On the Influence of Electric and Magnetic Fields on the Properties of Gases

A. A. Zaitsev, Moscow

It is known that the study of the thermal conductivity of a gas is a sensitive means of detecting the slightest changes in its state. In a number of cases it is of interest to know whether the thermal conductivity of a gas depends on those changes in its state that occur when the gas is subjected to the action of an external electric or magnetic field. At present, thanks to the work of a number of authors, above all Senftleben,^1 it may be regarded as firmly established that the flow of heat through a gas from a heated body to a cold one depends strongly on the action of an electric or magnetic field upon the gas. However, as experiments show, the actions of the electric and magnetic fields are of different nature. The magnetic field affects the thermal conductivity of gases by reducing its value. Up to now the experiments have been carried out mainly with the gases O$_2$ and NO, because here the effect reaches a considerable magnitude. The electric field, on the contrary, causes an increase in molecular heat transfer, but its action is associated with the possibility of the formation of convection currents in the gas and, in their absence, is not observed. Senftleben and Pitzner^2 also succeeded in showing, on the basis of experiments, that the relative change in thermal conductivity within the measured limits is a function of the argument $\dfrac{H\sqrt{T}}{P}$. If the pressure $P$ and the gas temperature $T$ are expressed through molecular constants, the argument takes the form $\dfrac{H \cdot \bar{l}\delta^2}{\bar{c}}$. Here $\bar{c}$ is the mean velocity, $\bar{l}$ the mean free path, $\delta$ the molecular diameter, and $H$ the magnetic-field strength. Such a form of the argument may be interpreted by the assumption that the action of the magnetic field consists in a change of the mean free path and of the effective cross section of the molecule. If this is so, then on the basis of the classical kinetic theory one may expect the magnetic field to influence also the internal friction of the gas. Experiments by Engelhardt and Sack,^3 intended to test this assumption, showed that such an influence does indeed exist. It is true that experiments of this kind have been carried out only with O$_2$ and NO, but even these few experiments

make it possible to be certain that the influence of a magnetic field on internal friction, at least in paramagnetic gases, is present. The regularity obeyed by the dependence of the electric effect on external conditions and, in particular, this dependence on the temperature of the gas, expressed by an exponential function, leads one to think that at the basis of this phenomenon lies a definite type of gas reaction, which is intensified when the gas is introduced into an electric field. The purpose of the present article is to give as complete a survey as possible of the works related to the questions touched upon here.

§ 1. Experimental Data

a) The influence of a magnetic field on the thermal conductivity of paramagnetic gases.

In all experiments the relative decrease of the thermal conductivity of the gas in a magnetic field was measured, i.e. the quantity

\[ -\frac{\Delta\lambda}{\lambda} = \frac{\lambda_H-\lambda_0}{\lambda_0}, \]

where \(\lambda_H\) is the thermal conductivity of the gas in the magnetic field, and \(\lambda_0\) is the thermal conductivity in the absence of the field. In what follows we shall, for brevity, call the quantity \(-\Delta\lambda/\lambda\) the effect and shall denote it by \(\varepsilon_1\). As has already been said, the gases \(O_2\) and \(NO\) give a considerable decrease of thermal conductivity in a magnetic field. Therefore accurate quantitative measurements of the effect were carried out precisely with these gases. The experimental technique in this case was very simple. The measurements were made in a tube, along the axis of which a filament was stretched, heated during the experiment by an electric current. A completely homogeneous magnetic field up to 300 gauss was obtained with the aid of a solenoid placed over the measuring tube. Larger fields, up to 12,000 gauss, were obtained with the aid of magnets with pole pieces of hemispherical form.

\[ -\frac{\Delta\lambda}{\lambda}\cdot 10^4 \]

Fig. 1.

