GUEST AUTHOR Dewey K. Carpenter
Georgia Institute of Technology Atlanta
Textbook Errors, 69
Kinetic Theory, Temperature, and Equilibrium
A
number of introductory books give a misleading impression about the status of temperature in physical chemistry.' Two cases where such an impression is given are (1) statements of the relationship of the postulates of the kinetic theory of gases to the concept of temperature, and (2) discussions of the molecular basis for the general validity of a proportionality between the mean molecular kinetic energy and the absolute temperature in systems a t equilibrium. Consider first the postulates of the kinetic theory of gases. A typical statement of these, as found in proper expositions (1,Z) of the theory is as follows: A gas consists of a collection of molecules which are in motion. The spatial locations and the velocities of these molecules are random; (the state of the gas is one of "molecular chaos"). Collisions of molecules with each other and with the container are elastic. The observable gas pressure is a time average of the rate of change of momentum of a unit area of the container due to impacts made by the gas molecules. The laws of classical mechanics may be used to cslculate the impulse given to the oontainer by a molecular impact.
A further assumption made in the interests of m a t h e matical simplicity is that: Intermolecular forces between the molecules bave a negligible effect on the properties of the gas (e.g., on the pressure).
These postulates are entirely mechanical in nature, i.e., the only variables required to give mathematical formulation to this model of a gas are those of mechanics: the mass of a molecule m, the number of molecules N, the volume of the container V , and the velocity of a molecule u. No reference is made in these postulates to the propManuscripts for this column, or suggestions of material suitable for it, are rarely solicited. These should be sent with as many details as possible, and particularly with references to modern textbooks, to W. H. Eherhardt, Department of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. 1 Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the source of errors discussed will not be cited. In order to be presented, an error must occur in a t least two independent recent standard books.
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erty of temperature. This is because the kinetic theory is a purely mechanical theory, whereas the concept of temperature belongs to the discipline of thermodynamics. It is a function introduced to account for the existence of the phenomenon of thermal equilibrium ( S , 4 ) . Thus, if a system A is brought into contact with a system C by means of a conducting wall, the proper: ties of each system will, in general, change until a state of equilibrium ensues. The same phenomenon occurs if a different system B is allowed to equilibrate in this manner with system C. It is a fact of experience (the "zeroth" law of thermodynamics) that if systems A and B are found to be in thermal equilibrium with the same system C, then they are also in thermal equilibrium with each other. Thus, it is appropriate to define a function of thevariables of a system (such as the pressure and the volume per unit mass) which has the property that systems in thermal equilibrium have the same values of this function. It is this function that is called the empirical temperature. One possible choice of a scale of temperature is based on the properties of ideal gases. From these remarks it is clear that temperature is a concept introduced in order to give quantitative expression to the experienced fact of thermal equilibrium between macroscopic systems. It is a function which necessarily remains abstract until some molecular significance can be assigned to it. This is of course the case with all of the functions defined in macroscopic thermodynamics (e.g., the entropy and free energy functions). Thus, the temperature is a property which may be defined in terms of mechanical quantities (as in the ideal gas equation) but it is, so to speak, a foreign element in mechanics. Let us now return to a consideration of the kinetic theory of gases. This theory is a statistical treatment of the model which conforms to the above postulates and is therefore a mechanical theory. Temperature finds no place in the kinetic theory. Several introductory texts nevertheless maintain that as an essential postulate of the kinetic theory one assumes that: The mean molecular kinetic energy of a molecule is proportional to the absolute temperature.
