Evaporation, condensation, and adsorption. Langmuir extends his well-known adsorption formula to the case when forces act between neighboring adsorbed particles (“adatoms”). In the old simple theory, the rate of evaporation of adatoms from a surface was taken to be equal to \(\nu_1\theta\), where \(\theta\) denotes the covered part of the surface, and \(\nu_1\) is a constant. If atoms condense only on the bare parts of the surface, then the rate of condensation is equal to \(\alpha_0\mu(1-\theta)\), where \(\alpha_0\) is a constant characterizing the capacity of the gas to condense on a bare surface, and \(\mu\) is the rate at which atoms strike unit area of the surface, with

\[ \mu=(2\pi m kT)^{-1/2}p=2.655\cdot 10^{19}p(MT)^{-1/2}, \tag{1} \]

where \(p\) is the pressure in bars, \(k\) is Boltzmann’s constant, \(m\) is the mass of the atom, and \(M\) is the molecular weight of the gas. In the stationary state the rates of evaporation and condensation are equal, whence

\[ \theta=\alpha_0\mu(\nu_1+\alpha_0\mu)=\alpha_0\tau\mu(\sigma_1+\alpha_0\tau\mu), \tag{2} \]

where \(\tau=\sigma_1/\nu_1\) is the mean life of an adatom, and \(\sigma_1\) is the number of adatoms falling on a unit of saturated surface \((\theta=1)\). Equation (2), with sufficient accuracy, agrees with the adsorption observed in a very large number of cases.

The vapor pressure of a liquid over a wide temperature interval can be represented by an equation analogous to Richardson’s formula for electronic emission,

\[ p=AT^\gamma e^{b/T}. \]

Plotting \(\ln p\) against \(1/T\), we obtain a straight line if \(\gamma=0\). If, however, \(\gamma\) is different from zero, then, generally speaking, a curve is obtained; but, since in practice the temperatures \(T_1,T_2\) increase, the interval of temperatures within which \(p\) can be observed is usually so limited that this curvature remains imperceptible. Therefore, for a limited interval of temperatures situated symmetrically above or below the mean temperature \(T_m\), we can always replace equation (3) by

\[ p=A_0 e^{-b_0/T}. \]

According to Clapeyron’s equation, the latent heat of evaporation \(\lambda\) per atom at constant pressure will be

\[ \lambda=-k\frac{d\ln p}{d(1/T)}=kb_0=k(b+\gamma T). \]

According to Trouton’s rule, the latent heat \(L\), in calories per gram-molecule, is proportional to the boiling temperature \(T_B\), and for most liquids the quantity \(L/T_B\) is usually equal to 20.7. Referring to equation (3a), it is evident that Trouton’s rule follows directly from this equation if \(A_0\) is considered a universal constant.

Hildebrand gave a rule different from Trouton’s rule. According to Hildebrand, for each series of substances it is not \(L/T_B\) that should be constant, but \(\lambda/T_0\), where \(T_0\) is the temperature at which the selected substances have the same vapor concentration. Hildebrand’s rule follows from equations (3) and (3a) if one sets \(\gamma=-1\) and assumes that \(A_0\) is not a universal constant, but that \(A_0\) is proportional to \(T_0\), i.e. that \(A_0=CT_0\), where \(C\) is a universal constant.

Comparison with experimental data for liquids shows that Hildebrand’s rule gives better results than Trouton’s rule.

ton, but still better agreement is given by the rule obtained if one puts \(i=1.5\), when \(\lg A_{1.5}=6.37\). The elasticities of the vapors of solids whose vapors have rigid molecules are also given by this equation with \(i=1.5\) and \(\lg A=6.9\), but much larger values of \(A\) are obtained if the molecules possess internal degrees of freedom. It is therefore assumed that such molecules in the vapor phase may possess high internal mobility (as in liquids), whereas at lower temperatures they may become rigid (as in solids). Such effects probably do not exist for the molecules of vapors and liquids.

The particular case of dividing the latent heat of fusion by the melting temperature has high values for large molecules such as stearic acid, showing an increase approximately proportional to the number of atoms in the molecule; in such cases a large part of the heat of fusion is the internal heat of fusion of the molecules themselves, which in the solid state are fixed in the lattice in an unchanged form, so that the molecule itself is solid, but when the solid melts the molecule also melts.

