A van der Waals gas is described by (P + (a)/(V_m^2))(V_m - b) = RT. Which of the following are correct?

Multiplying out gives a cubic equation in V_m for given P and T

Below the critical temperature the cubic can have three real positive roots for suitable P

Multiplying out gives a cubic equation in V_m for given P and T

Below the critical temperature the cubic can have three real positive roots for suitable P

The constant a has no effect on the equation at very high pressure where V_m -> b

The equation predicts that V_m can be smaller than b at very high pressure

JEE Advanced levelOne or more correct optionsDifficulty 3 of 3rg075

A van der Waals gas is described by $(P + \frac{a}{V _{m}^{2}}) (V_{m} - b) = RT$ . Which of the following are correct?

AMultiplying out gives a cubic equation in $V_{m}$ for given $P$ and $T$
BBelow the critical temperature the cubic can have three real positive roots for suitable $P$
CThe constant $a$ has no effect on the equation at very high pressure where $V_{m} \to b$
DThe equation predicts that $V_{m}$ can be smaller than $b$ at very high pressure

Expanding gives

P V_{m}^{3} - (P b + RT) V_{m}^{2} + a V_{m} - ab = 0

, a cubic in

V_{m}

(A true). Below

T_{c}

the isotherm has a loop and the cubic has three real roots for pressures between the vapour pressure limits (B true). As

V_{m} \to b

the term

a / V_{m}^{2}

remains finite and non-zero, so

a

still matters (C false).

V_{m} - b

must stay positive for finite

P

, so

V_{m}

cannot fall below

b

(D false).