Mastering Maxwell Relations in Thermodynamics: Derivation & Problem-Solving Tips

[BobaJ]

I'm studying Thermodynamics and I'm a little stuck at this problem.

1. Homework Statement

Starting with the first Maxwell relation, derive the remaining three by using only the relations:

$$\left(\frac{\partial x}{\partial y}\right) _{z} \left(\frac{\partial y}{\partial z}\right) _{x} \left(\frac{\partial z}{\partial x}\right) _{y} = -1$$

and

$$\left(\frac{\partial x}{\partial y}\right) _{f} \left(\frac{\partial y}{\partial z}\right) _{f} \left(\frac{\partial z}{\partial x}\right) _{f} = 1$$​

Homework Equations

The Maxwell relations are:

$$\left(\frac{\partial T}{\partial V}\right) _{S} = - \left(\frac{\partial P}{\partial S}\right) _{V}$$
$$\left(\frac{\partial T}{\partial P}\right) _{S} = \left(\frac{\partial V}{\partial S}\right) _{P}$$
$$\left(\frac{\partial S}{\partial V}\right) _{T} = \left(\frac{\partial P}{\partial T}\right) _{V}$$
$$\left(\frac{\partial S}{\partial P}\right) _{T} = - \left(\frac{\partial V}{\partial T}\right) _{P}$$

The Attempt at a Solution

My problem is, that I don't understand the second relation they give me to solve the problem. I'm not quite sure what would be f in this relation. I mean, in the book they define it as a function of x, y, and z, but I can't really use it. I don't know where to start. I'm sure that the problem is quite easy, but I need a little push to get started.

Any help would be appreciated

 

[Chestermiller]

I'm studying Thermodynamics and I'm a little stuck at this problem.

1. Homework Statement

Starting with the first Maxwell relation, derive the remaining three by using only the relations:

$$\left(\frac{\partial x}{\partial y}\right) _{z} \left(\frac{\partial y}{\partial z}\right) _{x} \left(\frac{\partial z}{\partial x}\right) _{y} = -1$$

and

$$\left(\frac{\partial x}{\partial y}\right) _{f} \left(\frac{\partial y}{\partial z}\right) _{f} \left(\frac{\partial z}{\partial x}\right) _{f} = 1$$​

Homework Equations

The Maxwell relations are:

$$\left(\frac{\partial T}{\partial V}\right) _{S} = - \left(\frac{\partial P}{\partial S}\right) _{V}$$
$$\left(\frac{\partial T}{\partial P}\right) _{S} = \left(\frac{\partial V}{\partial S}\right) _{P}$$
$$\left(\frac{\partial S}{\partial V}\right) _{T} = \left(\frac{\partial P}{\partial T}\right) _{V}$$
$$\left(\frac{\partial S}{\partial P}\right) _{T} = - \left(\frac{\partial V}{\partial T}\right) _{P}$$

The Attempt at a Solution

My problem is, that I don't understand the second relation they give me to solve the problem. I'm not quite sure what would be f in this relation. I mean, in the book they define it as a function of x, y, and z, but I can't really use it. I don't know where to start. I'm sure that the problem is quite easy, but I need a little push to get started.

Any help would be appreciated

You're starting from the wrong relationships. Are you familiar with the equation dU=TdS-PdV?

 

[BobaJ]

Yes. I'm familiar with this equation. I know that there are 3 more for the enthalpy, the Helmholtz function and the Gibbs function. But I thought that the Maxwell relations are the four I wrote down and as the problem says to start with the first Maxwell relation I didn't think much about them.

 

[Chestermiller]

Oh OK. I understand what you are being asked to do now.