States of Matter - Result Question 11
11. If $Z$ is a compressibility factor, van der Waals’ equation at low pressure can be written as
(2014 Main)
(a) $Z=1+\frac{R T}{p b}$
(b) $Z=1-\frac{a}{V R T}$
(c) $Z=1-\frac{p b}{R T}$
(d) $Z=1+\frac{p b}{R T}$
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Answer:
Correct Answer: 11. (b)
Solution:
To solve this problem, the stepwise approach required, i.e.
(l) Write the van der Waals’ equation, then apply the condition that at low pressure, volume become high,
$ \text { i.e. } \quad V-b \simeq V $
(ii) Now calculate the value of compressibility factor $(Z)$. $[Z=p V / R T]$
According to van der Waals’ equation,
$ \left(p+\dfrac{a}{V^{2}}\right)(V-b)=R T $
At low pressure, $\left(p+\dfrac{a}{V^{2}}\right) V=R T$
$ \Rightarrow \quad p V+\dfrac{a}{V}=R T \text { or } p V=R T-\dfrac{a}{V} $
Divide both side by $R T, \dfrac{p V}{R T}=1-\dfrac{a}{R T V}$
BETA