On Fig. 1 are given curves expressing the dependence of the effect on the field strength \(H\) at a gas temperature of \(293^\circ\). Here along the ordinate axis is plotted the quantity \(\varepsilon_1\cdot 10^4\), and along the abscissa axis the field strength in gauss. From the figure it is seen that at small fields the effect increases with the field more strongly; then, as the field strength increases, this growth...

decreases. The upper curve, taken at a pressure \(P=40\) mm Hg, becomes, at a field strength of 11,000 gauss, a straight line parallel to the abscissa axis. This shows that at this value of the field strength the effect reaches its greatest value and, with further increase of the field, does not change its magnitude—saturation sets in. The remaining curves, corresponding to higher gas pressures, at a field strength of 11,000 gauss continue to show an increase of the effect, and from the course of these curves it may be concluded that at sufficiently high fields they will merge with the upper curve and likewise, as does the latter, will run parallel to the abscissa axis. All the curves presented have a common form. The gas pressure affects only the absolute magnitude of the effect: with increasing pressure, at the same field strength and gas temperature, the effect decreases. It follows from this that at the same field strength and constant gas temperature, with decreasing pressure the magnitude of the effect must grow and, at sufficiently small pressure, reach its greatest value. With further decrease of the pressure the effect will not increase. It retains its constant value corresponding to saturation. A more detailed examination of the experimental data makes it possible to express quantitatively the dependence of the effect on the gas pressure and the field strength and to give an empirical formula. It turns out that, in small fields, within a certain pressure interval, the effect grows directly proportionally to the square of the field strength and inversely proportionally to the square of the pressure, i.e.

\[ -\frac{\Delta \lambda}{\lambda} \cong \mathrm{const}\,\frac{H^{2}}{P^{2}} . \]

In the region of validity of this law, the effect is thus a function of the argument \(\frac{H}{P}\). The dependence of the effect on \(\frac{H}{P}\) can be checked in the following way. At a pressure \(P\) equal, for example, to 40 mm, and at a field strength of 400 gauss, the ratio \(\frac{H}{P}=10\), and, as experiments show, the corresponding value \(\varepsilon_{1}\cdot 10^{4}\) will be equal to 3. If the effect is indeed a function of the argument \(\frac{H}{P}\), then for \(P=60\) mm and \(H=600\) gauss, \(P=70\) mm and \(H=700\) gauss, the effect must have the very same magnitude as in the first example. From Table 1, where the values of \(\varepsilon_{1}\cdot 10^{4}\) are given for arguments 1, 3, 5, 10, 15, 20, 50, it is seen that this requirement of the law for small arguments is, within the limits of measurement error, generally fulfilled.

At large arguments this regularity is observed up to a pressure of 300–400 mm, above which deviations begin: the effect has a value smaller than would follow from the regularity. Although these deviations are small, they are completely systematic. They cannot be explained by experimental errors.

At the same time it was established that temperature also affects the effect, but to a much smaller degree than pressure and field strength. The effect increases, at constant pressure and field strength, with increasing gas temperature. Upon reaching a certain tem-

perature, different for different \(P\) and \(H\), saturation occurs. In this sense there is a complete analogy between the dependence of the effect on \(P\) and \(H\), on the one hand, and the dependence of the effect on the temperature—

TABLE 1

of the gas \(T\), on the other. But since the change in the thermal conductivity of a gas in a magnetic field depends, in addition to the pressure \(P\) and the field strength \(H\), also on the temperature \(T\), the argument of the function expressing the connection of the effect with the gas pressure and the field and having, at least in the first approximation, the form \(\dfrac{H}{P}\), must be supplemented by introducing the temperature into it. Experimental data show that such an argument should be the expression \(\dfrac{H\sqrt{T}}{P}\). The applicability of this conclusion, however, is limited. At very low temperatures one must take the exponent of \(T\) to be greater than \(1/2\), and at high temperatures, on the contrary, less than \(1/2\).

