# Partial differentiation - Constants

1. Nov 25, 2012

### Jalo

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

Consider the following equality:

($\frac{∂S}{∂V}$)T = ($\frac{∂P}{∂T}$)V

If I rearrange the equality so that I write:

($\frac{∂S}{∂P}$)? = ($\frac{∂V}{∂T}$)?

What variables will be constant in each side?
I'm having some trouble in a few thermodynamics problems because of this...

2. Relevant equations

3. The attempt at a solution

I don't know how to do this. Normally I just ignore and assume that every variable except the variable of differentiation is a constant.
If anyone could confirm or correct me I'd be thankful.

Thanks.

2. Nov 26, 2012

### SammyS

Staff Emeritus

For the case in which three variables can be related by a function of the form f(x, y, z) = 0, then the following relations hold.

$\displaystyle \left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1$

$\displaystyle \left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{\left(\frac{\partial y}{\partial x}\right)_y}$

See the following two Wikipedia entries:

http://en.wikipedia.org/wiki/Triple_product_rule

http://en.wikipedia.org/wiki/Exact_differential#Cyclic_relation