)
| \(T\) in °C | \(k_A\) | Method of calculation A: collision yield | Method of calculation A: steric factor | \(k_B\) | Method of calculation B: collision yield | Method of calculation B: steric factor |
|---|---|---|---|---|---|---|
| 10 | 0.44 | \(2.7\cdot 10^{-7}\) | \(\dfrac{1}{9.1}\) | 0.5481 | \(3.4\cdot 10^{-7}\) | \(\dfrac{1}{7.3}\) |
| 50 | 1.80 | \(1.1\cdot 10^{-6}\) | \(\dfrac{1}{14.2}\) | 3.17 | \(2.0\cdot 10^{-6}\) | \(\dfrac{1}{7.9}\) |
| 100 | 4.12 | \(2.5\cdot 10^{-6}\) | \(\dfrac{1}{21.8}\) | 11.7 | \(7.3\cdot 10^{-6}\) | \(\dfrac{1}{7.7}\) |
The direct action of H atoms on \(p\mathrm{H}_2\) was studied by Gibb and Harteck \(^{35}\), who mixed H with a stream of active \(p\mathrm{H}_2\), obtained by the Bonhoeffer–Wood method, and determined the rate of transition_
The total pressure in this case reached 0.5 mm, and the temperature was 10, 57, and 100°C. The concentration of H atoms varied within the range from 3 to 18%. The reaction proceeded in a flask of volume about 1 l, and the residence time in this reaction space was determined by the flow rate. This residence time was also the reaction time, since the H atoms were destroyed immediately after passing through the reaction space in a U-shaped tube cooled with liquid air. The consumption of \(p\mathrm{H}_2\) was determined by means of control experiments in which \(p\mathrm{H}_2\) was added to the flowing gas after destruction of the H atoms. The constants calculated from the equation \(u_t = u_0 e^{-kt}\) (\(t\)—the residence time of the gas in the reaction space) for an H-atom content of 18.8% are given in Table 8, column 2, p. 379 (method of calculation A).
However, this method of calculating the constants is incorrect, since, owing to the high rate of diffusion of hydrogen, a more or less complete mixing of the fresh gas with the gas that has already partially reacted occurs.* Assuming complete mixing, the values given in column 5 (method of calculation B) are obtained for the reaction-rate constant. The collision yield (columns 3 and, respectively, 6) and the steric factor (columns 4 and 7) are calculated in the same way as in the case of the thermal reaction.
The good constancy of the steric factor in the second method of calculation, on the one hand, and comparison of the obtained values of the collision yield with the values determined by Farkas, on the other, make it possible to conclude that method of calculation B gives results very close to reality. From the temperature dependence of the reaction one obtains an activation heat of \(7000 \pm 500\) cal, which agrees very well with the value of the activation heat for the thermal reaction, *.
THEORETICAL WORKS ON THE REACTION
\( \mathrm{H} + \mathrm{H}_2 \longrightarrow \mathrm{H}_2 + \mathrm{H}. \)
Because of its simplicity, the reaction \(\mathrm{H} + \mathrm{H}_2\) has been considered in several theoretical works, and not only the heat of reaction but also the steric factor has thereby been satisfactorily calculated.
According to London, the binding energy of the molecule (\(-W_{12}\), in our case—the molecule \(\mathrm{H}_2\)) consists of two parts: \(-W_{12} = F(r) + f(r)\),
* See also M. Bodenstein, Z. phys. Chem. 61, 422, 1908.
** In the decomposition of \(p\mathrm{H}_2\) caused by an electric discharge at room temperature, the H atoms formed in the discharge also play a role; they act on \(p\mathrm{H}_2\), while, however, in addition to exchange reactions, recombination of H atoms also takes place.
*** H atoms obtained photochemically also cause the decomposition of \(p\mathrm{H}_2\), according to the mechanism described. In most cases the principal role is played by the exchange reaction, accompanied by dissociation of \(\mathrm{H}_2\) and recombination; the same occurred in the experiments of Senftleben⁶⁵, who investigated the reverse conversion of \(p\mathrm{H}_2\) in the presence of mercury vapor under the action of resonance mercury radiation.
where \(F(r)\) is the electrostatic part, and \(f(r)\) the resonance part of the binding energy. Both these expressions, in turn, are functions of the interatomic distances. For the interaction of three atoms with one another, the expression
\[ -W_{123}=F(r_{12})+F(r_{23})+F(r_{13})+ \sqrt{f^2(r_{12})+f^2(r_{23})+f^2(r_{13})-f(r_{12})f(r_{23})-f(r_{23})f(r_{13})-f(r_{13})f(r_{12})} \tag{43} \]
is approximately applicable.
The activation heat of the exchange reaction is that minimum energy which transfers the configuration \(r_{12}=R\) (\(R\) being the distance between nuclei in the molecule), \(r_{13}=r_{23}=\infty\), into the configuration \(r_{12}=r_{13}=\infty\), \(r_{23}=R\) (under the assumption of an adiabatic course of the reaction, i.e., a reaction without quantum jumps). Of all possible paths of the reaction, as London showed \(^{54}\), the least activation heat is possessed by that in which atom 3 approaches molecule \(1—2\) along the continuation of the line joining the nuclei, i.e., in which \(r_{12}+r_{23}=r_{13}\). If, in view of its smallness, we neglect the Coulomb term \(F(r)\), and also the interaction of the outer atoms \(f(r_{13})\), then we obtain:
\[ -W_{123}=\sqrt{f^2(r_{12})+f^2(r_{23})-f(r_{12})f(r_{23})}. \tag{44} \]
The energy reaches a maximum at constant \(r_{12}=R\) and with the approach of the 3rd atom. For \(f(r_{13})=\dfrac{f(r_{12})}{2}\) we have:
\[ -W_{123}=\sqrt{f^2(r_{12})+\frac{f^2(r_{12})}{4}-\frac{f^2(r_{12})}{2}}. \tag{45} \]
The heat of activation is equal to \(W_{123}-W_{12}=0.13\,f(r_{12})\) and for the reaction \(\mathrm{H}+\mathrm{H}_2=\mathrm{H}_2+\mathrm{H}\) reaches about \(13\,000\) cal.
Polanyi and Eyring \(^{23,24,61}\) further developed London’s theory, taking into account the interaction of the two outer atoms, not considered in that theory, and the Coulomb term. As a first approximation, the so-called resonance energy of a linear system as a function of the distances \(r_{12}\) and \(r_{23}\) was calculated by formula (43), where the Coulomb terms were omitted and instead of \(f(r)\) the total binding energy \(f(r)+F(r)\) was taken. It was calculated, as a function of distance, by the Morse formula*. In Fig. 16a are shown the energy ratios as functions of both distances \(r_{12}\) and \(r_{23}\) (in Fig. 16, the distances between atoms \(X\) and \(Y\), and, correspondingly, \(X\) and \(Z\)). The path leading through the lowest “pass” from the state \(\mathrm{H}+\mathrm{H}_2\) to \(\mathrm{H}_2+\mathrm{H}\) is indicated by arrows. As the H atom approaches the molecule along this path, the latter experiences increasing repulsion and at the same time will be stretched. For the symmetrical configuration \(\mathrm{H—H—H}\) (distance \(\mathrm{H—H}=0.91\ \text{Å}\)) the height of the barrier is the smallest and reaches 30 Cal. As soon as the approaching
\[ \text{* P. H. Morse, Phys. Rev. 34, 57. 1929.} \]
atom passes over the barrier, it will be attracted, and at the same time the external atom of the molecule will be repelled.
The height of this barrier, which represents the heat of activation, proves to be smaller if the Coulomb energy is taken into account. The latter is allowed for with the aid of formulas derived theoretically by Sugiura*. Superposition, in this way,
Fig. 16. Resonance energy of three rectilinearly arranged H atoms as a function of distance.
of the “Coulomb valley” on the resonance ridge (Fig. 16a) gives a heat of activation of about 20 cal. A further lowering of the heat of activation is achieved because the zero-point energy in the symmetrical configuration is considerably smaller than in the initial
* Sugiura, Z. Physik, 24, 484, 1927.
\(\left(\dfrac{1}{2}h\nu = 6.5\ \text{Cal}\right)\), and a considerable part of this energy reserve can be used to lower the activation energy. Taking this circumstance into account, the activation energy is found to be approximately 15 large calories, which is in satisfactory agreement with the experimental value of 6–7 Cal.