If one considers the evaporation of adsorbed atoms from a film containing \(\sigma\) atoms per unit surface, and assumes that the value \(\tau\)—the mean life—will be the same for all adatoms, then the rate of evaporation (atoms \(\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}\)) will be:

\[ \gamma=\sigma/\tau=\sigma_1\theta/\tau . \]

From equation (3), written in the form:

\[ p=A_{1,5}T e^{-b/T}, \quad \text{where } b=b_0-\frac{3}{2}T=\frac{\lambda}{k}-\frac{3}{2}T, \]

and from (1), where \(\mu=\nu_1=\sigma_1 k\), we find:

\[ \tau=(2\pi mk)^{\frac{1}{2}}(A_{1,5}T)^{-1}\sigma_1 e^{-b/T}, \]

where \(A_{1,5}=8\cdot 10^{5}\). Hence

\[ \gamma=A_{1,5}(2\pi mk)^{-\frac{1}{2}}\theta T e^{-b/T}. \]

This equation for the rate of evaporation of atoms or molecules from monomolecular films shows good agreement with the experimental values for films of thorium, oxygen, and cesium on the surface of tungsten. In equation (4) the forces of interaction between adatoms are included in the calculation through the quantity \(b\), and since \(b\), generally speaking, is a function of \(\theta\), the quantity \(\gamma\) will not be proportional to \(\theta\), except in cases of such small values of \(\theta\) that \(b\) is close to the limiting value at \(\theta=0\).

Although the conditions under which adsorbed films more than one molecule thick may form are rather unusual, they are also considered in the article. Adsorbed molecules on flat homogeneous solids are, generally speaking, subjected to the action of large forces originating from the solids. Therefore the adsorbed molecules become polarized and repel one another as dipoles with forces proportional to \(M^{2}r^{-4}\), where \(M\) is the dipole moment and \(r\) is the distance; the attractive forces prevail only when two differently polarized molecules are present next to each other, as, for example, cesium and oxygen on tungsten or salts on metals such as mercuric sulfate on mercury. However, in some cases the forces between the solid and the adatom are small, as, for example,

for example, when hydrogen molecules or helium atoms lose chemically saturated surfaces of tungsten covered with adsorbed oxygen. In such cases the mean life of the adatom is so small that it does not even attain thermal equilibrium with the solid body, so that the accommodation coefficient is much less than unity (0.1–0.2).

The equation of state of a two-dimensional gas constituting an adsorbed film can be found by the virial method for molecules that repel one another as dipoles:

\[ FA = RT + \frac{1}{2}\sum (rf). \]

The two-dimensional van der Waals equation, where the forces acting at long range are now regarded as repulsive, takes the form:

\[ (F-a/A^2)(A-A_1)=RT, \]

where \(A\) denotes the area containing one gram-atom, and \(A_1\), in the usual derivation, considering only first-order effects, turns out to be equal to only half the area actually covered by one gram-atom of adsorbed atoms. Experimental data for massive films at high surface concentrations show that \(A_1\) corresponds to a close-packed film, in which molecules cover the surface completely; this is also confirmed by a new theoretical derivation for the case of a high concentration of adatoms.

The choice of dipolar repulsive forces for the expression of the virial is justified by the results of experiments, namely, by the fact that adsorption of alkali-metal atoms occurs only when the electron affinity of the adsorbing metal exceeds the ionization potential of the alkali metal. The positive charge on the adatoms causes a change in the contact potential (by about 3 V) and a corresponding increase in electron emission. It is further shown that the moments \(M\) can be calculated as functions of \(\theta\). In the case of cesium on tungsten these moments have values from 16 debye for \(\theta=0\) to 6 debye for \(\theta=0.09\). A calculation is then made of the quantities of emission of electrons and of positive ions, and it is shown that they agree with experiment. In the first case the influence of the electron spin was taken into account and the Richardson equation for electron emission from pure tungsten was somewhat modified.

The results of experiments on the evaporation of cesium films prove to be very different from those that follow from the old formula (2), based on the assumption that no repulsion exists between adatoms; however, the general results are in agreement with the new equations.

Next, the influence of the inhomogeneity of the adsorbing surface, first predicted by Langmuir and then experimentally and theoretically studied by Taylor (H. S. Taylor) and others, is considered. The importance of the so-called “active areas” in determining the catalytic properties of surfaces, even flat ones, is well known. It has been possible, however, to show that the calculations lead to the result that the surface of tungsten is in general homogeneous, although they also show that on 0.5% of the surface cesium atoms are bound more strongly than on the remaining part. Probably the active spots consist of isolated elementary surfaces, each of which is capable of holding one adatom. [I. Langmuir, J. Amer. Chem. Soc., 54, 2798, July 1932, according to an abstract from Nature (No. 3283).]