Along with the investigations of the paramagnetic gases \(\mathrm{O}_2\) and \(\mathrm{NO}\), experiments were carried out with water vapor and with the gases \(\mathrm{H}_2\), \(\mathrm{He}\), \(\mathrm{Ar}\), \(\mathrm{Ne}\), in order to study the influence of the magnetic field on their thermal conductivity. It turns out that these gases either give no effect at all, or give one of such insignificant magnitude that it is still difficult to decide whether this effect belongs to the gases under investigation themselves or to \(\mathrm{O}_2\), with which the gases may have been contaminated.

At present there exists no theoretical conception that would make it possible to calculate the magnitude of the effect in each case. Therefore it was desirable to find as simple a function as possible giving the dependence of the effect on the field strength, the gas pressure, and its temperature. Such a function may be

\[ -\frac{\Delta\lambda}{\lambda}=\varepsilon_1=\frac{ax^2}{1+bx+cx^2}, \]

where \(x=\dfrac{H}{P}\sqrt{T}\). The temperature interval in which this applied—

function is extended if, instead of \(T^{1/2}\), one puts \(T^n\), where \(n\) varies from \(1/2\) to 1. Of course, the validity of applying this function goes in parallel with the fulfillment of the condition that, at constant temperature, the effect is a function of the argument \(\dfrac{H}{P}\). For small values of \(\dfrac{H}{P}\) the formula reproduces the experimental data with great accuracy, while for large values deviations of the formula from the experiments begin. This shows that the empirical formula given is only a first approximation to the truth.

b) The influence of a magnetic field on the internal friction of paramagnetic gases.

The experiments were carried out with Wheatstone’s aerodynamic bridge. In this latter, the unknown resistance in the ordinary Wheatstone bridge is replaced by the resistance of a capillary, which it offers to the gas under investigation passing through it under a pressure difference \(\Delta P\). Engelhardt and Zak observed that the quantity of gas flowing through a capillary placed in an electric field is greater than outside the field. But since this quantity is in inverse dependence on the coefficient of internal friction of the gas \(\eta\), the increase in \(\mu\) is expressed by a decrease in the coefficient \(\eta\). So far the experiments have been carried out only with the gases NO and \(O_2\). Comparing the results of Engelhardt and Zak’s experiments with the data of his own experiments on the study of the influence of the field on thermal conductivity, Senftleben\(^4\) shows that, at equal pressures and field strength, the relative change in thermal conductivity in a magnetic field is greater than the relative change in internal friction. In Table 2 are given the values \(\varepsilon_1=\dfrac{\Delta\lambda}{\lambda}\) and \(\varepsilon_2=\dfrac{\Delta\eta}{\eta}\) at different fields and, in addition, in the last column are given the values of the ratio \(\varepsilon_1:\varepsilon_2\).

TABLE 2

It is clear from the table that, as the field increases, the ratio \(\varepsilon_1:\varepsilon_2\) decreases.

A further comparison shows that the curve \(\varepsilon_1=f(H)\), taken at pressure \(P_1\), can be brought into coincidence with the curve \(\varepsilon_2=f(H)\), if for this latter one selects a corresponding pressure \(P_2\) smaller than \(P_1\). In Fig. 2 the solid curve expresses

\(\varepsilon_2=f(H)\) at pressure \(P_2=110\) mm Hg, and the separate points lying on this curve correspond to \(\varepsilon_1=f(H)\) at \(P_1=360\) mm Hg. The same coincidence of the curves will be obtained if, for the \(\varepsilon_2\) curve, we choose the pressure \(P_2^1=64\) mm, and for \(\varepsilon_1\), \(P_1^1=278\) mm Hg. It is easy to see that

\[ \frac{P_2}{P_2^1}=1.72,\qquad \frac{P_1}{P_1^1}=1.29. \]

But \(1.29\) is approximately \(\sqrt{1.79}\). Therefore we may write

\[ P_1=\operatorname{const}\sqrt{P_2}. \]

Since

\[ \varepsilon_1=f\!\left(\frac{H^2}{P^2}\right) \]

and the dependence of \(\varepsilon_1\) and \(\varepsilon_2\) on the field strength is the same, then

\[ \varepsilon_2=f\!\left(\frac{H^2}{P}\right). \]

If, at equal \(P\), \(T\), and \(H\), the relative change in the thermal conductivity \(\varepsilon_1\) is greater than the change in the internal friction \(\varepsilon_2\), then

Fig. 2.