The truth is that this statement is valid as a caclusion, but not as a postulate. To see this we consider the result of the derivation, given in standard works (5-7), of the kinetic theory expression for the pressure corresponding to the six postulates given above: Here 2 is the mean squared velocity of the molecules. The right hand member can be written (2/z)(N/V) (m3/2) to emphasize the occurrence of the mean molecular kinetic energy, m2/2. I n order to demonstrate some connection between this quantity and the absolute temperature i t is necessary to identify the purely mechanical expression for the pressure given in eqn. (1) with an expression for the pressure in the form of an equation of state, which relates the pressure to the temperature. The proper equation of state for this purpose is the ideal gas law, equ. (2), P = (N/N.)(RT/V)
(2)
since the experimental conditions under which it is valid (low densities) are those for which the sixth kinetic theory postulate is the most reasonable. I n eqn. (2) N. is the number of molecules in a mole and T is the ideal gas temperature. This temperature scale can be shown to be numerically identical to the absolute temperature scale which is defined in connection with the second law of thermodynamics (8). Identifying the pressures given in eqns. (1) and (2), we see that Thus, the proportionality of the mean molecular kinetic energy to the absolute temperature is a conclusion drawn by combining the kinetic theory expression with an equation which summarizes macroscopic "gas facts." The true significance of eqn. (3) is that it provides an interpretation in terms of molecular motion of the temperature concept, which is in itself an abstract and unedifying function. Now let us examine the molecular basis for the generality of the existence of the proportionality between the mean molecular kinetic euergy and the absolute temperature for systems a t equilibrium. The difficulty is that eqn. (3) has been demonstrated only for ideal gases. For other systems neither eqn. (1) nor eqn. (2) are valid, yet the assertion is commonly (and truthfully) made that eqn. (3) is valid. This is confusing to students when considering, for example, why it is that the mean molecular Emetic energies are equal for the molecules in a liquid phase and in a coexisting ideal vapor phase a t eqiulibrium, or why eqn. (3) should be valid for a non-ideal gas. A more general demonstration is thus required for the validity of eqn. (3) than that given above. For this purpose it is necessary to use some simple statistical mechanics. Some standard relationships will be used without derivation, but these are of the sort that have a certain plausability, and the argument should be comprehensible to students a t the level of a physical chemistry course, perhaps sooner. A system of interest in equilibrium with its environment can exist in a number of different quantum states. If E, is the total energy of one of these states, the proba-
bility P, that the system will be found in the state is given by the Boltzmann distribution law:
the energy values El are functions of N and V. The parameter fl is related to the possibility of the system and its environment exchanging energy (necessary if the system can have different values of E), but it depends in no way on the specific characteristics of the system of interest, e.g., whether the system is an ideal gas, an imperfect gas, a liquid, or solid. A quite general result of classical mechanics is that the translational energy of a system of interest is separable from the other forms of energy the system may possess. Thus, one of the total energy values E, can be written E;
=
El'
+ Ex0
(5)
where E,' is one of the translational energy values accessible to the system and E," is a term which includes all other contributions to the total energy, includmg internal energies (electronic, vibrational-rotational) and potential energy (which includes the intermolecular energies).% To find the average translational energy of the system, F, we require EI =
x ~ ~ l ~ ~ (6)
;
where P5is the probability that the system have the translational energy E;, irrespective of the value of the "other" energies E,O. This is found from eqn. (4) by summing the expression over all such values E,", but leaving a translational value E,' fixed.
I n eqn. (7) the sum is taken over translational energy states only. Thus, combining eqns. (6) and (7), =
xE5te-BEil/xe-BEjl
(8)
f
j
It is seen from this equation that the characteristics of the energy terms EnWo not affect the average translational energy of the system. The translational states E,' of any system are identical to those of a collection of mass points, for the energies of the system which arise from its intramolecular and intermolecular characteristics are all included in the terms Eao. It is straightforward to show (9) that for such a system that -
E'
=
3N/20
(9)
This expression is accordingly valid for any system which conforms to eqns. (4) and (5). The mean molecular kinetic energy mui/2 is thus given by eqn. (10) for
This separation is valid so long as the potential energy funotion of the system is independent of time. The complicatiom introduced by a time dependent potential energy me discussed in "Principles of Statistical Mechanics," by R. D. Tolman, Oxford University Press, 1938, p. 214. Volume 43, Number 6, June 1966
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any system.
Literature Cited
Since we have demonstrated that eqn. (3) is valid for an ideal gas, comparison witheqn. (10) reveals that for any system (recall that p is independent of the characteristics of a given system). Thus equation (10) becomes (V8)(rn;2/2) = (R/N.)T
(12)
The proportionality between the mean molecular kinetic energy and the absolute tenlpewture in eqn. (12) is thus valid for any system, provided merely that the Boltzmann distribution law eqn. (4) and the separability of the translational energy of the system eqn. (5) are valid.
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(1) GOLDEN, S., "Elements of the Theory of Gases," AddisonWesley Publishing Co., Reading, .Mass., 1964, p. 61. (2) LOEB,L. B., "Kinetic Theory of Gases," 2nd ed., McGrawHill Book Company, Inc., New York, 1934, p. 15. (3) DENBIGH,K. B., "Principles of Chemical Equilibrium," Cambridge University Press, Nev York, 1955, p. 9. (4) Kr~nwoon,J. G., AND OPPENHEIM, J., 'Chemical Thermodynamics," McGraw-Hill Book Company, Inc., New York, 1961, p. 3. ( 5 ) GOLDEN, S., o p . cit., p. 78. (6) BOLTZMANN, L., "Lecture8 an Gss Theory," University of Cdifornia Press, English Translation, 1964, p. 30. (7) KENNARD, E. H., "Kinetic Theory of Gases," McGraw-Hill Book Company, Inc., 1938, p. 7. (8) WALL,F. T., "Chemical Thermodynamics," 2nd ed., W. H. Freeman Co., San Francisco, Cal., 1965, p. 81. (9) DavmsoN, N., "Statistical Mechanics," McGraw-Hill Book Company, Inc., New York, 1962, p. 85.
