Page 260 - Advanced Thermodynamics for Engineers, Second Edition

12.2 CHEMICAL POTENTIAL, m 249

Hence, in terms of masses
n
X
m dm i (12.7a)
dG ¼ Vdp SdT þ i
i¼1
while in terms of amount of substance
n
X
m dn i (12.7b)
dG ¼ Vdp SdT þ
m i
i¼1
12.2 CHEMICAL POTENTIAL, m

vG
The term m is called the chemical potential and is defined as . The significance of m will
vm i
p;T;m jsi
now be examined. First, it can be considered in terms of the other derived properties.
By definition
dG ¼ dðH TSÞ¼ dH TdS SdT (12.8)
Hence
dH ¼ dG þ TdS þ SdT
X
m dm i þ TdS þ SdT
¼ Vdp SdT þ i
X
m dm i (12.9)
¼ Vdp þ TdS þ i
Considering each of the terms in Eqn (12.9), then these can be interpreted as the capacity to do
work brought about by a change in a particular property. The first term is the increase in capacity to do
work that is achieved by an isentropic pressure rise (cf. the work done in a feed pump of a Rankine
cycle) and the second term is the increased capacity to do work that occurs as a result of reversible heat
transfer. The third term is also an increase in the capacity of the system to do work, but this time it is
brought about by the addition of a particular component to a mixture. For example, if oxygen is added
to a mixture of carbon monoxide, carbon dioxide, water and nitrogen (the products of combustion of a
hydrocarbon fuel), then the mixture could further react to convert more of the carbon monoxide to
carbon dioxide, and more work output could be obtained. Thus m i is the increase in the capacity of a
system to do work when unit mass (or, in the case of m , unit amount of substance) of component i is
m i
added to the system. m i can be considered to be a ‘chemical pressure’ because it is the driving force in
bringing about reactions.
Assuming that H is a continuous function,

vH vH vH vH
dm n (12.10)
vp S;m vS p;m vm 1 S;p;m is1 vm n S;p;m isn
dH ¼ dp þ dS þ dm 1 þ :::: þ
By comparison of Eqns (12.9) and (12.10),

vH
i : (12.11)
m ¼
vm i
s;p;m jsi