An exact justification of the adiabatic course of the reaction \(\mathrm{H} + \mathrm{H}_2 \to \mathrm{H}_2 + \mathrm{H}\) was given by Pelzer and Wigner \({}^{60}\), and they indicated that a necessary condition for this is a sufficiently
TABLE 9
Theoretical values of the rate constant of the reaction
\(\mathrm{H} + \mathrm{H}_2 \to \mathrm{H}_2 + \mathrm{H}\)
| \(T\) | \(k\) in \(\text{mol}^{-1}\cdot \text{l}\cdot \text{sec}^{-1}\) |
\(C\) | \(C_{\text{quant}}\) |
|---|---|---|---|
| 283 | \(8.6\cdot 10^{4}\) | \(2.0\cdot 10^{6}\) | \(2.0\cdot 10^{6}\) |
| 373 | \(2.7\cdot 10^{6}\) | \(2.5\cdot 10^{6}\) | \(3.0\cdot 10^{6}\) |
| 873 | \(1.37\cdot 10^{9}\) | \(1.7\cdot 10^{6}\) | \(2.6\cdot 10^{6}\) |
| 1023 | \(2.4\cdot 10^{9}\) | \(1.9\cdot 10^{6}\) | \(2.7\cdot 10^{6}\) |
large distance between the lower quantum level and the next one after it, which is fulfilled in the present case. It was further possible to calculate the rate constants given in Tables 7 and 8. For this, in addition to the “straight-line” displacements already mentioned, cases were considered in which the H atom moves at an angle \(\delta\) to the line connecting the nuclei, which gives an activation heat greater by approximately \(\delta^2\cdot 10\ \text{Cal}^*\). The range of variation of this angle is calculated from the equality \(\delta^2\cdot 10\ \text{Cal} = RT\), where \(RT\) is the mean activation energy released in the collision. For the rate constant we obtain
\[ k = C T^{3/2} e^{-\frac{W}{RT}}, \tag{45}** \]
where \(C\), calculated from molecular data, is approximately equal to \(10^6\). If, using the experimentally found values of the constants \(k\) and taking the activation heat to be 6000 cal \({}^{***}\), one calculates the quantities \(C\) from formula (45), then one obtains about \(2\cdot 10^6\) (see Table 9, column 3), which is in satisfactory agreement with the theoretical value, equal to \(1\cdot 10^6\).
* This quantity depends on the configuration at the moment of collision, and not on the direction of the velocities.
** This formula differs from the formula usually used in reaction kinetics for the temperature dependence of the reaction rate by a factor \(T\). In determining the value from experimental data, in view of the smallness of the temperature range investigated, this factor does not have an effect.
*** Owing to the presence in (45) of the factor \(T^{3/2}\), the activation heat here is larger by \(RT\) than in A. Farkas.
The constants of column 2 and the quantities \(C\) refer to the rate of the general exchange reaction
\[ \mathrm{H}+\mathrm{H}_2 \rightleftarrows \mathrm{H}_2+\mathrm{H}, \]
irrespective of whether a transition has occurred or not. The ratio of the transition reaction to the pure exchange reaction reaches, in the case of the reaction \(\mathrm{H}+p\mathrm{H}_2\), the value \(3:1\), since the nuclear spin \(+\tfrac{1}{2}\) (of the free atom) can combine with \((\tfrac{1}{2},-\tfrac{1}{2})\) (the \(p\mathrm{H}_2\) molecule) in four different ways while conserving the nuclear multiplicity; three combinations give the \(o\)-state \((1,0,-1)\) and one the \(p\)-state \((0)\). (The distinction between the newly formed \(p\)-molecule and the \(o\)-state with nuclear spin angular momentum equal to zero cannot be interpreted without special consideration.) The yield of the exchange reaction \((ARk)\) in this case is
\[ Z_{ARk}=\frac{4}{3}Z_{p\to o}=Z_{k_1+k_2} \]
(Table 7), i.e. it is numerically equal to the sum of the probabilities of the spin reorientation reaction upon collision of \(\mathrm{H}_2\) with \(\mathrm{H}\). The same is also true for the reaction \(\mathrm{H}+o\mathrm{H}_2\), where the transition occurs in one of the four exchange reactions.
Here
\[ Z_{ARk}=4Z_{o\to p}=Z_{k_1+k_2}, \]
and thus we have grounds for using the constants \((k_1+k_2)\) for the exchange reaction, as was done in column 2 of Table 9.*
In one of his later works Wigner \(^{78}\) gave a quantum correction to the values cited, proceeding from a consideration of a nonmechanical transition (tunnel effect). According to quantum mechanics, particles whose energy lies below the heat of activation can also react with finite probability. However, this quantum correction hardly changes the constants \(C\) (Table 9, column 4).
c) Conversion by means of paramagnetic bodies. The increase in the rate of the mutual transition of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) by paramagnetic bodies was discovered by L. Farkas and H. Sachs \(^{32,33}\). In this case the conversion in the transitions \(p\to o\) (and conversely) will be facilitated by the fact that the combination prohibition will be, to a known degree, lifted under the action of a nonuniform magnetic field. It is essential here that the acting field be nonuniform over molecular dimensions, as indeed takes place in the collision of \(\mathrm{H}_2\) with a paramagnetic gas molecule or ion. In particular, the transition was investigated under the influence of \(\mathrm{O}_2\) (gaseous) in the range from 77 to 780°K. It turned out that the transition reaction throughout the investi—
* The above ratios \(4:3\) and \(4:1\) of the pure exchange reaction to the transition reaction for the elementary reactions \(p\to o\) and \(o\to p\) are applicable at all temperatures. At low temperatures, when
\[ \frac{p\mathrm{H}_2}{o\mathrm{H}_2} \]
is not greater than \(1/\infty\), equilibrium is reached owing to the fact that in this case the factors
\[ e^{-\frac{W}{RT}}, \]
which determine the individual rates, are not quite equal. At low temperatures the transition \(p\to o\), in contrast to the transition \(o\to p\), is on the average (over the various rotational states) endothermic.
in the important pressure range from 1 to 300 mm \(O_2\) and from 10 to 760 mm \(H_2\) proceeds according to the scheme
\[ pH_2 + O_2 \rightleftarrows oH_2 + O_2 \tag{46} \]
bimolecularly and homogeneously*. The reaction scheme corresponds to the following equation for the rate:
\[ -\frac{dx[H_2]}{dt}=k_1x[H_2][O_2]-k_2(1-x)[H_2][O_2], \tag{47} \]
Fig. 17. Time course of the reaction \(pH_2+O_2\rightleftarrows OH_2+O_2\).
integration of which, quite analogous to that carried out for the thermal reaction, gives:
\[ u_t=u_0 e^{-(k_1+k_2)[O_2]t}. \tag{48} \]
Since \([O_2]\) does not change during the experiment, we have an exponential time course (a reaction of first order with respect to \(H_2\)). Fig. 17 shows, on a logarithmic scale, the course of the transformation with time at various \(O_2\) pressures.
The temperature dependence of the reaction-rate constant is given in Table 10 in \(\mathrm{mole}^{-1}\cdot l\cdot \mathrm{min}^{-1}\). The collision yield of the overall transformation reaction and of the separate reactions \(p\to o\) and \(o\to p\) was calculated by means of Sutherland’s rule for the dependence of the effective cross section on temperature:
\[ d_{O_2}=2.4\sqrt{1+\frac{C_{O_2}}{T}}\ \text{Å}, \]
\[ d_{H_2}=2.0\sqrt{1+\frac{C_{H_2}}{T}}\ \text{Å}, \]
where \(C_{O_2}=130\), and \(C_{H_2}=80\).
* The transition of \(pH_2\) under the influence of \(O_2\) was also investigated in aqueous solutions of both gases. For \((k_1+k_2)\) here one obtains \(10.5\ \mathrm{mole}^{-1}\cdot l\cdot \mathrm{min}^{-1}\), which is very close to the constant for gases. This leads to the same collision yield for aqueous solutions, which is of interest for calculating the number of collisions in solutions. It is planned to carry out the experiment also in other solvents.
