# State equations for a thermodynamic substance/system

• I
• cianfa72
cianfa72
TL;DR Summary
About the state equations for a thermodynamic substance/system.
Hi, as follow up to this thread I believe for any substance/thermodynamic system there exists actually a set of 3 state equations between the 5 variables ##(U,T,S,p,V)##.

For example in the case of ideal gas which are the 3 equations ? Thanks.

What are your thoughts on this? Please also articulate your understanding of the definition of a "state equation."

Chestermiller said:
Please also articulate your understanding of the definition of a "state equation."
For an ideal gas, I'm aware of there are two state equations, namely $$pV=nRT$$ and $$U=\frac 3 2 nRT$$ From that thread it should be another equation in which enters the entropy ##S##. What is this third equation ?

Last edited:
cianfa72 said:
For an ideal gas, I'm aware of there are two state equations, namely $$pV=nRT$$ and $$U=\frac 3 2 nRT$$ From that thread it should be another equation in which enters the entropy ##S##. What is this third equation ?
How about $$dS=\frac{3}{2}nR\frac{dT}{T}+nR\frac{dV}{V}$$

Yes, integrating it we get $$S=\frac 3 2 nR \,lnT + nR\, lnV$$

That cannot be true, because you have dimensionful quantities in the logarithm. The correct Sackur-Tetrode formula for the entropy of an ideal gas is
$$S=\frac{5}{2} k_{\text{B}} N +k_{\text{B}} N \ln \left [ \frac{V}{N} \left (\frac{m U}{3 \pi \hbar^2 N} \right)^{3/2}\right].$$

vanhees71 said:
That cannot be true, because you have dimensionful quantities in the logarithm
Sorry, we cannot simply integrate the differential form in post #4 ?

[EDIT: Correted typos in formulae in view of #9]

From this you can only get the entropy differences, i.e.,
$$S-S_0=\frac{3}{2} n R \ln(T/T_0)+n R \ln(V/V_0)=n R \ln \left [\frac{V}{V_0} \left (\frac{T}{T_0} \right)^{3/2}\right] .$$
Now ##U=3 N k_{\text{B}} T/2## and ##n R=N k_{\text{B}}##. So you can write the above result as
$$S-S_0=N k_{\text{B}} \ln \left [\frac{V}{V_0} \left (\frac{U}{U_0} \right)^{3/2} \right].$$
Thus this is, of course, consistent with the Sackur-Tetrode formula for the absolute entropy, but the latter can only be derived by semi-classical quantum considerations, not from phenomenological classical thermodynamics.

You need in addition to the "classical fundamental Laws 0-2 of thermodynamics" also Nernst's theorem (3rd Law) as well as the indistinguishability of particles and the "natural measure" for phase-space volumes, which is determined by QT in terms of Planck's action constant, ##h=2 \pi \hbar##.

For details of a semi-classical argument for the entropy, see Sect. 1.5 in

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf

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vanhees71 said:
So you can write the above result as
$$S-S_0=N k_{\text{B}} \ln \left [\frac{V}{V_0} \left (\frac{U}{U_0} \right) \right].$$
From what you said, it should be actually: $$S-S_0=N k_{\text{B}} \ln \left [\frac{V}{V_0} \left (\frac{U}{U_0} \right)^{\frac 3 2} \right]$$

Last edited:
vanhees71
Of course. I correct it in the original posting.

## What is a state equation in thermodynamics?

A state equation in thermodynamics, also known as an equation of state, is a mathematical equation that describes the state of a thermodynamic system by relating its state variables, such as temperature, pressure, and volume. One of the most well-known examples is the Ideal Gas Law, which relates these variables for an ideal gas.

## Why are state equations important in thermodynamics?

State equations are crucial in thermodynamics because they provide a way to predict the behavior of a system under different conditions. By knowing the relationship between state variables, we can determine how a system will respond to changes in temperature, pressure, volume, and other properties, which is essential for designing and optimizing various processes and equipment.

## What is the Ideal Gas Law?

The Ideal Gas Law is a simple state equation that describes the behavior of an ideal gas. It is expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the universal gas constant, and T is the temperature in Kelvin. This equation assumes that the gas molecules do not interact with each other and that the volume of the gas molecules themselves is negligible.

## What are some common state equations for real gases?

For real gases, which do not behave ideally, more complex state equations are used. Some common ones include the Van der Waals equation, the Redlich-Kwong equation, and the Peng-Robinson equation. These equations account for intermolecular forces and the finite volume of gas molecules, providing a more accurate description of real gas behavior under various conditions.

## Can state equations be applied to liquids and solids?

Yes, state equations can also be applied to liquids and solids, though they are often more complex due to the stronger intermolecular forces and less compressibility. For liquids and solids, state equations such as the Tait equation for liquids and the Birch-Murnaghan equation for solids are used to describe their behavior under different conditions of pressure and temperature.

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