# How do I deduce some basic thermodynamic identities using multivariate calculus?

• CharlieCW
In summary, the author is trying to solve the equations for parts (b), (c), and (d) using differentials and the inverse function theorem. He is having trouble with part (d). He is able to find part (b) using the chain rule and differentials, but is struggling with part (d). He is able to find part (c) using the chain rule and differentials, but is struggling with part (d). He is able to find part (b) using the chain rule and differentials, but is struggling with part (d).
CharlieCW

## Homework Statement

Let x, y and z satisfy the state function ##f(x, y, z) = 0## and let ##w## be a function of only two of these variables. Show the following identities:

$$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial x}{\partial z}\right )_w$$

$$\left(\frac{\partial x}{\partial y}\right )_w=\frac{1}{\left(\frac{\partial x}{\partial y}\right )_w}$$

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

$$\left(\frac{\partial x}{\partial y}\right )_z=\left(\frac{\partial x}{\partial y}\right )_w + \left(\frac{\partial x}{\partial w}\right )_y \left(\frac{\partial w}{\partial}\right )_z$$

## Homework Equations

All relationships must be deduced only by assuming ##f(x,y,z)=0## and ##w## function only of two of these variables. Cannot introduce topology or complex analysis theorems (multivariate calculus basics (i.e. chain rule, exact differentials) are allowed).

## The Attempt at a Solution

These are some thermodynamic identities that I've been using for a while, and while I have seen the proofs of some of this in GR and QFT (for example, the inverse function theorem), I'm struggling a bit on how deduce them from more basic calculus theorems used in statistical physics.

To begin with, I noted that since I have ##f(x,y,z)=0## then I can write, for example, ##x=x(y,z)##. Moreover, assuming ##w=w(x,y)##, I can also write ##w=w(x(y,z),z)=w=(y,z)##, and so on.

For part (a), I used the chain rule to show straightforward that:

$$\left(\frac{\partial x}{\partial z}\right )_w=\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w$$

However, I'm not entirely sure if this is an acceptable proof or if there's a way to develop it a bit more.

For part (b), I tried, for example:

$$\left ( \frac{\partial x}{\partial y} \right )_w=\left ( \frac{\partial x}{\partial z} \right )_w \left ( \frac{\partial z}{\partial y} \right )_w=\left ( \frac{\partial x}{\partial y} \right )_w \left ( \frac{\partial y}{\partial z} \right )_w \left ( \frac{\partial z}{\partial y} \right )_w$$

Were I applied the chain rule again to the left term. The above expression reduces to:

$$1=\left ( \frac{\partial y}{\partial z} \right )_w \left ( \frac{\partial z}{\partial y} \right )_w \ \ \ \longrightarrow \ \ \ \left ( \frac{\partial y}{\partial z} \right )_w=\frac{1}{\left ( \frac{\partial z}{\partial y} \right )_w}$$

I have seen several proofs online using complex analysis or topology, so I'm not sure if this derivation is also valid or I just forced it.

For part (c), the proof is pretty straightforward using differentials and the inverse function theorem, so I'll skip it (a simple proof can be found here: https://en.wikipedia.org/wiki/Triple_product_rule)

For part (d), I'm having the most trouble. I tried using differentials and the chain rule:

$$\left ( \frac{\partial x}{\partial y} \right )_z=\left ( \frac{\partial x}{\partial w} \right )_z \left ( \frac{\partial w}{\partial y} \right )_z$$

$$dy=\left ( \frac{\partial y}{\partial x} \right )_z dx + \left ( \frac{\partial y}{\partial z} \right )_x dz$$

$$dx=\left ( \frac{\partial x}{\partial y} \right )_z dy + \left ( \frac{\partial x}{\partial z} \right )_y dz$$

Substituting the first and second expressions into the third one:

$$dx=\left ( \frac{\partial x}{\partial w} \right )_z \left ( \frac{\partial w}{\partial y} \right )_z \left ( \left ( \frac{\partial y}{\partial x} \right )_z dx + \left ( \frac{\partial y}{\partial z} \right )_x dz \right ) + \left ( \frac{\partial x}{\partial z} \right )_y dz$$

The coefficient for ##dx## should be equal to ##1##, while the coefficient for ##dz## should be zero. Therefore, we get the following equalities:

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

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

However, this doesn't seem to lead anywhere, so this is probably the one I have the least idea how to proceed.

Last edited:
I think I've seen these in Callen's "Thermodynamics and Introduction to Thermostatics" in the appendix. Although, I'm not sure if (d) can be found there.

DrClaude
CptXray said:
I think I've seen these in Callen's "Thermodynamics and Introduction to Thermostatics" in the appendix. Although, I'm not sure if (d) can be found there.

Thanks, I'll check if we got it in the library after the morning lectures and I'll update you if I find the solution.

Edit: I checked the book, they have the proofs I needed. As for the last one, I found it here in page 4: http://mutuslab.cs.uwindsor.ca/schurko/introphyschem/handouts/mathsht.pdf

Last edited:
DrClaude

## 1. What are thermodynamic identities?

Thermodynamic identities are mathematical equations that relate different thermodynamic properties of a system, such as temperature, pressure, and energy. They are derived from the laws of thermodynamics and are used to analyze and predict the behavior of physical systems.

## 2. How are thermodynamic identities derived?

Thermodynamic identities are derived from the fundamental laws of thermodynamics, such as the first and second laws. They are also derived from the definitions of thermodynamic properties, such as internal energy, entropy, and enthalpy.

## 3. What is the significance of thermodynamic identities in science?

Thermodynamic identities are essential in understanding and predicting the behavior of physical systems, such as chemical reactions, phase changes, and heat transfer. They allow scientists to analyze and design systems for maximum efficiency and to make accurate predictions about their behavior.

## 4. Can thermodynamic identities be applied to all systems?

Yes, thermodynamic identities can be applied to all systems, whether they are simple or complex, open or closed. However, the equations may need to be modified or adapted for specific systems or conditions.

## 5. How can thermodynamic identities be used in practical applications?

Thermodynamic identities are used in various practical applications, such as in the design of engines, refrigeration systems, and power plants. They are also used in chemical engineering to optimize processes and in materials science to understand the behavior of materials under different conditions.

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