TABLE 10
Dependence of the reaction-rate constant on temperature for the reaction
\(p\mathrm{H}_2 + \mathrm{O}_2 \rightleftarrows o\mathrm{H}_2 + \mathrm{O}_2\)
| \(T\) | \(k_1 + k_2\), in \(\mathrm{mol}^{-1}\cdot l\cdot \mathrm{min}^{-1}\) | \(Z_{k_1+k_2}\) | \(Z_{k_1}(p \to o)\) | \(Z(o \to p)\) |
|---|---|---|---|---|
| 77 | 2.81 | \(2.38\cdot 10^{-13}\) | \(1.19\cdot 10^{-13}\) | \(1.19\cdot 10^{-13}\) |
| 86 | 2.94 | \(2.61\cdot 10^{-13}\) | \(1.44\cdot 10^{-13}\) | \(1.17\cdot 10^{-13}\) |
| 143 | 7.10 | \(5.86\cdot 10^{-13}\) | \(4.34\cdot 10^{-13}\) | \(1.52\cdot 10^{-13}\) |
| 293 | 9.16 | \(6.80\cdot 10^{-13}\) | \(5.10\cdot 10^{-13}\) | \(1.70\cdot 10^{-13}\) |
| 373 | 10.00 | \(6.85\cdot 10^{-13}\) | \(5.13\cdot 10^{-13}\) | \(1.72\cdot 10^{-13}\) |
| 493 | 11.10 | \(7.13\cdot 10^{-13}\) | \(5.25\cdot 10^{-13}\) | \(1.77\cdot 10^{-13}\) |
| 623 | 12.75 | \(7.61\cdot 10^{-13}\) | \(5.71\cdot 10^{-13}\) | \(1.90\cdot 10^{-13}\) |
| 773 | 23.90 | \(13.40\cdot 10^{-13}\) | \(10.05\cdot 10^{-13}\) | \(3.35\cdot 10^{-13}\) |
In Fig. 18 the collision yields \(Z_{k_1+k_2}\), \(Z_{k_1}\), and \(Z_{k_2}\) are shown graphically. The increase of the rate constant in the temperature region above \(600^\circ\mathrm{K}\) depends on the thermal reaction between \(\mathrm{O}_2\) and \(\mathrm{H}_2\), which above this temperature—proceeding homogeneously or heterogeneously—causes, in addition to the paramagnetic action, the transition reaction.
Fig. 18. Dependence of the collision yield of the reaction
\(p\mathrm{H}_2 + \mathrm{O}_2 \rightleftarrows o\mathrm{H}_2 + \mathrm{O}_2\) on temperature.
We shall return below to a discussion of the temperature dependence of the collision yield between \(77.3\) and \(623^\circ\).
The transition reaction has the same mechanism also in the presence of other paramagnetic gases, such as \(\mathrm{NO}\) or \(\mathrm{NO}_2\). But in the case of these gases it must be taken into account that not all their molecules are paramag-
are, for example, in NO, only those which are in the state \({}^{2}\Pi_{3/2}\) paramagnetic; in the case of \(\mathrm{NO_2}\), only the actually dissociated ones are, since the diamagnetic molecules \(\mathrm{N_2O_4}\) are inactive.
The ratio of the number of paramagnetic NO molecules to the total number is obtained from:
\[ \mathrm{NO}^{*}=\frac{e^{-\frac{354}{RT}}}{1+e^{-\frac{354}{RT}}}, \tag{49} \]
since the paramagnetic state \({}^{2}\Pi_{3/2}\) lies 354 cal higher than the lower \({}^{2}\Pi_{1/2}\) state (their statistical weights are equal).
Fig. 19. Temperature dependence of the collision yield for the reaction \(p\mathrm{H}_2+\mathrm{NO}^{*}\rightleftarrows o\mathrm{H}_2+\mathrm{NO}^{*}\) (on the ordinate axis the collision yield, multiplied by \(10^{12}\), is plotted).
Table 11 and Fig. 19 contain the results for NO. The collision yield was calculated from
\[ d_{\mathrm{NO}}=2.4\sqrt{1+\frac{130}{T}}\ \text{Å}. \]
For \(\mathrm{NO_2}\) the reaction was studied only at 293 and 373°. The rate constants, referred to pure \(\mathrm{NO_2}\), are equal to \(12.5\) and \(13.05\ \mathrm{mol}^{-1}\cdot l\cdot min^{-1}\).
Under the influence of paramagnetic ions, conversion of \(p\mathrm{H}_2\) also occurs in the dissolved state. Experiments with the series \(\mathrm{Zn}^{++}\), \(\mathrm{Cu}^{++}\), \(\mathrm{Ni}^{++}\), \(\mathrm{Co}^{++}\), \(\mathrm{Fe}^{++}\), \(\mathrm{Mn}^{++}\) show that with increa-
TABLE 11
Rate constants of the reaction \(p\mathrm{H}_2 + \mathrm{NO}^* \rightleftarrows o\mathrm{H}_2 + \mathrm{NO}^*\)
| \(T\) | \(\%\ \mathrm{NO}^*\) | \(k_1 + k_2\) in \(\mathrm{mol}^{-1}\cdot l\cdot \mathrm{min}^{-1}\) |
\(Z_{k_1+k_2}\) | \(Z_{k_1}(p \to o)\) | \(Z_{k_1}(o \to p)\) |
|---|---|---|---|---|---|
| 143 | 20,8 | 40,6 | \(3{,}40\cdot 10^{-12}\) | \(2{,}40\cdot 10^{-12}\) | \(1{,}00\cdot 10^{-12}\) |
| 199 | 28,0 | 41,6 | \(3{,}34\cdot 10^{-12}\) | \(2{,}46\cdot 10^{-12}\) | \(0{,}90\cdot 10^{-12}\) |
| 293 | 34,0 | 34,9 | \(2{,}59\cdot 10^{-12}\) | \(1{,}94\cdot 10^{-12}\) | \(0{,}65\cdot 10^{-12}\) |
| 373 | 37,4 | 40,0 | \(2{,}76\cdot 10^{-12}\) | \(2{,}07\cdot 10^{-12}\) | \(0{,}69\cdot 10^{-12}\) |
| 493 | 40,2 | 39,0 | \(2{,}54\cdot 10^{-12}\) | \(2{,}54\cdot 10^{-12}\) | \(0{,}64\cdot 10^{-12}\) |
| 793 | 43,5 | 49,4 | \(2{,}75\cdot 10^{-12}\) | \(2{,}06\cdot 10^{-12}\) | \(0{,}69\cdot 10^{-12}\) |
with an increase in the paramagnetic moment the transition is accelerated *; quantitative experiments have so far been carried out only for certain trivalent rare-earth ions.
First, experiments were carried out with neodymium chloride, for which the transition rates proved to lie in the concentration range from 2.5 to 21.4 millimoles/l and to be exactly proportional to the concentration; moreover, the time course of the reaction obeyed the exponential formula \(u_t = u_0 e^{-(k_1+k_2)[X]t}\), as was to be expected by analogy with \(\mathrm{O}_2\). Table 12 gives the rate constants and half-conversion times for the ions studied.
TABLE 12
Rate constants of the reaction for the \(p\mathrm{H}\)-conversion under the action of rare-earth ions
| Ion | Half-conversion time in min. at \(10^{-3}\ \mathrm{mol}/l\) | \(k_1 + k_2\) in \(\mathrm{mol}^{-1}\cdot l\cdot \mathrm{min}^{-1}\) |
Magnetic moment \(\mu\) ** | \(\dfrac{k_1+k_2}{\mu^2}\) |
|---|---|---|---|---|
| \(\mathrm{Pr}^{+++}\) | 304 | 2,26 | 3,62 | 0,181 |
| \(\mathrm{Nd}^{+++}\) | 290 | 2,37 | 3,68 | 0,168 |
| \(\mathrm{Sm}^{+++}\) | 1080 | 0,64 | \(\sim 1{,}60\) | 0,250 |
| \(\mathrm{Gd}^{+++}\) | 39,4 | 17,50 | 7,94 | 0,276 |
| \(\mathrm{Er}^{+++}\) | 18,0 | 38,20 | 9,70 | 0,416 |
| \(\mathrm{Yb}^{+++}\) | 67,5 | 10,20 | 4,50 | 0,502 |
* The rate constants of the reaction have so far been determined only approximately; they are equal to 0, 1.0, 1.3, 3.0, 3.3, and \(5.3\ \mathrm{mol}^{-1}\cdot l\cdot \mathrm{min}^{-1}\).
** In units of the Bohr magneton \(= 9{,}17\cdot 10^{-21}\) CGS.
Although in the gaseous state, under the influence of diamagnetic gases, such as, for example, \(N_2\), \(NH_3\), \(CO_2\), \(CO\), \(SO_2\), \(HJ\), \(Fe(CO)_5\), etc., no appreciable conversion is observed in the course of a day at room temperature and at a pressure of \(300\) mm, it is nevertheless possible to find a number of diamagnetic solvents containing H atoms under whose influence dissolved \(pH_2\) is converted. The rate constants of these reactions are all approximately \(10^5\) times smaller than, for example, those observed in the case of dissolved \(O_2\). The cause of these conversions should be regarded as the nuclear magneton of the proton, which possesses a (though very small) paramagnetic moment.
