Property Relation From Gibbs and Maxwell Relations

Prequisite Knowledge
Maxwell Relations

Problem
Part a.) Use the Gibbs relation: du = T ds – P dv and one of Maxwell's relations to find an expression for (∂u/∂P)T that only has properties P, v, and T involved. Part b.) What is the value of that partial derivative for an ideal gas?

Solution
First divide the Gibbs relation by dP, as this already results in the left-hand side equaling the part that we want:

Part a.)

du
dP
 =
T ds
dP
 –
P dv
dP

while holding the temperature T constant results in:

(
∂u
∂P
)
 
 
T
 =  T(
∂s
∂P
)
 
 
T
 –  P(
∂v
∂P
)
 
 
T

substituting the Maxwell relation:

(
∂v
∂T
)
 
 
P
 =  −(
∂s
∂P
)
 
 
T

results in:

(
∂ u
∂ P
)
 
 
T
 =  −T(
∂ v
∂ T
)
 
 
P
 –  P(
∂ v
∂ P
)
 
 
T


Part b.)

For an ideal gas:

Pv = RT

rearranging for v yields:

v =
RT
P

the derivates follow:

(
∂v
∂T
)
 
 
P
 =  
R
P

and:

(
∂v
∂P
)
 
 
T
 =  –RTP -2

substituting these for the derivative of u above:

(
∂u
∂P
)
 
 
T
 =  −T
R
P
 – P(–RTP –2) = 0


Note: (big letter) = (mass)*(small letter), ex: W = mw, this is applicable for work (W, w), heat (Q, q), enthalpy (H, h), entropy (S, s). The pattern here is the energy terms with the exception of volume/specific volume (V, v)