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So I am having real trouble understanding what equations of state are and how we form them. My issue stems from reading multiple sources. So I understand that an equation of state is used to build a relationship between variables to describe a state of a system.

For example $P=P(V,T)$, where $P=NKT/V$, is a function of state, but then I started reading about Gibbs, Helmholtz, enthalpy, etc and suddenly am very confused.

So looking at Gibbs, for example, we have two equations $$G=E+PV-TS$$ and $$dG=-SdT+VdP$$ but both describe the state of the system. I did not think an equation of state could be a differential equation but according to one book I have read, $dE$ is related to equation of state and this has caused me great confusion.

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I think you might be confusing two different concepts.

  1. Thermodynamic potentials (Helmholtz, Gibbs, enthalpy etc.) describe the available energy of a system subject to certain external factors. A system maintained at constant temperature by a heat bath can be described by the Helmholtz potential ($A=E-TS$). If in addition to this, a pressure can be applied from the outside, Gibbs must be used ($G=E-TS+PV$). If $A$ or $G$ are expressed via the partition function, the differentials tell you how to calculate quantities of interest. In your case,

$$dG/dT=-S$$

  1. Equations of state. These specify the relationship between various measurable properties of the system. Deriving them from the partition function of the system once it is known which potential should be used is possible but complicated.

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Equations of state are literally just equations relating state variables. State variables are values that just depend on the current state of the system and not on how the system got to that state. Contrast this with things like work done on the system or heat leaving the system, which depend on the process a system undergoes.

So, if values in your equation depend on the "path" or history of the process, then it isn't an equation of state.

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An equation of state is simply an equation that shows the relationship between the properties of a system when the system is in equilibrium, that is, when the properties (Temperature, Pressure, Volume, etc.) are not changing in time. For example, the equation of state of an ideal gas is given by

$$PV=NRT$$

Where $P$ is the absolute pressure, $V$ is the volume, $N$ is the number of moles of gas, $T$ is the absolute temperature in deg Kelvin, and R is the universal gas constant.

As has been pointed out by Ole Krarup, Helmlholtz free energy, Gibbs free energy, Enthalpy are not equations of state. They are called thermodynamic potentials. To get more info on these, check out the Hyperphysics web site.

Hope this helps.

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  • $\begingroup$ @Anton Why do you ask, and what does it have to do with my answer? $\endgroup$
    – Bob D
    Sep 13, 2021 at 14:59
  • $\begingroup$ I apologize. I was just curious if we should call them equations or functions. Anyways I will remove the comment as it doesn't relate with your answer. $\endgroup$
    – Anton
    Sep 13, 2021 at 15:09
  • $\begingroup$ @Anton No problem and no need to apologize. Just wasn't sure why you were asking. $\endgroup$
    – Bob D
    Sep 13, 2021 at 15:14
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An equation of state is a relation between intrinsic quantities of a system in equilibrium.

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