TABLE 13
Time of half-conversion of \(pH\) in various solvents
| Solvent | Observed half-conversion time in min. | \(k_1 + k_2 = \dfrac{0.69}{\tau[\mathrm H]}\ \dfrac{l}{mole}\) |
|---|---|---|
| \(H_2O\) | 134* | \(4.64\cdot 10^{-5}\) |
| Benzene \((C_6H_6)\) | 350 | \(2.92\cdot 10^{-5}\) |
| Aniline \((C_6H_5NH_2)\) | 270 | \(3.32\cdot 10^{-5}\) |
| Methyl alcohol \((CH_3OH)\) | 230 | \(3.04\cdot 10^{-5}\) |
| Cyclohexanol \((C_6H_{11}OH)\) | 450 | \(1.36\cdot 10^{-5}\) |
| Carbon disulfide \((CS_2)\) | 1000 |
Table 13 contains the results of experiments belonging here. The reaction here too exhibits an exponential course in time and proceeds bimolecularly according to the equation \(pH_2 + A = oH_2 + A\), where \(A\) denotes a molecule of the dissolving medium. We see that the rate constants, referred to identical \([\mathrm H]\), have the same order of magnitude, which speaks in favor of the correctness of the view that the conversion is caused by nuclear magnetons. In water the conversion proceeds fastest of all, while in \(CS_2\) it is, on average, one order of magnitude smaller; and in this case, where there are no nuclear moments, it depends either on impurities \([O_2]\), or else on the magnetic moment that appears when \(pH_2\) and \(CS_2\) approach very closely at the moment of impact.
A theoretical calculation of the paramagnetic transition \(p \to o\) was undertaken by Wigner\({}^{79}\). The course of his reasoning is as follows: If a \(pH_2\) molecule is in a homogeneous magnetic field, then the interaction energy of the system magnetic field \(+\ pH_2\) will not change if the Cartesian and spin coordinates of both nuclei are interchanged.
* The conversion of \(pH_2\) caused by \(H_2O\) was taken into account in all experiments with aqueous solutions.
If, however, the field is so inhomogeneous that the field strength changes appreciably over a segment equal to the distance \(R\) between the nuclei of \(\mathrm{H}_2\), then the energy of the system also changes, and for the matrix element of the energy of interaction of the nuclear spins \(\mu_P\) with the dipole field \(\dfrac{\mu_A}{r^3}\) of a paramagnetic particle we obtain:
\[ M_{jm,j-1,m}=\frac{3R\mu_A\mu_P}{r^4} \sqrt{\frac{j^2-m^2}{(2j+1)(2j-1)}}, \tag{50} \]
\[ M_{jm,j+1,m}=\frac{3R\mu_A\mu_P}{r^4} \sqrt{\frac{(j+1)^2-m^2}{(2j+1)(2j+3)}}. \tag{51} \]
The probability \(\left|c_{j+1}\right|^2\) of the state \(j+1\), arising as a result of a collision of \(\mathrm{H}_2\) with a paramagnetic molecule in the rotational quantum state \(j\), is given by the equation:
\[ \frac{ih}{2\pi}\frac{\partial c_{j+1}}{\partial t} = M_{jm,j+1,m} e^{-\frac{i(E_{j+1}-E_j)}{h}t}. \tag{52} \]
The collision is here replaced by the following process: the paramagnetic molecule is brought infinitely rapidly to a place close to the place of collision, remains at this distance for a time equal to the mean duration of the collision, and is then removed infinitely rapidly. During the time of the collision the occurrence of the new state is considered to be described by the Schrödinger equation (52).
Solving equation (52) and substituting the mean interaction time \(\dfrac{a_s}{3v}\) (where \(v\) is the relative velocity at the moment of collision), squaring and dividing by \(m\), we obtain approximately for the probability of the transition \(p\to o\) (for small \(j\)):
\[ W_{j,j+1} = \frac{8\pi^2J\mu_A^2P}{9h^2a_s^6kT} \cdot \frac{j+1}{2j+1} = W_{01}\frac{j+1}{2j+1}; \tag{53} \]
\[ W_{j,j-1} = W_{01}\frac{j}{2j+1}. \tag{54} \]
If we are dealing with the transition \(o\to p\), then an additional factor \(1/3\) enters. For transitions proceeding with absorption of energy \((E)\), in order to find the yield of collisions it is necessary to multiply the above probability by \(e^{-\frac{E}{RT}}\). Numerically, for \(T=300^\circ\), \(a_s=1\ \text{\AA}\), and \(\mu_A=1\) Bohr magneton, \(W_{01}\) is equal to \(3\cdot10^{-12}\).
For comparison of the theoretical reaction rate with experiment it is necessary to start from the following reaction equation:
\[ -\frac{dp[\mathrm{H}_2]}{dt} = \bigl(k_{01}a_0+(k_{21}+k_{23})a_2+(k_{43}+k_{45})a_4+\ldots \]
\[ -(k_{10}+k_{12})a-(k_{32}+k_{34})a_3-\ldots\bigr)[\mathrm{X}], \]
which combines the various transitions with rate constants \(k_{lm}\). \(a_0, a_1, a_2\ldots\) are the concentrations of \(\mathrm{H}_2\) in the individual rotational states.
Integration of these equations, taking into account that \(a_0, a_2, a_4\ldots\) and \(a_1, a_3\ldots\) are each in thermal equilibrium with one another (see in more detail L. Farkas and G. Sachs\(^{32}\)), gives the following expression for the constants \(k_1\) and \(k_2\) of equation (48):
\[ k_1=\frac{k_{01}+(k_{21}+k_{23})5e^{-\frac{E_2}{kT}}+\ldots} {1+5e^{-\frac{E_2}{kT}}+9e^{-\frac{E_4}{kT}}+\ldots} \]
\[ k_2=\frac{(k_{10}+k_{12})9e^{-\frac{E_1}{kT}}+(k_{32}+k_{34})21e^{-\frac{E_3}{kT}}+\ldots} {9e^{-\frac{E_1}{kT}}+21e^{-\frac{E_3}{kT}}+\ldots} \]
At low temperatures it is sufficient to consider transitions occurring among three thermal states. Substituting the theoretical expressions for \(k_{01}\), \(k_{10}\), and \(k_{21}\) (the transitions \(0\to 1\) and \(1\to 2\) are endothermic and are accompanied by effects of \(337.2\ \mathrm{cal}\) and \(674.4\ \mathrm{cal}\)), we obtain:
\[ Z_{k_1+k_2}= \frac{ e^{-\frac{337.2}{RT}}W_{10}+\frac{2}{5}W_{01}5e^{-\frac{1011.6}{RT}} }{ 1+5e^{-\frac{1011.6}{RT}} } + \]
\[ + \frac{ \frac{1}{9}W_{01}9e^{-\frac{337.2}{RT}}+e^{-\frac{674.4}{RT}}\frac{2}{9}W_{01}9e^{-\frac{337.2}{RT}} }{ 9e^{-\frac{337.2}{RT}} }. \tag{55} \]
From (55), substituting \(a_s\) equal to from \(1.0\) to \(3\cdot 10^{-8}\ \mathrm{cm}\), we obtain for \(Z_{k_1+k_2}\) from \(8.8\cdot 10^{-12}\) to \(1.6\cdot 10^{-14}\), whereas the experimental value is \(6.8\cdot 10^{-13}\) (Table 10, column 3). Some uncertainty in \(a_s\) is explained by the fact that the gaskinetic collision cross section need not necessarily coincide with the effective cross section relevant for the paramagnetic transformation. Since in the theoretical formula \(a_s\) enters as \(a_s^{-6}\), while in the experimental one—for the yield of collisions—as \(a_s^{-2}\) (the number of collisions), and consequently their ratio contains \(a_s^4\), it is easy, by a suitable choice of \(a_s\), to obtain complete agreement between theory and experiment.
