Partial Derivatives for P, S, T in Differential Calculus

[Bill Foster]

Homework Statement

I'm looking for $$\frac{\partial{P}}{\partial{V}}$$ at fixed T and fixed S.

Homework Equations

$$P=\frac{TS}{4V}$$

The Attempt at a Solution

$$\frac{dP}{dV}=\frac{\partial{P}}{\partial{V}}+\frac{\partial{P}}{\partial{T}}\frac{dT}{dV}+\frac{\partial{P}}{\partial{S}}\frac{dS}{dV}$$

$$\frac{\partial{P}}{\partial{V}}=-\frac{TS}{4V^2}$$

$$\frac{\partial{P}}{\partial{T}}=\frac{S}{4V}$$

$$\frac{\partial{P}}{\partial{S}}=\frac{T}{4V}$$

$$\frac{dP}{dV}=\frac{\partial{P}}{\partial{V}}+\frac{\partial{P}}{\partial{T}}\frac{dT}{dV}+\frac{\partial{P}}{\partial{S}}\frac{dS}{dV}=-\frac{TS}{4V^2}+\frac{S}{4V}\frac{dT}{dV}+\frac{T}{4V}\frac{dS}{dV}$$

At constant T, I get this: $$\frac{dP}{dV}=-\frac{TS}{4V^2}+\frac{T}{4V}\frac{dS}{dV}$$

At constant S, I get this: $$\frac{dP}{dV}=-\frac{TS}{4V^2}+\frac{S}{4V}\frac{dT}{dV}$$

What do I do about the other differentials: $$\frac{dS}{dV}$$ and $$\frac{dT}{dV}$$?

Wouldn't this also be true?

$$\frac{dS}{dV}=\frac{\partial{S}}{\partial{V}}+\frac{\partial{S}}{\partial{T}}\frac{dT}{dV}+\frac{\partial{S}}{\partial{P}}\frac{dP}{dV}$$

$$\frac{dT}{dV}=\frac{\partial{T}}{\partial{V}}+\frac{\partial{T}}{\partial{S}}\frac{dS}{dV}+\frac{\partial{T}}{\partial{P}}\frac{dP}{dV}$$

 

[Office_Shredder]

You want to find $$\frac{\partial{P}}{\partial{V}}$$

and your second line is

$$\frac{\partial{P}}{\partial{V}}=-\frac{TS}{4V^2}$$

So... you're done?

 

[Bill Foster]

You want to find $$\frac{\partial{P}}{\partial{V}}$$

and your second line is

$$\frac{\partial{P}}{\partial{V}}=-\frac{TS}{4V^2}$$

So... you're done?

Yes. That's at constant T and constant S.

Is it the same if T is constant and S is not constant? Or vice-versa?