Fig. 3.

the ratio \(\lambda:\eta\), equal without a field to \(K C_v\), must change in a magnetic field.

c) Influence of an electric field on heat transfer in gases.

The measuring vessel was a cylindrical metal tube. Between the filament passing along the axis of the cylinder and the cylinder a potential difference was applied, so that in the tube, near the filament, there arose an inhomogeneous but strong radial field. When there is the possibility in the gas for the occurrence of convective currents, with the same amount of electrical energy supplied to the filament, the temperature of the filament in an electric field proves, as shown by the experiments of Senftleben\(^5\), to be lower than without a field. Thus, in an electric field there occurs an increase of heat transfer from the filament to the wall, and owing to this additional cooling of the filament occurs. Here also the relative change of heat transfer in the electric field was measured. The measurements show that

the effect increases proportionally to the square of the pressure at constant field \(E\) and temperature \(T\), and proportionally to the square of the field strength at constant \(P\) and \(T\). At constant \(P\) and \(E\) the dependence of the effect on temperature can be expressed by an exponential function. In Fig. 3 the curve shows the course of the change in the calculated values of the effect under the assumption that the effect increases proportionally to the square of the pressure, while the individual points lying exactly on the obtained curve depict the measured value of the effect as a function of pressure. The measurements were made with \(\mathrm{CO_2}\).

§ 2. Theoretical considerations

As was said at the beginning, the dependence of the influence of an electric field on the heat transfer of a gas on external conditions indicates that this phenomenon is connected with a definite type of gas reaction. The very same regularity that this dependence follows is exhibited by the chemical reaction of the form \(A + A = A_2\). In a pure gas this reaction corresponds to the formation of a complex molecule from simpler ones. The increase of the effect in proportion to the square of the pressure, and the strong decrease of its value with increasing temperature, which can be represented by an exponential curve, become intelligible if it is assumed that the action of the electric field on the gas is based on a reaction of the indicated type. Further evidence in favor of such a conception is given by the results of investigations of the influence on the effect of admixtures of foreign gases. If a gas showing an especially strong effect, for example acetone, is mixed with air, in which the effect is incomparably smaller, then the effect in acetone does not change. This is as it should be if the formation of double molecules takes place in acetone. Zenftleben\(^6\) at first recognized these data as sufficient for interpreting the effect on the basis of the hypothesis that in an electric field an association of gas molecules takes place and that the heat of association determines the observed effect. If this is so, if in an electric field an association of molecules occurs, then on the basis of the law of mass action one may write

\[ \frac{p}{(P-p)^\nu}=\mathrm{const}, \tag{1} \]

where \(p\) is the partial pressure of the \(\nu\)-fold molecule and \(P\) is the total pressure. For weak reactions \(\nu\) may be taken equal to two. Since then \(p\) is very small in comparison with \(P\), the first relation may be rewritten in the form

\[ \frac{p}{P^2}=K . \]

If the equilibrium constant \(K\) is expressed in terms of the heat of reaction \(Q\), then for the number of double molecules \(n\) in a unit volume we obtain

\[ n=\mathrm{const}\,\frac{P^2}{T}\,e^{\frac{Q}{kT}} \quad (k\text{ is Boltzmann’s constant}). \]

In gases with ready-made dipoles, the heat of reaction, at least in part, reduces to the interaction of dipoles. Therefore

\[ Q=Q_0+Q_1, \]

where \(Q_1\) is determined by the dipole moment \(\mu\). In addition, in each gas, under the influence of an external electric field, induced dipoles arise, which also have a noticeable influence on the heat of reaction \(Q\). If the part of the heat of reaction \(Q_2\) due to these induced dipoles is taken to be proportional to the square of the field strength \(E\), then we obtain

\[ Q=Q_0+Q_1(\mu)+aE^2. \]