The theoretical temperature dependence of the collision yield \(Z_{k_1+k_2}\) is given by the expression:
\[ \frac{1}{T a_s^6} \left\{ \frac{ e^{-\frac{337.2}{RT}}+2e^{-\frac{1011.6}{RT}} }{ 1+5e^{-\frac{1011.6}{R}} } +\frac{1}{9}+2e^{-\frac{674.4}{RT}} \right\}, \tag{56} \]
in which the temperature dependence of \(a_s\) can be introduced into the calculations by means of the factor
\[ \sqrt{1+\frac{C_O}{T}}+\sqrt{1+\frac{C_H}{T}}, \]
Table 14 gives this theoretical temperature dependence; its course agrees well with experiment. For NO one likewise obtains a qualitative agreement with experiment if one makes the assumption that, in a collision accompanied by conversion, NO passes from the paramagnetic state \({}^{2}\Pi_{3/2}\) into the diamagnetic \({}^{2}\Pi_{1/2}\).* The energy acting here causes the temperature dependence of the collision yield in this case, in first approximation, to be given by the expression:
TABLE 14
Theoretical dependence of the collision yield on temperature in the paramagnetic \(pH\)-conversion
| \(T\) | \(\mathrm{O}_2\) | NO |
|---|---|---|
| 77.3 | \(1.9\cdot10^{-13}\) | — |
| 143 | \(5.3\cdot10^{-13}\) | \(3.5\cdot10^{-12}\) |
| 193 | \(6.1\cdot10^{-13}\) | \(3.3\cdot10^{-12}\) |
| 293 | \(6.8\cdot10^{-13}\) | \(2.6\cdot10^{-12}\) |
| 273 | \(7.4\cdot10^{-13}\) | \(2.5\cdot10^{-12}\) |
| 493 | \(6.8\cdot10^{-13}\) | \(2.3\cdot10^{-12}\) |
\[ \frac{1}{T a_s^6}\cdot \left( \frac{1+2e^{-\frac{1011.6}{RT}}}{1+5e^{-\frac{1011.6}{RT}}} +\frac{1}{9} +2e^{-\frac{320.4}{RT}} \right). \]
Experiments with rare elements show, in addition to the influence of \(a_s\), also a quadratic dependence of the collision yield in conversions on the effective moment. If for this case we form
\[ \frac{k_1+k_2}{\mu^2}, \]
then this quantity increases with increasing atomic number, although \(\mu\) itself does not change monotonically (see Table 12, column 5). This can be explained only by a decrease of the “magnetic collision distance,” which changes with increasing atomic number. The fact that the “magnetic effective cross section” is mono-
* It is probably this “resonance,” and not a substantial difference in the radii of action, that explains the larger collision yield for NO in comparison with \(\mathrm{O}_2\).
…decreases monotonically with the ordinal number, is in qualitative agreement with the fact that the radii of the atoms of the rare-earth elements, determined by other methods, also decrease in this direction. Over still wider limits the quadratic dependence of the reaction rate on the magnetic moment is shown by transitions in solutions under the influence of nuclear magnets. These rates, as has been established experimentally, are approximately \(10^{-5}\) times smaller than the rate for \(\mu = 1\) Bohr magneton, whereas theoretically one would expect a ratio equal to \(10^{-6}\). This insignificant discrepancy in this case can also be explained by the difference in the values of \(a_s\).
Finally, it must be mentioned that the transition \(p\mathrm{H}_2\) into \(g\mathrm{H}_2\) upon collision of \(p\mathrm{H}_2\) with \(o\mathrm{H}_2\), or with another \(p\mathrm{H}_2\) molecule, must also occur in the gaseous phase. In the latter case we have no paramagnetic moment; but the fact that at the transition a moment appears makes matters easier, and this virtual moment makes the transition possible.* An approximate calculation shows that the half-conversion time at atmospheric pressure under these conditions should reach several years, which is not in contradiction with experience.
Fig. 20. Time course of the reaction \(n\mathrm{H}_2 \longrightarrow p\mathrm{H}_2\) for the liquid and solid states.
d) Conversion of liquid and solid hydrogen. The enrichment of liquid hydrogen with \(p\mathrm{H}_2\) was observed already by Bonhoeffer and Harteck.** The kinetics of the transition in the liquid and solid phase was studied in more detail by Cremer and Polanyi\(^{13}\). Fig. 20 and Table 15 show the time course of the reaction found by them in the liquid and solid phases (their results for liquid \(\mathrm{H}_2\) agree well with the data of Bonhoeffer and Harteck\(^{4,5}\), and also of Keesom\(^{52}\)). The rate of the reaction is well expressed by the formula
\[ \frac{dx}{dt} = -kx^2 \tag{57} \]
(see columns 3 and 5 of Table 14, where \(x\) denotes the concentration of \(o\mathrm{H}_2\) in percent, and \(t\) is expressed in hours). It is noteworthy that in the solid phase the reaction constant is larger than in the liquid, and that at concentrations below \(26\%\) \(o\mathrm{H}_2\) it decreases noticeably.
* That the reaction \(p\mathrm{H}_2 + p\mathrm{H}_2 \to p\mathrm{H}_2 + o\mathrm{H}_2\) must occur follows from the fact that the reaction \(p\mathrm{H}_2 + o\mathrm{H}_2 \to p\mathrm{H}_2 + p\mathrm{H}_2\) exists.
** The heat liberated during the transition causes accelerated evaporation of the condensate. Therefore condensate rich in \(p\mathrm{H}_2\) is preserved much better than liquid or solid \(n\mathrm{H}_2\).
The “transition-reaction rate” constant, both in the solid (4–14° K) and in the liquid (14–20° K) state, is completely independent of temperature. As the mechanism of the reaction, according to equation (57), a reaction between two ortho molecules was assumed,
TABLE 15
Rate constants of the reaction \(p\mathrm{H}_2 \rightleftarrows o\mathrm{H}_2\) in the liquid and solid phase
| Time | Transformation in solid phase \(x\) | Transformation in solid phase \(k \cdot 10^5\) | Transformation in liquid phase \(x\) | Transformation in liquid phase \(k \cdot 10^5\) |
|---|---|---|---|---|
| 5 | 69.5 | 20 | 72.3 | 10 |
| 15 | 61.6 | 18 | 67.0 | 11 |
| 25 | 55.2 | 18 | 62.5 | 12 |
| 40 | 47.8 | 20 | 56.5 | 12 |
| 60 | 40.0 | 20 | 50.0 | 12 |
| 80 | 34.2 | 21 | 44.4 | 12 |
| 100 | 30.0 | 20 | 40.0 | 13 |
| 115 | 27.7 | 19 | 37.5 | 12 |
| 125 | 26.5 | 14 | 35.8 | 12 |
| 135 | 25.6 | 12 | 34.3 | 11 |
| 145 | 24.7 | 11 | 32.8 | 13 |
which leads to the formation of \(2p\mathrm{H}_2\) or \(1p\mathrm{H}_2 + 1o\mathrm{H}_2\) molecules. However, apart from an exchange reaction of the chemical type (such a reaction would have to have an activation energy of \(100\) cal), and also apart from a nonmechanical exchange of protons in the \(\mathrm{H}_2\) molecules under consideration (tunnel effect; the transition probability, owing to the unfavorable configuration, is very small), the transition will be caused mainly by forces that begin to act when the molecules come into close contact. If it is assumed that \(o\mathrm{H}_2\) molecules take part in the reaction and if one takes into account that the reaction rate is independent of temperature, then it becomes very probable that here we are dealing with a magnetic transition under the influence of a nuclear magneton*, as, for example, occurs at room temperature in \(p\mathrm{H}_2\) dissolved in \(\mathrm{H}_2\mathrm{O}\), when the transition is caused by the magnetic moments of the protons.
The “enhancement” of the reaction at a considerable concentration of \(o\mathrm{H}_2\) probably depends on the fact that individual \(o\mathrm{H}_2\) molecules located in the lattice, as a result of a considerable rate of interchange of place
* Two \(o\mathrm{H}_2\) molecules with nuclear moments \(1 + 1\) (three values are possible: \(1, 0, -1\)) can, while preserving nuclear multiplicity, give 9 states: \((+2, +1, 0, -1, -2)\), \((1, 0, -1)\), and \((0)\), of which the first group yields \(o\mathrm{H}_2 + o\mathrm{H}_2\) (i.e. in this case no reaction has occurred, probability \(5/9\)), the second—\(1o\mathrm{H}_2 + 1p\mathrm{H}_2\) (probability \(3/9\)), and the third—\(1p\mathrm{H}_2 + 1p\mathrm{H}_2\) (probability \(1/9\)).
do not find in their vicinity reaction partners (see, in this connection, the report by F. Kremer). This is confirmed by the fact that the conversion begins again at the former rate when the solid H₂ is melted.
