Concise Physical Chemistry

c05 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

76 ENTROPY AND THE SECOND LAW

We know from the second law that at constant pressure, we obtain

dS=

dqp

T

=

Cp

T

so

dS=

Cp

T

dT+

(

∂S

∂p

)

T

dp

By the Euler reciprocity relation, for exact differentialsduwritten in differential form

du=M(x,y)+N(x,y)

we have the equality

∂M(x,y)

dy

=

N(x,y)

dx

In the case of the Gibbs thermodynamic function (next chapter)μ=f(S,p), we

have

dμ=−SdT+Vdp

so

−

(

∂S

∂p

)

T

=

(

∂V

∂T

)

p

which leads to

dS=

Cp

T

dT+

(

∂S

∂p

)

T

dp=

Cp

T

dT−

(

∂V

∂T

)

p

dp

Both of these coefficientsCp/Tand(∂V/∂T)pcan be measured, so the infinitesimal

dScan be found at anyTandV. The finite change
Sis

S=

∫T 2

T 1

Cp

T

dT−

∫p 2

p 1

(

∂V

∂T

)

p

dp

Starting with the Helmholtz free energy in place of the Gibs function, a comparable

derivation yields

S=

∫T 2

T 1

Cp

T

dT−

∫V 2

V 1

(

∂p

∂T

)

V

dV