Question

The given graph represents the variation of compressibility factor $ZvsP$ for three gases $A,B$ and $C$.


Identify the incorrect statements.

• A. For gas $A,a=0$ and its dependence on $P$ is linear at all pressures.
• B. For gas $B,b=0$ and its dependence on $P$ is linear at all pressures.
• C. For gas $C$, which is a typical real gas, neither $a$ nor $b=0$. By knowing the minima and power of intersection with $Z=1,a$ and $b$ can be calculated.
• D. At high pressure the slope is positive for all real gases.

Answer 1

The correct option is B For gas $B,b=0$ and its dependence on $P$ is linear at all pressures.

Compressibility factor, $Z=\frac{PV}{nRT}$
where
$P=$ Pressure,
$V=$ Volume
$n=$ Number of moles
$T=$ Temperature
$R=$ Ideal gas constant
At high pressure, volume occupied by a gas is very small. We have the Vander wall equation as
$(p+\frac{a}{V^2})(V-b)=RT$ (Let $n=1$)
At high value of pressure, $b$ cannot be neglected but $\frac{a}{V^2}$ can be neglected as compared to $P$.
$\Rightarrow \frac{a}{V^2}<<P$
$\Rightarrow P(V-b)=RT$
$\Rightarrow PV=RT+Pb$
$\Rightarrow \frac{PV}{RT}=1+\frac{Pb}{RT}$
$\Rightarrow Z=1+\frac{Pb}{RT}$.
This tells us that if $Z>1$, then it should increase linearly with pressure so option (b) is wrong.