2. Heterogeneous Catalysis of the Transition
The heterogeneous catalysis of the process of establishing equilibrium between the concentrations of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) was discovered by Bonhoeffer and Harteck \(^{4,5}\). This phenomenon can occur both at high and at low temperatures and can be caused by various substances. Such catalysts may be activated charcoal, platinum black, powders of various metals, metal oxides, etc. In this case the catalytic activity naturally depends on the state of the surface.* If, as a measure of catalytic activity, we take the quantity of H₂ converted by the given catalyst in 1 sec, then the quantity \(A\), defined in this way, shows a very peculiar dependence on temperature.
Fig. 21. Temperature dependence of the activity of sugar charcoal.
At low temperatures \(A\) has a negative temperature coefficient, whereas at high temperatures it has a positive one. Fig. 21 shows the typical course of the curve of the conversion rate under the influence of activated charcoal. Very similar curves are obtained for powders of Cu and NaCl, although in some cases the negative temperature coefficient is not so sharply expressed (see Bonhoeffer and A. Farkas \(^{8,9}\), Bonhoeffer, A. Farkas and Rummel \(^{10}\)).
The observed temperature dependence of the reaction rate \(A\) shows a known similarity to the temperature dependence of the sorption of H₂ observed for various adsorbents,** when the absorption of H₂ in a certain definite temperature interval, with increasing temperature, first decreases and then, upon further—
* At high pressures the establishment of equilibrium, observed in a glass or metallic vessel, in the temperature range from \(-190\) to \(20^\circ\mathrm{C}\) (Bonhoeffer and Harteck \(^{4,5}\), Eucken and Hiller \(^{20}\)), also proceeds as a heterogeneous reaction (Farkas and Bonhoeffer \(^{29}\)). The time of half-conversion reaches 1–14 days, depending on temperature and pressure, but it is very poorly reproducible. It is possible, however, that traces of O₂, by means of paramagnetic catalysis, also influence the transition (see Chapter IV, 1c).
** Adsorption of gases by solid substances. General discussion held at the Faraday Society, Trans. Far. Soc. 28, 1932.
further increase, increases again. This behavior was explained by Taylor\(^{71}\) by assuming two kinds of adsorption: one kind is van der Waals adsorption, having a (large) temperature-independent rate and in which, with increasing temperature, the amount of adsorbed substance decreases; the other kind of adsorption proceeds as a chemical reaction, has a definite heat of activa-
TABLE 16
Sugar charcoal; \(T = -183^\circ\mathrm{C}\)
| Pressure in mm | \(\tau_{\frac{1}{2}} =\) Time of half-transformation in sec |
\(k = \dfrac{0.69}{\tau_{\frac{1}{2}}}\) |
|---|---|---|
| 7 | 1 140 | \(6.05 \cdot 10^{-4}\) |
| 60 | 1 200 | \(5.75 \cdot 10^{-4}\) |
| 102 | 1 170 | \(5.90 \cdot 10^{-4}\) |
| 170 | 1 260 | \(5.48 \cdot 10^{-4}\) |
| 760 | 1 200 | \(5.75 \cdot 10^{-4}\) |
TABLE 17
Coconut-shell charcoal (benzene-free). Pressure 760 mm
| \(T\) in °K | \(\tau_{\frac{1}{2}}\) in sec | \(k = \dfrac{0.69^*}{\tau_{\frac{1}{2}}}\) | \(k_1\) \(o \to p\) |
\(k_2\) \(p \to o\) |
|---|---|---|---|---|
| 106 | 1260 | \(5.48 \cdot 10^{-4}\) | \(2.00 \cdot 10^{-4}\) | \(3.48 \cdot 10^{-4}\) |
| 90 | 1200 | \(5.75 \cdot 10^{-4}\) | \(2.48 \cdot 10^{-4}\) | \(3.27 \cdot 10^{-4}\) |
| 62 | 1140 | \(6.05 \cdot 10^{-4}\) | \(3.71 \cdot 10^{-4}\) | \(2.34 \cdot 10^{-4}\) |
tion (activated adsorption) and manifests itself only when, owing to the rise in temperature, its rate reaches an appreciable magnitude; this also explains the renewed increase in adsorption.**
In what follows it will be shown that the noted similarity between the temperature dependence of the absorption of \(\mathrm{H_2}\), on the one hand, and the rate of transition, on the other, is caused by the fact that in these two kinds of adsorption there occur two different indepen-
* Since the equilibrium between \(p\mathrm{H_2}\) and \(o\mathrm{H_2}\) changes with temperature, the rate constant of the individual reactions \(p \to o\) and \(o \to p\) also changes.
** In this kind of adsorption the measured “adsorption isotherms” express no equilibrium state.
distinct from one another, mechanisms of the transition reaction. Both of these mechanisms were investigated separately and will be denoted by us as the low- and high-temperature mechanisms.
a) Low-temperature mechanism. The low-temperature mechanism has been studied best of all on activated charcoal (K. P. Bonhoeffer, A. Farkas, and K. W. Rummel^10, Bonhoeffer and Rummel^11, Rummel^64). It is clear in advance that the activity \(A\), defined above, must depend on two factors: first, on the concentration of \(\mathrm{H}_2\) in the adsorbed layer, and second, on the rate constant of the reaction taking place at the phase boundary. Of these two factors, the first, as is known, has
Fig. 22. Apparatus for measuring the transition of \(\mathrm{pH}_2\) in the adsorbed state.
Fig. 23. Catalysis of the transition on various kinds of charcoal.
a negative temperature coefficient (the amount of adsorbed substance decreases with temperature). For investigation of the second factor, activated charcoal is especially convenient, since, owing to its great adsorption capacity, it is possible from time to time, for studying the course of the reaction at the phase boundary, to take samples without substantially changing the concentration in the adsorption layer. The apparatus for investigations of this kind is shown schematically in Fig. 22.
Different kinds of charcoal catalyze at different rates. Figure 23 shows the time course of the establishment of equilibrium at \(88^\circ\ \mathrm{K}\) for six different charcoal preparations. The remaining experiments were carried out with ash-free varieties V and VI.
The time course follows an exponential law \(u_t = u_0 e^{-kt}\) (here \(k\), as in the case of the thermal transformation, represents the sum of the rate constants of the forward and reverse reactions).
The experiments show (see Tables 16 and 17) that the rate constant of the reaction \(k\) does not depend on pressure (a first-order reaction) nor on temperature. From the independence of \(k\) from temperature it follows that the decrease in activity with increasing temperature is explained only by the decrease in the concentration of \(\mathrm{H}_2\) in the adsorption layer. The rate of the reaction
under constant conditions it is well reproducible, but it decreases when the carbon is treated with H₂ or O₂ at high temperature. Conversely, adsorption of O₂ at −185°C causes an increase in \(k\), sometimes up to a tenfold value.
This latter effect, together with the independence of the temperature and the first order of the reaction with respect to H₂, makes it probable to suppose that the conversion of adsorbed H₂ molecules is caused by the magnetic forces of the substrate, just as occurs in the homogeneous phase under the influence of O₂ and other paramagnetic substances (Chapter IV, 1c). The magnetic forces depend predominantly on carbon atoms with free valences. At low temperatures O₂ is adsorbed molecularly, and its accelerating action should be attributed to its paramagnetic moment. Under
TABLE 18
pH-conversion on Pt at a pressure of 30 mm
| \(T\) in °C | \(\tau_{\frac{1}{2}}\) in sec. | \(k=\dfrac{0.69^*}{\tau_{\frac{1}{2}}}\) |
|---|---|---|
| 100 | 420 | \(1.64\cdot10^{-3}\) |
| 120 | 126 | \(5.47\cdot10^{-3}\) |
| 140 | 48 | \(1.44\cdot10^{-2}\) |
| 170 | 18 | \(3.83\cdot10^{-2}\) |
| 210 | 8 | \(7.63\cdot10^{-2}\) |
| 245 | 5 | \(1.38\cdot10^{-1}\) |
TABLE 19
pH-conversion on W at a pressure of 50 mm
| \(T\) in °C | \(\tau_{\frac{1}{2}}\) in sec. | \(k=\dfrac{0.69^*}{\tau_{\frac{1}{2}}}\) |
|---|---|---|
| 110 | 920 | \(7.50\cdot10^{-4}\) |
| 100 | 340 | \(2.03\cdot10^{-3}\) |
| 75 | 59 | \(1.17\cdot10^{-2}\) |
| 50 | 24 | \(2.87\cdot10^{-2}\) |
| 25 | 15 | \(4.60\cdot10^{-2}\) |
| 0 | 8 | \(7.63\cdot10^{-2}\) |
the action of O₂ at high temperatures it combines chemically with the carbon, and the slowing of the reaction is explained in this case, probably, by saturation of the free C valences.
