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I was reading up on the derivation of the values of various critical state functions( $\ce{Vc}$, $\ce{pc}$ and $\ce{Tc}$) from the Van der Waals equation (Physical Chemistry by G.W. Castellan). There, he was about to express a and b in terms of these critical state functions, when I came across this line:

Since experimentally it is hard to determine $\ce{Vc}$ accurately, it would be better if a and b could be obtained from $\ce{pc}$ and $\ce{Tc}$ only.

What is exactly the difficulty in finding $\ce{Vc}$ accurately? I found this source stating that:

Of the three constants, measurement of critical temperature and critical pressure is very easy. The measurement of critical volume is very difficult because it varies considerably for slightest variation of temperature and pressure near the critical region; hence for the measurement of critical volume, the substance must be kept in perfectly critical temperature and pressure.

My two cents

If $\ce{Vc}$ is so short-lived and unstable, then any variation in the same would almost instantly affect the vapour pressure of the gas phase above it, and thereby $\ce{pc}$ would also not be a very reliably measured quantity. Then maybe we would end up with two variables(a and b) and one equation($\ce{Tc}$).

Can somebody explain the procedure of measurement of critical state function here, and why only $\ce{Vc}$ causes problems,and the other two remain appreciably accurate?

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  • $\begingroup$ Around the critical point, a variation in volume has very little effect on pressure. Related: chemistry.stackexchange.com/questions/97671/… $\endgroup$
    – Ivan Neretin
    Commented Feb 18, 2020 at 20:34
  • $\begingroup$ @IvanNeretin Yes $\endgroup$
    – Yusuf Hasan
    Commented Feb 18, 2020 at 20:38
  • $\begingroup$ Well, here we go: the first and second derivatives are both zero, hence... $\endgroup$
    – Ivan Neretin
    Commented Feb 18, 2020 at 20:40
  • $\begingroup$ @IvanNeretin Ok, I think I got the idea, that since the situation is a saturated vapour pressure, volume change doesn't change the limiting value of pressure,I feel. But then, why is it saturated vapours at critical point? $\endgroup$
    – Yusuf Hasan
    Commented Feb 18, 2020 at 20:42
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    $\begingroup$ It is not about the saturated vapor. It is about the critical point, which (in a way) is even worse. The volume is "unstable" precisely for the reason that the pressure depends on it very little. As for reaching the critical point, well, see, there is no road sign that says "CRITICAL POINT" in huge letters made with reflective paint. How do you know whether you are there? $\endgroup$
    – Ivan Neretin
    Commented Feb 18, 2020 at 20:53

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