Physical Chemistry , 1st ed.

For ideal gases, we can use the ideal gas law and substitute for V:VnRT/p,

so

G

pi

pf



nR

p

T

dp

pi

pf

nRT

d

p

p

nRT

pi

pf



d

p

p



From calculus, we know that (dx/x) ln x. Applying this to the integral in

the above equation and evaluating at the limits, we get

GnRTln 

p

p

f

i

 (4.45)

which is applicable only for isothermal changes.

Example 4.12

What is the change in Gfor a process in which 0.022 mole of an ideal gas

goes from 2505 psi (pounds per square inch) to 14.5 psi at a room tempera-

ture of 295 K?

Solution

Direct application of equation 4.45 yields

G(0.022 mol)8.314 

mo

J

l K

(295 K) ln 

2

1

5

4

0

.5

5

p

p

s

s

i

i



G278 J

Would this be considered a spontaneous process? Since the pressure is not

kept constant, the strict application ofGas a spontaneity condition is not

warranted. However, gases do tend to go from high pressure to low pres-

sure, given the opportunity. We might expect that this process is in fact

spontaneous.

4.8 The Chemical Potential and Other Partial

Molar Quantities

So far, we have focused on changes in systems that are measured in terms of

the system’s physical variables, like pressure and temperature and volume and

the like. But in chemical reactions, substances change their chemical form. We

need to begin to focus on the chemical identity of a material and how it might

change during the course of a process.

It has been assumed that the number of moles,n, of a substance has re-

mained constant in all of the changes considered so far. All of the partial de-

rivatives should also have an nsubscript on the right side to indicate that

the amount of material remains constant: for example, ( U/ V)T,n. How-

ever, there is no reason that we can’t consider a derivative with respect to

amount,n.

Because of the importance of the Gibbs free energy in spontaneity consid-

erations, the majority of derivatives with respect to nconcern G.The chemical

potentialof a substance,, is defined as the change in the Gibbs free energy

with respect to amount at constant temperature and pressure:



G

n



T,p

(4.46)

108 CHAPTER 4 Free Energy and Chemical Potential