A force acts on the dipolar molecules of a gas in the direction of the electric field. Therefore the number of collisions of molecules in which the electric moments are parallel increases. If all this is taken into account and the action of the field is taken to be proportional to the square of the field strength, then for the increase in the number of double molecules \(\Delta n\) we obtain

\[ \Delta n=\mathrm{const}\,\frac{P^2}{T}\,E^2\,e^{\frac{Q_1}{kT}} \quad \text{(for gases without dipoles)} \]

\[ \Delta n=\mathrm{const}\,\frac{P^2}{T}\,b\,E^2\,e^{\frac{Q_1}{kT}} \quad \text{(for gases with rigid dipoles).} \]

Here it is assumed that the field strength increases from \(O\) to \(E\). The constant \(b\) depends on \(\mu\) and must be different for different gases. The heat released in the formation of double molecules will be equal to \(\Delta n Q\). Thus, for the quantity of heat liberated in a unit volume of a gas with a rigid dipole in an electric field, we obtain

\[ q=\mathrm{const}\,E^2\cdot\frac{P^2}{T}\,Q_1 e^{\frac{Q_1}{kT}}. \tag{2} \]

Exactly the same dependence of the released heat \(q\) on the field, pressure, and temperature of the gas is given, as Debye\(^7\) showed, by the thermodynamic calculation of the electrocaloric effect, which arises every time a dielectric is introduced into an electric field, the dielectric constant of which will depend on temperature. On the basis of thermodynamic considerations one obtains the relation

\[ q=T\left(\frac{\partial \chi}{\partial T}\right)\cdot\frac{E^2}{2}, \]

where \(\chi\) denotes the dielectric constant, and \(q\) is the quantity of heat that arises in a unit volume when the field increases from \(0\) to \(E\). If one uses the relation that exists between the dielectric constant and the polarization of molecules, and for the latter uses

…give an expression which, indeed, for the case of molecules with rigid dipoles, gives for \(q\) the relation

\[ \frac{\pi}{54}\cdot \frac{\delta^{12}}{\mu^{4}K} E^{2}\cdot \frac{P^{2}}{T}\cdot \frac{2\mu^{2}}{\delta^{3}} e^{\frac{2\mu^{2}}{\delta^{3}kT}}, \]

where \(\delta\) is the diameter of the molecule. Introducing the abbreviation

\[ \frac{\pi}{54}\cdot \frac{\delta^{12}}{\mu^{4}K}=\mathrm{const} \quad\text{and}\quad \frac{2\mu^{2}}{\delta^{3}}=Q_{1}, \]

we finally have

\[ q=\mathrm{const}\cdot E^{2}\cdot \frac{P^{2}}{T} Q_{1}\cdot e^{\frac{Q_{1}}{kT}}, \]

i.e., we thus obtain the same expression as in 2.

This shows that the assumption of the formation in the gas of double molecules under the action of the electric field, at least for gases with rigid dipoles, may be recognized as correct. Thus, the process by which the effect arises may be represented in the following way. In a gas placed in an electric field, double molecules are formed. Owing to the presence of convection, these associated molecules move in large numbers toward the heated filament and, coming close to it, dissociate. The energy of dissociation is taken from the filament, which thereby undergoes additional cooling. If this is so, then the dependence of the cooling effect of the filament on the pressure \(P\), temperature \(T\), and field strength \(E\) must obey the same law as that given for the quantity \(q\) by formula (2). Below is given Table 3, from which it is easy to see that the value of the effect calculated from the formula agrees fairly well with the experimental data.

