States of Matter 1 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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Solution:
- PLAN 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,
$$ \left(p+\frac{a}{V^{2}}\right)(V-b)=R T $$
At low pressure, $\left(p+\frac{a}{V^{2}}\right) V=R T$
$$ \Rightarrow \quad p V+\frac{a}{V}=R T \text { or } p V=R T-\frac{a}{V} $$
Divide both side by $R T, \frac{p V}{R T}=1-\frac{a}{R T V}$
