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JEE Advanced levelOne or more correct optionsDifficulty 3 of 3rg042

For a gas obeying the van der Waals equation, which of the following are correct at the critical point?

A $(\frac{\partial P}{\partial V _{m}})_{T} = 0$ and $(\frac{\partial ^{2} P}{\partial V _{m}^{2}})_{T} = 0$
B $V_{c} = 3 b$
C $\frac{R T _{c}}{P _{c} V _{c}} = \frac{3}{8}$
D $P_{c} = \frac{a}{27 b ^{2}}$

Answer: A, B, D

The critical point is a horizontal point of inflection on the

P

-

V

isotherm, so both derivatives vanish (A true). Solving these conditions gives

V_{c} = 3 b

,

T_{c} = 8 a /27 R b

,

P_{c} = a /27 b^{2}

(B and D true). The ratio

P_{c} V_{c} / R T_{c} = 3/8

, so its reciprocal

R T_{c} / P_{c} V_{c} = 8/3

, not

3/8

(C false).

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