Thus the mechanism of the transition at low temperatures consists, apparently, in the fact that H₂ first is molecularly adsorbed, and then in the adsorbed phase, under the influence of magnetic forces, without coming into contact with another H₂ molecule, after a certain and temperature-independent** “holding” time, the transition occurs.
b) High-temperature mechanism. The high-temperature mechanism was investigated on various materials, such as metals (in the form of wire and powder), salts, and metallic
* The time course of the reaction here too obeys the exponential law \(u_t=u_0e^{-kt}\).
** Recently Taylor and Diamond showed that the heterogeneous conversion \(p\mathrm{H}_2\to o\mathrm{H}_2\) at low temperatures on paramagnetic adsorbents (rare-earth oxides) proceeds with a much greater velocity.
oxides. Whereas the activity of catalysts in the low-temperature mechanism changes little with an increase in the amount of adsorbed gas and for other reasons, at high tempera—
TABLE 20
\(p\mathrm{H}_2\)-conversion on NaCl. Gas volume \(50\ \mathrm{cm}^3\); pressure, \(0.02\ \mathrm{mm}\); \(2\ \mathrm{g}\) of substance
| \(T\) in °C | \(\tau_{\frac{1}{2}}\) in sec. | \(k=\dfrac{0,69^*}{\tau_{\frac{1}{2}}}\) | \(T\) in °C | \(\tau_{\frac{1}{2}}\) in sec. | \(k=\dfrac{0,69^*}{\tau_{\frac{1}{2}}}\) |
|---|---|---|---|---|---|
| \(-183\) | 108 000 | \(6,39\cdot 10^{-5}\) | 205 | 348 | \(1,98\cdot 10^{-3}\) |
| \(-80\) | 36 000 | \(1,92\cdot 10^{-5}\) | 246 | 192 | \(3,59\cdot 10^{-3}\) |
| \(-120\) | 50 000 | 300 | 66 | \(1,04\cdot 10^{-2}\) | |
| \(-146\) | 1 000 | \(6,57\cdot 10^{-4}\) | 340 | 42 | \(1,64\cdot 10^{-2}\) |
tures, on the contrary, the reaction is already “poisoned” by very small impurities of certain substances (\(\mathrm{O}_2\), \(\mathrm{H}_2\mathrm{S}\), fats, etc.), and the activity is lowered by whole orders of magnitude. The apparatus for experiments with powdered catalysts in this
TABLE 21
Catalysis by oxides at a pressure of \(200\ \mathrm{mm}\)
(according to Taylor and Sherman \(^{72}\))
| Catalyst | Amount | \(T\) °K | Duration of contact, min. | % conversion |
|---|---|---|---|---|
| ZnO | 15 g | 298 | 15 | 1,4 |
| ZnO | 15 g | 373 | 15 | 79,5 |
| ZnO·Cr\(_2\)O\(_3\) | 10 g | 298 | 15 | 100 |
| CdO | 45 g | 373 | 15 | 0 |
| MnO·Cr\(_2\)O\(_3\) | 15 g | 273 | 5 | 17 |
| MnO·Cr\(_2\)O\(_3\) | 15 g | 285 | 5 | 68 |
| CuO·Cr\(_2\)O\(_3\) | 45 g | 373 | 5 | 55 |
| Al\(_2\)O\(_3\) | 10 g | 674 | 15 | 14 |
case is the same as for experiments with charcoal**. In studying the conversion of \(p\mathrm{H}_2\) on metal wires, they were placed in a vessel resembling an electron tube and could be heated by an electric current. In this case it was possible to compare with one another the catalytic activities of different metals (at identical catalyzing surfaces).
* The first three numbers indicate negative temperature coefficients.
** It goes without saying that, as the initial gas, gas rich in \(p\mathrm{H}_2\) is used (from 45 to 46% \(p\mathrm{H}_2\)).
These experiments showed that Pt, W (see 8), Ni and Fe (see 28) catalyze the transition well; conversely, Cu, Ag, Au (see 9) are almost completely inactive up to a temperature of 500–600°C. Tables 18 and 19 give the temperature dependences for Pt and W. The volume of the vessel reached approximately 1 l, the surface area of the wire \(1/2\ \text{cm}^2\).
For NaCl and for the oxides the following temperature dependences were obtained (Tables 20 and 21). From the dependence of the time of half-conversion on pressure, using the already known relation \(\tau_{1/2}\cdot \text{pressure}^{\text{order}-1}=\text{const}\), one obtains an apparent order* of about 0.3–1, depending on the temperature, pressure range, and catalyst (see Tables 22 and 23).
TABLE 22
Dependence of the rate of the \(pH\)-conversion—on a tungsten wire—on pressure, at 100°C
| Pressure in mm | \(\tau_{1/2}\) |
|---|---|
| 25 | 150 |
| 50 | 240 |
| 100 | 510 |
| 200 | 720 |
| 400 | 1100 |
| Order 0.3 | Order 0.3 |
TABLE 23
Dependence of the rate of the \(pH\)-conversion—on a nickel tube—on pressure, at 12°C
| Pressure in mm | \(\tau_{1/2}\) |
|---|---|
| 0.004 | 138 |
| 0.04 | 294 |
| 0.44 | 780 |
| 4.5 | 1980 |
| Order 0.6 | Order 0.6 |
The order found indicates that at high temperatures the transition takes place entirely in the adsorbed layer and that no reaction occurs between adsorbed molecules and molecules arriving at the surface of the adsorbent, since in that case the apparent order would be greater than 1. The circumstance that the high-temperature mechanism manifests itself in such a temperature region in which activated adsorption is also noticeable, first discovered by Taylor \(^{71,72,73}\), makes the following interpretation of the processes probable: when a molecule strikes a catalyst, it is either reflected, or van der Waals adsorption occurs. The adsorbed molecule either evaporates after a comparatively short interval—
* The apparent order of a heterogeneous reaction gives the dependence of the reaction rate on gas pressure, whereas the true order gives the number of molecules participating in the reaction in the adsorbed layer. The explanation of high-temperature catalysis is also complicated by the fact that the dependence of the concentration in the adsorption layer for these catalysts is not known in a form permitting application.
...over time, or after absorption of a certain heat of activation, passes into an activated adsorbed state. In this activated adsorbed state the bond between the atoms is so disrupted that it is no longer possible to distinguish \(p\)- and \(o\)-molecules. Thus activated (atomic) adsorption leads directly to the transition. Conversely, in short-lived adsorption of the first kind the transition has no time to occur before reflection. The reason for the dependence of the reaction rate on temperature is therefore the presence of a heat of activation in that kind of adsorption which leads to the transformation.
Both reaction mechanisms (high-temperature and low-temperature) are illustrated by the scheme shown in Fig. 24.
V. Applications
1. Measurement of the concentration of H-atoms
The exchange reaction
\[ \mathrm{H} + p\mathrm{H}_2 \rightleftarrows o\mathrm{H}_2 + \mathrm{H} \]
can be used for detecting and measuring the concentration of H-atoms in a mixture of reacting gases, provided that \(p\mathrm{H}_2\) can be added to the reacting substances without disturbing the course of the reaction. The presence of H-atoms will be indicated by the destruction of \(p\mathrm{H}_2\), whereas for concentration measurements one may make use of the decrease in the concentration of \(p\mathrm{H}_2\) under definite temperature conditions.
Labels in Fig. 24 diagram: Molecules in the gas; van der Waals adsorption; activated adsorption.
Fig. 24. Scheme of high- and low-temperature catalysis: \(q\) — heat of van der Waals adsorption, \(E\) — heat of activation in activated van der Waals adsorption, \(Q\) — heat of adsorption in van der Waals activated adsorption.
