How Does Enthalpy Relate to Heat Capacity at Constant Composition?

[fluidistic]

Homework Statement

Demonstrate that C_{Y,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{Y,N} where H is the enthalpy and Y is an intensive variable.

Homework Equations

(1) C_{Y,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N}
(2) T= \left ( \frac{ \partial U}{\partial S } \right ) _{X,N} where X is an extensive variable.

The Attempt at a Solution

Using (1) and (2) I reach that C_{Y,N}=T \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N}+ P \left ( \frac{ \partial V}{\partial T } \right ) _{Y,N}. I don't know how to proceed further, I'm really stuck here.

 

[I like Serena]

Which intensive variables are we talking about?
What do you get if you pick the first one that springs to mind?

 

[fluidistic]

Which intensive variables are we talking about?
What do you get if you pick the first one that springs to mind?

Usually the pressure, but it is not specified.
C_{P,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{P,N}.
Where \left ( \frac{ \partial S}{\partial T } \right ) _{P,N}=\left ( \frac{ \partial U}{\partial T } \right ) _{P,N}\left ( \frac{ \partial S}{\partial U } \right ) _{P,N}=\frac{1}{T} \left ( \frac{ \partial U}{\partial T } \right ) _{P,N}.
Thus C_{P,N}= \frac{1}{N} \left ( \frac{\partial U }{\partial T } \right ) _{P,N }.
Now I use the relation U=H-PV to get \left ( \frac{ \partial U}{\partial T } \right ) _{P,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{P,N} - \left [ \underbrace { \left ( \frac{ \partial P}{\partial T } \right ) _{P,N} V }_{=0} + P \left ( \frac{ \partial V}{\partial T } \right ) _{P,N} \right ].
Therefore I'm left with C_{P,N}=\frac{1}{N} \left [ \left ( \frac{ \partial H}{\partial T } \right ) _{P,N} - P \left ( \frac{ \partial V}{\partial T } \right ) _{P,N} \right ].

 

[I like Serena]

Hmm, let's start with H=U+PV, or rather dH=TdS+VdP.

When I take the partial derivative with respect to T, and with P,N constant, I almost get what you're looking for (typo?).

 

[fluidistic]

Hmm, let's start with H=U+PV, or rather dH=TdS-VdP.

When I take the partial derivative with respect to T, and with P,N constant, I almost get what you're looking for (typo?).

Hmm I don't think there's a typo.
Anyway you took the partial derivative of "dH"? I'm having some troubles to figure this out :)

 

[I like Serena]

dH=TdS+VdP
So:
$$\left({\partial H \over \partial T}\right)_{P,N}=\left({T\partial S + V\partial P \over \partial T}\right)_{P,N}$$

Factor out and replace with $C_{P,N}$ where applicable...