# Thermodynamics, entropy representation problem

• fluidistic
In summary, the problem is asking for the three equations of state in the entropy representation for a system with the fundamental equation u = A*s^(5/2)/v^(1/2). The equations are T = (5A*s^(3/2))/(2*N^2*v^(1/2)), P = (A*s^(5/2))/(2*N^2*v^(3/2)), and mu = -(A*s^(5/2))/(N^3*v^(1/2)). The next question asks to graph T(V) for a fixed P, which may be confusing as it seems to require finding T(V,P) instead of T(S,V,N). The correct equations for T(V,P) are (partial
fluidistic
Gold Member

## Homework Statement

The problem is taken from Callen's book (page 36).
Find the three equations of state in the entropy representation for a system with the fundamental equation $u =A\frac{s^{5/2}}{v^{1/2}}$.
Show by a diagram (drawn to abitrary scale) the dependence of temperature on volume for fixed pressure. Draw two such "isobars" corresponding to two values of the pressure and indicate which isobar corresponds to the higher pressure.

## Homework Equations

$dU=TdS-PdV+\mu dN$.

## The Attempt at a Solution

I've been looking in the book for the "entropy representation" and what I understood is that they ask for $T(S,V,N)$, $P(S,V,N)$ and $\mu (S,V,N)$. Google didn't give me a better clue on the "entropy representation" either.
So I've found out the 3 equations of state to be $T=\frac{5AS^{3/2}}{2N^2V^{1/2}}$, $P=\frac{AS^{5/2}}{2N^2V^{3/2}}$ and $\mu =-\frac{AS^{5/2}}{N^3V^{1/2}}$.
What destroys me is the next question. They ask me to graph T(V) for a fixed P; as if they had asked me first to find $T(V,P)$ instead of $T(S,V,N)$. Did I get the "entropy representation" wrong?

Apparently what I did is wrong.
The 3 equations of state are $\left ( \frac{\partial S }{\partial U } \right ) _{V,N}=\frac{1}{T}$, $\left ( \frac{\partial S }{\partial N } \right ) _{V,U}=\frac{\mu}{T}$ and $\left ( \frac{\partial S }{\partial V } \right ) _{U,N}=\frac{-P}{T}$. I calculated them to be worth $\frac{2N^{2/5}V^{1/5}}{5A^{2/5}U^{3/5}}$, $\frac{2U^{2/5}V^{1/5}}{5A^{2/5}N^{3/5}}$ and $\frac{N^{2/5}U^{2/5}}{5A^{2/5}V^{4/5}}$ respectively.
I am not sure how to do the diagram though.

## 1. What is thermodynamics?

Thermodynamics is the branch of physics that deals with the relationship between heat, energy, and work. It studies how energy is transferred and transformed within a system, and how these processes affect the overall properties of the system.

## 2. What is entropy representation problem?

The entropy representation problem is a concept in thermodynamics that deals with the difficulty in representing the microscopic behaviour of a system in terms of its macroscopic properties. It is essentially the challenge of bridging the gap between the microscopic and macroscopic descriptions of a system.

## 3. What is the difference between entropy and energy?

Entropy and energy are both important concepts in thermodynamics, but they are not the same thing. Energy is a measure of a system's ability to do work, while entropy is a measure of the disorder or randomness in a system. In other words, energy describes the quantity of a system, while entropy describes the quality.

## 4. How is the second law of thermodynamics related to the entropy representation problem?

The second law of thermodynamics states that the total entropy of an isolated system will always increase over time. This law is closely related to the entropy representation problem because it highlights the difficulty in predicting the exact behaviour of a system due to the constant increase in entropy.

## 5. Are there any real-life applications of thermodynamics and the entropy representation problem?

Yes, thermodynamics and the entropy representation problem have many practical applications in various fields, including engineering, chemistry, and biology. Some examples include designing more efficient engines, understanding chemical reactions, and studying the behaviour of complex biological systems.

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