# Finding the equations of state via the fundamental relation

• hijasonno
In summary, the conversation discusses the partial derivatives of entropy with respect to internal energy and volume in terms of temperature and pressure. It also introduces the concept of homogeneous functions and how to determine if a function is homogeneous of order zero. The conversation ends with a question about continuing the problem.
hijasonno

## Homework Equations

$$\frac{\partial S}{\partial u}\Bigr|_{v} = \frac{1}{T}$$
$$\frac{\partial S}{\partial v}\Bigr|_{u} = \frac{-P}{T}$$[/B]

## The Attempt at a Solution

a.) $$\frac{S}{R} = \frac{UV}{N} - \frac{N^3}{UV}$$ $$\frac{S}{R} = \frac{UV}{N} - \frac{N}{uv}$$ $$\frac{S}{NR} = \frac{UV}{N^2} - \frac{1}{uv}$$ $$\frac{s}{R} = uv - \frac{1}{uv}$$ where ##s = \frac{S}{N}##, ##u = \frac{U}{N}##, and ##v = \frac{V}{N}##.

Equations of state(in entropy representation):

$$\frac{\partial S}{\partial u}\Bigr|_{v} = \frac{1}{T} = R(v + \frac{1}{uv^2})$$

$$\frac{\partial S}{\partial v}\Bigr|_{u} = \frac{P}{T} = R(u + \frac{1}{vu^2})$$

And this is where I get stuck; when I try and multiply ##u## and ##v## by a constant, i.e. ##\lambda v + \frac{1}{\lambda u (\lambda v)^2}##, this is obviously not homogeneous of order zero. I have an idea about how to continue the rest of the parts, but I want to be sure that the first one is correct be for I move on.
My values for ##P## and ##T## in the entropy representation seem right, maybe I'm not quite understanding what a homogeneous function actually is. Any help is appreciated.[/B]

Last edited by a moderator:
I can't figure out why the image won't load, so I have to put it in a reply to the thread.
https://www.physicsforums.com/attachments/210922

## 1. What is the fundamental relation in finding equations of state?

The fundamental relation is a mathematical expression that describes the thermodynamic properties of a system in terms of its state variables, such as temperature, pressure, and volume. It is used to derive equations of state, which relate these state variables to each other.

## 2. How do you derive equations of state from the fundamental relation?

To derive equations of state, the fundamental relation is differentiated with respect to the state variables and manipulated using mathematical techniques such as calculus and thermodynamics. This results in equations that describe the relationship between the state variables in a specific system.

## 3. What types of systems can be described using equations of state?

Equations of state can be used to describe the behavior of various systems, including gases, liquids, and solids. They are also used to model complex systems such as mixtures and solutions.

## 4. Why is it important to find equations of state?

Equations of state are crucial in understanding the behavior of a system and predicting its properties under different conditions. They are also used in the design and optimization of industrial processes, such as in the production of energy and pharmaceuticals.

## 5. Are there different methods for finding equations of state?

Yes, there are several methods for finding equations of state, including the virial equation, the van der Waals equation, and the Peng-Robinson equation. Each method has its own advantages and limitations, and the choice of method depends on the specific system being studied.

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