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Q.

If the virial equation is written as $Z = 1 + A_{1} P + A_{2} P^{2} + \dots$ for van der Waals' equation then which of the following will be true under limiting state of zero pressure?

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a

$\begin{aligned} A_{1} = (b - \frac{a}{R T}) \cdot \frac{1}{R T} \end{aligned}$

b

$\begin{aligned} Z \to 1 \end{aligned}$

c

$A_{2} = \frac{a}{(R T)^{3}} (2 b - \frac{a}{R T})$

d

$A_{2} = {(\frac{b}{R T})}^{2} - A_{1}^{2}$

answer is B, D.

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Detailed Solution

Here,

$\begin{array}{l} 1 + \frac{B}{V_{m}} + \frac{C}{V_{m^{2}}} + \dots = 1 + \frac{B . P}{R T Z} + C {(\frac{P}{R T Z})}^{2} + \dots \\ \Rightarrow 1 + (b - \frac{a}{R T}) \cdot \frac{P}{R T Z} + (\frac{b}{R T Z}) \cdot P^{2} + \dots \\ \Rightarrow A_{1} = (b - \frac{a}{R T}) \cdot \frac{1}{R T} \end{array}$

Now,

$A_{2} = {(\frac{b}{R T})}^{2} - (A_{1} - \frac{A_{1}}{Z}) \cdot \frac{1}{P}$

$= {(\frac{b}{R T})}^{2} - A_{1}^{2}$ $(∵ \frac{Z - 1}{P} = A_{1})$

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