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$V_m^3-\dfrac{RT+bP}{P}V_m^2+\dfrac{a}{P}V_m-\dfrac{ab}{P}=0$

So, At $T<T_c$ the above equation has three real roots say $V_1,V_2,V_3$, my doubt is what does this mean physically because for some particular pressure and temperature gas must have a fixed volume , so what does these three roots mean.

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    $\begingroup$ 1 of 3 roots for T < Tc is just the artefact of the mathematical model, using the cubic equation, to archive 2 V values for the gas and the liquid phase. Be aware the equation is not much usable near the critical point. Note that multiple phenomena use models where only some of solutions have physical significance. $\endgroup$
    – Poutnik
    Commented May 17, 2022 at 9:38
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    $\begingroup$ Cross posted on Physics SE. $\endgroup$ Commented May 17, 2022 at 16:31
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    $\begingroup$ It is discussed at Wikipedia: van der Waals' equation: Maxwell equal area rule $\endgroup$
    – Poutnik
    Commented May 18, 2022 at 7:17

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The van der Waals equation describes the isotherms of a gas with respect to pressure and volume. The first van der Waals root (the first solution of the cubic equation) will give you the dew point, which is the point where the first drop of liquid forms from the gas, the other root of the equation is the so-called bubble point, which is when the first liquid drop of the compound is formed. The intermediate root will always be found in a region where liquid and vapor coexist, but just as it is a region and not a point, that root itself has no physical meaning.

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