Page 259 - Advanced Thermodynamics for Engineers, Second Edition

248 CHAPTER 12 CHEMICAL EQUILIBRIUM AND DISSOCIATION

Equations (12.2) and (12.3) are based on the assumption that G ¼ mg ¼ mgðp; TÞ, and this is quite
acceptable for a single component system, or one of fixed composition. If the system has more than one
component and these components can react to form other compounds, e.g. if the system contained
carbon monoxide, oxygen and carbon dioxide as defined in Eqn (12.1), then it is necessary to define the
P
m i . The
Gibbs energy as G ¼ mg ¼ mgðp; T; m i Þ where m i is the mass of component i and m ¼
significance of changes of composition on the value of the Gibbs energy of a mixture will now be
investigated.
If
G ¼ mg ¼ mgðp; T; m i Þ; (12.4)

and if it is assumed that G is a continuous function with respect to p and T and the masses
of constituents comprising the mixture, then the change of G with changes in the independent
variables is

vG vG vG vG
dm 1 þ .. (12.5a)
dG ¼ dp þ dT þ dm n
vp T;m vT p;m vm 1 p;T;m is1 vm n p;T;m isn
where dm 1 .dm n are changes in mass of the various constituents. A similar equation can be written in
terms of amount of substance, and is

vG vG vG vG
dn 1 þ :::::: dn n (12.5b)
dG ¼ dp þ dT þ
vp vT vn 1 vn n
T;n p;n p;T;n is1 p;T;n isn
For the initial part of the development of these equations, the mass-based relationship will be used.
dm 1 represents the ‘quantity’ of Gibbs energy introduced by the transfer of
The term ðvG=vm 1 Þ
p;T;m is1
mass dm 1 of constituent 1 to the system. (This can be more readily understood by considering the
dm 1 has a more readily appreciated
change in internal energy, dU, when the term ðvU=vm 1 Þ
p;T;m is1
significance.)
The significance of the terms on the right of Eqn (12.5a) is:
1. The first term denotes the change of Gibbs energy due to a change in pressure; the
temperature, total mass and composition of the system remaining constant. This is equivalent
to the term derived when considering a system of constant composition and is Vdp.
2. The second term denotes the change of Gibbs energy due to a change in temperature, the
pressure and total mass of the system remaining constant. This is equivalent to SdT derived
previously.
3. The third term shows the change of Gibbs energy due to a change in the mass (or amount of
substance if written in terms of n) of constituent m 1 , the pressure, temperature and masses of
other constituents remaining constant. It is convenient to define this as

vG
1 (12.6)
m ¼
vm 1
4. The fourth term is a general term of the form of term (3) in Eqn (12.6).