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In the book of Kondepudi, Modern Thermodynamics, at page 21, it is given that

Every gas has a characteristic temperature, volume and pressure; $T_c$,

$V_c$, $p_c$ which depend on molecular size and intermolecular forces.In view of this, one can introduce dimensionless reduced variables defined by

$$T_{\mathrm{r}}=\frac{T}{T_{\mathrm{c}}}, \quad V_{\mathrm{r}}=\frac{V_{\mathrm{m}}}{V_{\mathrm{mc}}}, \quad p_{\mathrm{r}}=\frac{p}{p_{\mathrm{c}}}$$

Van der Waals showed that, if his equation is rewritten in terms of these reduced variables, one obtains the following ‘universal equation’ (Exercise 1.18), which is independent of the constants a and b:

$$p_{\mathrm{r}}=\frac{8 T_{\mathrm{r}}}{3 V_{\mathrm{r}}-1}-\frac{3}{V_{\mathrm{r}}^{2}}$$

This is a remarkable equation because it implies that gases have corresponding states: at a given value of reduced volume and temperature, all gasses have the same reduced pressure [...] (**which is not necessarily the value given by the above equation **)

I'm confused, if the above equation is valid for all gases, so that at a given value of reduced volume and temperature, all gasses have the same reduced pressure, how can that value of the pressure not be given by the above equation ?

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  • $\begingroup$ I don't see in the book where the which is not necessarily the value given by the above equation is. Where is that sentence? (Maybe I'm looking at the wrong version). $\endgroup$ – Aaron Stevens Aug 18 at 12:39
  • $\begingroup$ @AaronStevens 2nd edition of the book, almost the last sentence in the page 21 $\endgroup$ – onurcanbektas Aug 18 at 12:43
  • $\begingroup$ @AaronStevens "(which is not necessarily the value given by Equation (1.5.6)"; I edited the original sentence since I didn't put equation numbers $\endgroup$ – onurcanbektas Aug 18 at 12:44
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    $\begingroup$ I can't confirm this, but I bet they are just saying that the model isn't exact, so the experimental value doesn't always match the equation. $\endgroup$ – Aaron Stevens Aug 18 at 12:48
  • $\begingroup$ @AaronStevens that makes sense actually, thanks. By the way, just to be complete: Dilip Kondepudi, Ilya Prigogine, modern Thermodynamics: From Heat Engines to Dissipative Structures, 2nd Edition, Wiley. $\endgroup$ – onurcanbektas Aug 18 at 13:07
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The question is" "if the above equation is valid for all gases, so that at a given value of reduced volume and temperature, all gasses have the same reduced pressure, how can that value of the pressure not be given by the above equation ?"

First, it is is not true that all gasses have the same reduced pressure at the same reduced temperature and pressure. This statement, also known as the principle of corresponding states, is based on the experimental observation that many (but not all) gases have approximately (but not exactly) the same reduced pressure at the same reduced temperature and volume. This is usually the case for symmetric molecules that interact primarily via non-polar interactions such as noble gases, methane and other simple molecules. It breaks down (i.e., the approximation becomes poorer) for polar molecules such as water, ethanol, sulfuric acid and many others.

The molecular explanation is that the behavior of spherical molecules interacting via a Lennard-Jones type of potential (see for example wikipedia) can be described with two independent variables (minimum value of the potential, position at which this minimum appears) which can then be mapped into reduced volume and reduced temperature.

If a class of molecules is found to obey this scaling, that does not imply that the equation of state is the van der Waals equation. The van der Waals equation is one of many that satisfy the principle of corresponding states (other examples are SRK and the Peng-Robinson equations).

Equations that satisfy the principle of corresponding states can be expressed entirely as a relationship between $P_r$, $V_r$ and $T_r$, or as an equation that involves the critical temperature ($T_c$) and critical pressure ($P_c$) as the only substance-specific parameters. The equations I mentioned above (SRK, Peng Robinson) make use of the additional parameter $\omega$ (acentric factor) in an effort to improve their accuracy.

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