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Given:

  • $V \propto 1/P$
  • $V \propto T$
  • $V \propto n$

Prove that:

$V \propto \frac{nT}{P}$

Here is the intuitive argument I have: We can see that

  1. $V = \langle \text{mystery meat} \rangle  1/P$
  2. $V = \langle \text{mystery meat} \rangle  T$, and
  3. $V = \langle \text{mystery meat} \rangle  n$

but it's possible to interpret it so that the mystery meat in (1) is actually equal to $nT$, and the mystery meat in (2) is equal to $\frac{n}{P}$, and the mystery meat in (3) is equal to $\frac{T}{P}$ and so it ends up looking like the blind men touching the elephant.

...But I don't know how to make this intuitive argument formal. How would this be turned into a formal proof?

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1 Answer 1

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In my humble opinion, I think that we must start with historical data

  • In year $1662$, Boyle experimentally observed that, at constant temperature, the product $PV$ is almost a constant

  • In year $1787$, Charles experimentally observed that, at constant pressure, the change of volume is proportional to the change of temperature.

  • In year $1834$, Clapeyron combined the two facts and proposed $PV=kT$ and, at that time, $k$ was supposed to be component dependent before becoming the universal $R$.

  • Volume is an extensive property.

These are experimental facts which have been turned into the first equation of state ($PV=RT$ is not a law).

It is from here that all theoretical work started.

If you are concerned by the next steps, I could add (even a lot !).

Edit

Very early, thermodynamicists point ted out that this was not good since $PV=RT$

  • first aof all, traduces only repulsion between molecules (and, if there is repulsion, there is attration somewhere !)

  • would lead to a zero volume at constant $T$ and infinite $P$ which is not possible.

Already, in year $1865$, Dupré introduced the concept of the covolume (the limit of volume at infinite pressure and constant temperature) and proposed o rewrite $$P(V-b_T)=RT$$

Back to experimantal, in these years, in order to have nice plots, physicists used to plot pressure as polynomials of the reciprocal of volume $$P=\sum_{n=1}^p \frac{a_n(T)}{V^n}$$ which was already called as virial expansion.

Back to physics, the jey year is $1873$ when Van der Waals combined all of the above and proposed $$\large\color{blue}{P=\frac{RT}{V-b}-\frac a{V^2}}$$ which, expanded as a series, gives $$P=\frac {R T}V-\frac{a-b R T}{V^2}+RT \sum_{n=3}^\infty \frac{b^{n-1}}{V^n}$$ which, written as $$P=\frac {R T}V+\sum_{n=2}^\infty \frac{A_n(T)}{V^n}$$ is the standard form of the virial equation of state.

Van der Waals equation of state is so simple that all modern developments of cubic equations of state started from here.

To be continued if you wish.

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    $\begingroup$ Thank you for clarifying. And yes, I am concerned by the next steps. $\endgroup$
    – Fomalhaut
    Nov 1, 2023 at 11:50
  • $\begingroup$ Please excuse me, but this is flying waay over my head. I thought that the simple change of the = symbol to the $\propto$ symbol would result in a simple argument. I didn't expect to become embroiled in this type of deep magic in the process of combining three proportional relationships. I came up with an intuitive argument that sounds correct, and I was hoping that perhaps there would be a simple geometric argument or algebraic technique that would make that argument formal. $\endgroup$
    – Fomalhaut
    Nov 2, 2023 at 13:53

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