C. J. G. Raw'
I
-'~emical Equilibria in Imperfect Gases
M a n y elementary textbooks of physical chemistry2 discuss the question of chemical equilibrium i n a reacting mixture of imperfect gases in terms of the fugacit~,but no attempt is usually made to correlate the equilibrium constant with the equation of state of the gas. I t will be pointed out here how this may be done in a very simple, general manner suitable for inclusion in an elementary course. This will lead to a het,ter appreciation of the relation between equilibrium constants in gases and the deviations from gas ideality. The fugacity of the itb component of a gas mixture is a quantit,~entirely equivalent in meaning to the act,ivity. It is defineda by: where pi is the chemical potential (prorefers to an arbitrary standard state), p,* is the fugacity, R is the molar gas constant, and T is the absolute temperature. If the bB gases react according to the equation: aA . . . mM nN . . ., then the thermodynamic equilibrium constant is:
+
+
+
+
In the limiting rase of low pressures, the fugacity becomes equal to t,he partial pressure of the gas, and we find the corresponding relations for ideal gases:
where pi is now the partial pressure of component i. Let us now extend the above to include explicitly t,he equation of state of imperfect gases in equation (2). First of all, we shall writ,e the equation of state in the general virial form: @ = I + - B(T) Rl' + Y- + . C(T) . V.
(3)
where B(T)s . ' . are the seco0d2 third, ' . ' virial coefficients, which are functions of temperature only. These virial coefficientsare related by statistical mechanics to the forces between the gas molecules. The word "virial" is derived from the Latin for "force." Xow, it is shown in most textbooksa on chemical 'The author is indebted to Professor Bryre Crawford, Jr. for making it possible for him to spend a year at the University of hlinnesota. Preeent address: Saint Louis University, 1402 South Grand Boulevard, Saint Louis 4, Missouri. 'See, for example, MOORE,W. J., "Physical Chemistry," 3rd ed., Prentice-Hall, Inc., New York, 1957. a See, for example, GLASSTONE, S., 'Thermodynamics for chemist,^," D. Van Nostrand Co., New York, 1947.
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Journal o f Chemical Education
thermodynamics or physical chemistry that the fugacity is given by:
Inserting eq. (3) in the integral of eq. (4), and ignoring terms involving the third and higher virial coefficients, we arrive at the following relation:
If we now take t,he logarit,hm of the equilibrium constant as given by eq. (2), and if we insert eq. (5) for each p,*, we have: In K = m In p ~ + * n In p ~ + * . . . - a In pn* - h In ps* -...
Using equation (2') in equation ( G ) ,
The terms involving summation in equations (6) and (7) do not explicitly show the stoichiometrlc coefficients; i.e., reactants i and products j need not be all different in these equations. If the second virial coefficients of the individual (gaseous) reactants and products at t,he temperature T are known, equation (7) may be used to evaluat,e the accurate thermodynamic equilibrium constant a t the temperature T from the "ideal gas" equilibrium constant in terms of the partial pressures. Equation (7) is not new to the student of statistical therm~dynamics,~ but its introduction in the simple manner here out,lined makes it possible to acquaint the elementary student with an imvortaut and useful equation in the thermodynamic thkory of gas equilibria. If greater accuracy is desired, the higher virial coefficients could be considered, '
but very few extensive and accurate measurements of the higher virial coefficientshave been made and their inclusion is thus hardly justified. As is well known, the in terms of deviations from gas ideality are best the second "irial coefficient,"and much accurate equation of state data have been accumulated and written in this form, It is consequently of importance to know how to use the virial coefficient data directly in t,he study of equilibria involving imperfect gases. R. H., AND GUGOENEE~M, E. A,, "Ststi~ticdTher'FOWLER, modynamics," Cambridge Univ. Press, 1949. S H ~ ~J. 0.) s CCRTISS, ~ ~ C. ~F., AND ~ BIRD, ~ ~R. B., ~ "Molecular Theory of Gases and Liquids," John U'iley & Sons, Ine., New York, 1954.
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