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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Solution:

  1. 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 $R T, \frac{p V}{R T}=1-\frac{a}{R T V}$



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