TABLE 3

The calculated values correspond to a quadratic dependence of the effect, according to formula (2). As regards the influence of temperature on the effect, here likewise the result of measurement falls within the limits of the formula. From measurements of the electric effect at dif-

temperatures, one can compute from formula (2) the heat of association, or, what is the same thing, the heat of dissociation. If the results obtained in this way agree with measurements made by other methods, this will also testify that the considerations set forth here are correct. To carry out such a comparison, a series of measurements with NO₂ was made. Generally speaking, formula (2) is not applicable to NO₂. At room temperature this gas contains a considerable percentage of double molecules, but in deriving the formula for \(q\) it was assumed that the degree of association is so small that the quantity \(p\) may be neglected in comparison with \(P\). It turned out that the heat of association \(Q\) is the smaller, the greater the gas pressure. When the pressure is decreased from 670 to 260 mm, the value of \(Q\) increases from 8,300 to 11,100 cal. Thermal measurements give for the same gas a value of \(Q\) equal to 12,700. With decreasing pressure the degree of dissociation of the molecules decreases. Therefore, applying formula (2) to NO₂ at sufficiently small pressure values, we make only a small error. But at low pressures the quantity \(Q\), calculated from measurements of the electrical effect, tends to 12,700 cal, which is obtained from thermal measurements. Such was Zenftleben’s first attempt to interpret theoretically the results of the first experimental investigations on the influence of an electric field on heat transfer in gases. However, the most recent and most refined experiments in the same direction show that the field effect under certain circumstances can become so large that heat transfer is thereby doubled. Such a phenomenon cannot be based solely on the mere formation of double molecules. Therefore Zenftleben, in his second work devoted to the same questions of the theoretical interpretation of the measured effect,³ puts forward a second hypothesis, on the basis of which, in his opinion, one could construct a theoretical interpretation of the phenomenon. He says that the occurrence of the effect is apparently determined by the presence of double molecules in the gas even before the electric field is switched on. Nernst⁹ had already shown that in gases which contain double molecules, molecular heat transfer has a greater magnitude than follows from the ordinary thermal conductivity of the gas: to the ordinary thermal conductivity is added the transport of dissociation energy. But without a field, dissociation occurs to such a small extent that the change in heat transfer caused by it is very insignificant. The increase of the effect in an electric field can be explained by the fact that it causes an increase in the number of double molecules reaching the filament under the action of the forces of the nonuniform field, where dissociation occurs. Without a field, the only factor causing the motion of gas particles toward the filament is diffusion.

§ 3. Conclusion

From what has been set forth it is clear that, under certain circumstances, the influence of magnetic and electric fields on the properties of certain gases

is indeed observed. It is also evident that, in view of the absence of a rigorous theory explaining the regularities observed in the experiments, at present one must, in the necessary cases, make use of empirical formulas. That the nature of the influence of an electric field on the properties of gases differs from the influence of a magnetic field follows clearly from the difference in the regularities obeyed by the changes of the effects under external conditions. For a more complete clarification of the question of the influence of external fields on the properties of gases, it is necessary to carry out experiments with as large a number of gases as possible. For constructing a theory of the question, knowledge of the dependence of the effect on the angle between the direction of the heat flux and the direction of the field should be of considerable importance. Senftleben points out that the absolute magnitude of the effect caused by the action of a magnetic field on the thermal conductivity of gases depends on this angle.

Literature

1. H. Senftleben, Physik. Z. 31, 961; 1930; 32, 550, 1931; Z. physik 74, 757, 1932.
2. H. Senftleben u. Pietzner, Ann. d. Phys., 16, 907, 1933.
3. H. Engelhardt u. H. Sack, Physik. Z. 33, 724, 1932.
4. H. Senftleben, Physik. Z. 34, 141, 1932.
5. H. Senftleben, Physik. Z. 34, 230, 1933.
6. H. Senftleben, Physik. Z. 33, 947, 1932.
7. P. Debye, Hadbuch d. Radiologie, VI, 637, 1925.
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9. W. Nernst, Ann. d. Phys. 1904: Boltzmann-Festschrift, S. 907.