Physical Chemistry , 1st ed.

we remember that Sis defined by the partial derivative in equation 4.40.

Substituting:

GH T

G

T



p

where the minus signs have canceled. We rearrange this by dividing both sides

of the equation by T, and get



G

T



H

T

 (^) 

G

T



p

Now we further rearrange by bringing all terms in Gto one side:



G

T



G

T



p



H

T

 (4.41)

Although this might look intractable, we will introduce a simplifying substitu-

tion in a roundabout way. Consider the expression G/T. The derivative of this

with respect to Tat constant pis



T



G

T



p



T

G

 (^2) 

T

T



p



T

1



G

T



p

by strict application of the chain rule. T/ Tequals 1, so this expression sim-

plifies to



T



G

T



p



T

G

 2 T

1



G

T



p

If we multiply this expression by T,we get

T 

T



G

T



p



G

T



G

T



p

Note that the expression on the right side of the equation is the same as the

left side of equation 4.41. We can therefore substitute:

T 

T



G

T



p



H

T



or



T



G

T



p



T

H

 2 (4.42)

This is an extremely simple equation, and when expanding our derivation to

consider changes in energy, it should not be too difficult to derive, for the over-

all process:



T





T

G



p





T

H

 2 (4.43)

for a physical or chemical process. Equations 4.42 and 4.43 are two expressions

of what is called the Gibbs-Helmholtz equation.By using substitution [that is,

let u1/T,du(1/T^2 ) dT, and so on], you can show that equation 4.43

can also be written as



TG

 pH (4.44)

T

(^1) 
The form given in equation 4.44 is especially useful. By knowing Hfor a
process, we know something about G. A plot ofG/Tversus 1/Twould be
equal to Has a slope. (Remember that a derivative is just a slope.) Further, if
106 CHAPTER 4 Free Energy and Chemical Potential