Question

At low pressure, the van der Waals' equation is reduced to:

• A. $Z=\frac{P{V}_m}{RT}=1-\frac{aP}{RT{V}_m}$
• B. $Z=\frac{P{V}_m}{RT}=1+\frac{bP}{RT}$
• C. $P{V}_m=RT$
• D. $Z=\frac{P{V}_m}{RT}=1-\frac{a}{RT}$

Answer 1

The correct option is A $Z=\frac{P{V}_m}{RT}=1-\frac{aP}{RT{V}_m}$

At low pressure, the van der Waals' equation is reduced to
$Z=\frac{P{V}_m}{RT}=1-\frac{a}{RT{V}_m}$
The van der Waals' equation is
$(P+\frac{a}{V_m^2})(V_m-b)=RT$
At low pressure, volume is very large and the correction term b (a constant of small value) can be neglected in comparision to very large value of the volume.
$(P+\frac{a}{V_m^2})\left(V_m\right)=RT$
$P{V}_m+\frac{a}{V_m}=RT$
$P{V}_m=RT-\frac{a}{V_m}$
Divide both sides of the equation with RT
$\frac{P{V}_m}{RT}=1-\frac{a}{RT{V}_m}$
But $Z=\frac{P{V}_m}{RT}$
Hence, $\frac{P{V}_m}{RT}=1-\frac{a}{RT{V}_m}$