I
GUEST AUTHOR P. J. Robinson University of Monchester Manchester, England
I
I
Textbook Errors,
Chemical Potentials in an Ideal Mixture of Ideal Gases
Errors arise in some textbooks1 in the mathematical derivation of equation (1) below for the chemical potential of a component in an ideal gas mixture. The purpose of this article is to examine the errors involved and to suggest a simple but correct derivation. An ideal mixture of ideal gases obeys both Amagat's Law and Dalton's Law. Thus the total volunle V and pressure P of the mixture at temperature Tare given by the equations: V
=
X V s a n d ~= X P d i
i
where V , = volume occupied by the ith component alone a t pressure P and temperature T ; and P , = partial pressure of the ith con~ponent,i.e., the pressure which it alone would exert in volume V a t temperature T. It follows that the chemical potential p< of the ith component of such a mixture is given by the equation: pi = pio
+ RTlnPi
54
(1)
where p,' is an integration constant and R is the gasconstant. This theorem is proved incorrectly, however, in the majority of the undergraduate textbooks which attempt the proof. The correct result is obtained in these texts by the cancellation of two independent errors. Fallacious Proof
The first stage in the fallacious proof is the following derivation of an incorrect expression for the partial molar volume of a component in the mixture. The partial molar volume V , of the ith component is defined by the equation:
respect ton," to give the equation
This differentiation is clearly incorrect, however, since P , is not constant under the required conditions of constant temperature, total pressure, and amounts of the other components. Qualitatively, the volume of the system increases while all the n j remain constant, so that all the P, must therefore decrease. Since the total pressure is also constant, P, must therefore increase, and it can be shown,2in fact, that
where x, is the mole fraction of the itb component in the mixture. The second stage of the proof is the derivation of an equation for the chemical potential of the ith component from the well-known formula
6pj = Vj6P
at e a s t a n t femperalure and eompositia
(4)
The fallacious argument states that 6P can be replaced by 6P,in this equation, so that: 6pi = pj8Pi
(5)
This is clearly incorrect if the composition remains constant as required for the validity of equation (4); under these conditions P , = xiP and SP, = xJP. Combination of the two incorrect equations (2) and (5) leads t o the correct relationship
and it follows from the definition of partial pressure that and hence the correct integrated equation (1). where nt is the number of moles of the ith component in the mixture. This equation is "differentiated with Suggestions of material suitable for this column, and guest columns suitable for publication direetlv, should he sent with as many details as possible, and particularly with references to modern textbooks, to Karol J. Mysels, Department of Chemistry, University of Southern California, Los Angela 7, California. 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 at least two independent r e cent standard books.
654
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Journal of Chemical Educdion
a This useful result, which does not appear to have been quoted elsewhere, may be derived as follows: Since P, = niRT/V
Equation (6) gives Vd = RT/P, and since PV = nRT, where n is the total number of moles of mixture, it fallows that VJV = l / n , whence
To summarize, the two incorrect steps in the argument are (a) Pi is not constant under the conditions of the differentintion used to derive equation (2); (b) Equation (4) is correct only if the composition remains constant, and does not, therefore, lead to equation (5).
Each error arises from the disregard of the conditions attached to a partial differential coefficient, and the resulting fallacies illustrate strikingly the importance of keeping these conditions clearly in mind.
Law implies that the partial molar volume of a component in the mixture is equal to the molar volume of the pure component a t pressure P and temperature T , namely R T / P and not RTIP,. The correct equations (4) and (6) may now be combined to give RT 6pi = -6P at emslant temperature and eornpositim P
This equation can be written in terms of P,, since under these conditions P , = x,P and SP, = xJP, whence:
Correct Proof
A correct expression for V , may be derived by adding to the right hand side of equation (2) a term
and using equation (3). A simpler derivation is obtained, however, by differentiating the equation V
=
nRT (where n P
=
Ci ni)
with respect t o n , a t constant T , P , and n,.
Thus
L equal to Sn,when the ith component is added since S ~ is a t constant n,. This result, which is given by several text-books ( I S ) , is obviously correct since Amagat's
and
It should be noted, incidentally, that the constant p t 0 is the chemical potential of the ith component in a mixture of the same compositia, a t the same temperature, when the total pressure is such that P , is unity. It cannot be equated, without further argument, to the chemical potential of the pure component a t unit pressure. Litemture Cited (1) Fowmn AND GUGGENHEIM, "Statistical Thermodynamics," Cambridge University Press, New York, 1949, p. 72. (2) GLASSTONE,"Thermodynamics for Chemists," D. van Nostrand Co., Ine., Princeton, N. J., 1947, p. 219. (3) DENBIGH,"The Principles of Chemical Equilibrium," Cambridge University Press, New York, 1961, p. 112.
Volume
41, Number 12, December 1964
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655
