Question

If $Z$ is a compresibility factor, van der waals equation at low pressure can be written as:

• A. $Z=1+\frac{RT}{Pb}$
• B. $Z=1-\frac{a}{VRT}$
• C. $Z=1-\frac{Pb}{RT}$
• D. $Z=1+\frac{Pb}{RT}$

Answer 1

The correct option is B $Z=1-\frac{a}{VRT}$

Van der waals equation can be given as:
$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$
For $n=1$, the equation can be written as:
$(P+\frac{a}{V^2})(V-b)=RT$
At low pressure, $V>>b$
Hence, the equation becomes
$(P+\frac{a}{V^2})V=RT$
Rearranging this equation, we get
$PV+\frac{a}{V}=RTPV=RT-\frac{a}{V}$
Dividing both sides by RT, we get
$\frac{PV}{RT}=1-\frac{a}{VRT}$ (as $\frac{PV}{RT}=Z$)
$Z=1-\frac{a}{VRT}$