# Enthalpy, partial derivatives

1. Apr 29, 2012

### fluidistic

1. The problem statement, all variables and given/known data
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.

2. Relevant 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.

3. 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.

Last edited: Apr 29, 2012
2. Apr 29, 2012

### I like Serena

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

Last edited by a moderator: Apr 29, 2012
3. Apr 29, 2012

### fluidistic

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 $C_{P,N}=\frac{1}{N} \left [ \left ( \frac{ \partial H}{\partial T } \right ) _{P,N} - P \left ( \frac{ \partial V}{\partial T } \right ) _{P,N} \right ]$.

Last edited by a moderator: Apr 29, 2012
4. Apr 29, 2012

### I like Serena

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

Last edited: Apr 29, 2012
5. Apr 29, 2012

### fluidistic

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 :)

6. Apr 29, 2012

### 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...