Question

Which of the following options is correct about the plot between compressibility factor $Z$ and $\frac{1}{T}$? (T= temperature in $K$)

• A. At high pressure, slope= $-\frac{a}{V_{m}R},Z<1$
• B. At low pressure, slope= $\frac{a}{V_{m}R},Z>1$
• C. At low pressure, slope= $-\frac{Pb}{R},Z<1$
• D. At high pressure, slope= $\frac{Pb}{R},Z>1$

Answer 1

The correct option is D At high pressure, slope= $\frac{Pb}{R},Z>1$

According to Van der Waals' equation:
$(P+\frac{a}{V_m^2})(V_m-b)=RT$
When the pressure is high the above equation can be written as,
$\left(P\right)(V_m-b)=RT$
$\frac{P{V}_m}{RT}=1+\frac{Pb}{RT}$
$Z=1+\frac{Pb}{RT}$
So,
$Z>1andslope=\frac{Pb}{R}$

When the pressure is low Van der Waals' equation can be written as:
$(P+\frac{a}{V_m^2})\left(V_m\right)=RT$
$(P{V}_m+\frac{a}{V_m})=RT$
$\frac{P{V}_m}{RT}=1-\frac{a}{V_{m}RT}$
So,
$Z<1andslope=\frac{-a}{V_{m}R}$