In the case of the photochemical reaction chlorine—hydrogen, the liberation of H-atoms was established by Geib and Harteck\({}^{36}\) by the following method: into a flask filled with \(p\mathrm{H}_2\) to a pressure of several hundred millimeters, chlorine was slowly introduced through a capillary while shaking and under intense illumination, so that the stationary concentration of chlorine inside the flask was extremely small. The chlorine entering the vessel was decomposed by the light into atoms, and these latter reacted according to the equation \(\mathrm{Cl} + \mathrm{H}_2 = \mathrm{HCl} + \mathrm{H}\). The H-atoms, which, because of the very small concentration of \(\mathrm{Cl}_2\), only in rare cases could react according to the scheme \(\mathrm{H} + \mathrm{Cl}_2 = \mathrm{HCl} + \mathrm{Cl}\), converted \(p\mathrm{H}_2\) into \(g\mathrm{H}_2\).*
* Since the yield of collisions of the reaction \(\mathrm{H} + p\mathrm{H}_2 \rightleftarrows o\mathrm{H}_2 + \mathrm{H}\) at room temperature reaches approximately \(3 \cdot 10^{-7}\), the yield of collisions of the reac-
From the decrease in the concentration of \(pH_2\) with time we obtain the stationary concentration of H-atoms, according to the formula \(2.3 \lg \frac{u_0}{u_t}=k[\mathrm{H}]\,t\); if \(t\) is expressed in seconds and \([\mathrm{H}]\) in mm Hg, then at \(20^\circ\) \(k=8\). Under the specified experimental conditions the concentration of H-atoms is \(1—5\cdot 10^{-5}\) mm. By the same method, L. Farkas and P. Harteck\(^{33}\) studied the concentration of H-atoms liberated in the photochemical decomposition of \(\mathrm{NH_3}\), since this makes it possible to obtain important data on the kinetics of this reaction. The determination of the concentration of H-atoms here is possible because the admixture of \(pH_2\) has no effect on the quantum yield and kinetics of the decomposition of \(\mathrm{NH_3}\). The most important consequences obtained in this way are, first, the establishment of the fact that the photochemical reaction proceeds according to the scheme
\[ \mathrm{NH_3}+h\nu\to \mathrm{NH_2}+\mathrm{H}, \]
and, secondly, that the H-atoms liberated in this process are destroyed not in a triple collision, but in a bimolecular reaction proceeding with a large collision yield. The simplest reaction of this kind is
\[ \mathrm{NH_2}+\mathrm{H}\to \mathrm{NH}+\mathrm{H_2}. \]
The concentration of H-atoms obtained in the reaction under consideration is fairly considerable and, with an absorbed amount of light of \(3\cdot 10^{14}\) quanta/\(\mathrm{cm^3}\) sec., reaches \(5\cdot 10^{-5}\) mm Hg.
From the temperature dependence of the conversion of \(pH_2\) in the presence of \(\mathrm{NH_3}\), an activation heat of 7000 cal is obtained, which is in agreement with the results of Geib and Harteck\(^{35}\), as well as A. Farkas\(^{27}\). This is an indication that the reactions in which H-atoms are destroyed proceed essentially without heat of activation, which also agrees with the temperature dependence of the decomposition of \(\mathrm{NH_3}\).
2. Energy exchange at the boundary surface metal—\(\mathrm{H_2}\).
With regard to energy exchange at the boundary surface metal—\(\mathrm{H_2}\), the application of \(pH_2\) also gives important results. In the investigation of the catalysis of \(pH_2\) by metal wires heated by an electric current, it turns out that the heat consumption in active wires above a certain temperature is considerably greater than in inactive ones (Bonhoeffer and A. Farkas\(^{8}\)). Fig. 25 shows the behavior of various platinum wires [the results for tungsten are very close to this (A. Farkas\(^{28}\), Bonhoeffer and A. Farkas\(^{9}\)), which was also established by Langmuir and Blodgett\(^{2}\)]. Comparison
of the reaction \(\mathrm{H}+\mathrm{Cl_2}=\mathrm{HCl}+\mathrm{Cl}\) is approximately \(10^{-2}\) (Bodenstein, Trans. Farad. Soc. 123, 413, 1913), then in the chlorine—hydrogen reaction the liberated H-atoms normally do not cause a noticeable conversion of \(pH_2\). In the work of F. Haber and F. Oppenheimer\(^{42}\) this method was applied to determine the content of H-atoms in a stream of active hydrogen.
with the temperature dependence of the rate of transition of \(p\mathrm{H}_2\) into \(n\mathrm{H}_2\), shows that this effect—a difference in heat exchange—appears when the number of molecules that have transformed is comparable with the initial number of molecules. Since the transition of \(p\mathrm{H}_2\) at this temperature depends on activated adsorption, and in this case, evidently, complete energy exchange (accommodation) occurs, the smaller heat release by inactive wires is explained by the absence of this process.
At temperatures below \(300^\circ\mathrm{C}\), heat exchange occurs mainly by way of reflections. In this temperature region the heat release on active and inactive wires differs very little, although the accommodation coefficient* of the reflection process for active and inactive wires, naturally, need not be exactly the same. A comparison of the accommodation coefficients of \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\) at low temperatures provides further possibilities for studying the energy exchange under consideration. According to the experiments of Rowley and Bonhoeffer\({}^{63}\), \(p\mathrm{H}_2\) has a coeffi-
Fig. 25. Dependence of heat release on temperature for various wires.
* The accommodation coefficient \(a\), introduced by Knudsen (Ann. Phys. 34, 593, 1931), is given by the expression
\[ a=\frac{T_g-T_g'}{T_g-T_f}, \]
where \(T_g\), \(T_f\), and \(T_g'\) are the temperatures of the incident molecule, the solid substrate, and the molecule, respectively, delayed at the surface. It can be calculated from the heat release \((W)\) of a wire heated by an electric current; thus, for example, for small pressures we have
\[ a=\frac{W}{nc\Delta T}, \]
where \(n\) is the number of moles of gas arriving each second at the surface of the wire, \(c\) is the molecular heat capacity, and \(\Delta T\) is the temperature difference between the wire and the vessel wall.
cient of accommodation is approximately 10% greater than that of \(n\mathrm{H}_2\) (Fig. 26). This relation is explained by the difference in the accommodation coefficients of translational and rotational energy. When the ratio of rotational and translational energy is considered in the overall balance of the energy carried by \(p\mathrm{H}_2\) and \(o\mathrm{H}_2\), the separate accommodation coefficients give:
Fig. 26. Accommodation coefficient of \(\mathrm{H}_2\) with different contents of \(p\mathrm{H}_2\).
\[ 140^\circ\mathrm{K}\quad a_{\mathrm{transl}}=0.43 \qquad a_{\mathrm{rot}}=0.26 \]
\[ 170^\circ\mathrm{K}\quad a_{\mathrm{transl}}=0.44 \qquad a_{\mathrm{rot}}=0.18 \]
These results indicate that, in this temperature region, accommodation is caused mainly by reflection processes, since with a single adsorption both forms of energy would probably be exchanged with equal probability.
3. Determination of the self-diffusion coefficients of \(\mathrm{H}_2\)
The self-diffusion coefficient of \(\mathrm{H}_2\), which is usually calculated indirectly from the diffusion constants of three diffusing pairs, according to P. Harteck and W. Schmidt\(^{45}\), can be determined exactly by studying the diffusion of \(p\mathrm{H}_2\) in \(n\mathrm{H}_2\). This is important because it then becomes possible to determine, on the basis of the kinetic expression \(D=f\frac{\eta}{\rho}\) (where \(\eta\) is the viscosity coefficient and \(\rho\) the density), the numerical factor \(f\), which depends on the laws of action of the forces between molecules (theoretically, for a force law \(kr^{-s}\), for elastic spheres at \(s=\infty\), \(f=1.2\); at \(s=5\), \(f=1.55\)).
TABLE 24
Diffusion constant of H at various temperatures
| Method | \(T\) | \(D\) | \(f\) |
|---|---|---|---|
| Indirect . . . . . . . | 273 | 1.26—1.34 | 1.34—1.43 |
| Tube . . . . . . . | 273 | \(1.28_5 \pm 0.0025\) | 1.37 |
| Reverse diffusion . . . | 273 | 1.28 (calibration) | 1.37 |
| Reverse diffusion . . . | 85 | 0.172 | 1.32 |
| Reverse diffusion . . . | 20.4 | 0.00816 | 1.28 |
The self-diffusion coefficient was determined by two methods. In the first, at \(20^\circ\mathrm{C}\), two precisely identical tubes separated by stopcocks were filled with \(p\mathrm{H}_2\) and \(n\mathrm{H}_2\) and connected with each other by opening the stopcock. Then, after a definite time, concentration measurements were made. In the second case, was used
PARA- AND ORTHOHYDROGEN
the reduction method of Hertz*, consisting in the fact that another gas was mixed into the stream of some gas, and the changes in concentration caused by this admixture were measured at some definite place in the stream. This method, because of the inexact specification of the boundary conditions, does not give such accurate results as the first, but it does permit investigation over a wide temperature interval. Table 24 gives the results obtained in this way; for \(T = 273^\circ\) and \(f = 1.37\), \(s = 15\).
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