The van der Waals equation written as a cubic in V_m is V_m^3 - (b + (RT)/(P))V_m^2 + (a)/(P)V_m - (ab)/(P) = 0. Which statements are correct?

The sum of the three roots equals b + RT/P

The product of the three roots equals ab/P

At the critical point all three roots are equal to 3b

The sum of the three roots equals b + RT/P

The product of the three roots equals ab/P

At the critical point all three roots are equal to 3b

For T well above T_c the equation has three distinct real positive roots

JEE Advanced levelOne or more correct optionsDifficulty 3 of 3rg187

The van der Waals equation written as a cubic in $V_{m}$ is $V_{m}^{3} - (b + \frac{RT}{P}) V_{m}^{2} + \frac{a}{P} V_{m} - \frac{ab}{P} = 0$ . Which statements are correct?

For a monic cubic

x^{3} + p x^{2} + q x + r = 0

, the sum of roots is

- p

and the product is

- r

; here these give

b + RT / P

and

ab / P

(A and B true). At the critical point the three roots coincide at

V_{c} = 3 b

(C true). Above

T_{c}

only one root is real; three real roots occur only below

T_{c}

(D false).