Question

The following graph is plotted between compressibility factor (Z) versus pressure of a gas at different temperatures.

Which of the following statements is /are correct?

• A. $T_1$ is more than the Boyle temperature of the gas
• B. $T_2$ is the Boyle temperature of the gas in the range $P_1$ to $P_2$
• C. $T_3$ is higher than the Boyle temperature of the gas where it cannot be liquified by applying any pressure
  
• D. At the Boyle temp, the net attractive force is counterbalanced by the net repulsive force, so there will be no net force and gas obeys the ideal gas equation

Answer 1

Answer: (B), (C), (D)

The correct options are
B $T_2$ is the Boyle temperature of the gas in the range $P_1$ to $P_2$
C $T_3$ is higher than the Boyle temperature of the gas where it cannot be liquified by applying any pressure

D At the Boyle temp, the net attractive force is counterbalanced by the net repulsive force, so there will be no net force and gas obeys the ideal gas equation

From the Virial equation of a real gas at low pressures
Z = 1 + $\frac{1}{V_m}$$\left(b-\frac{a}{\mathrm{RT}}\right)$
At Boyle tempertature $T_b$ = $\frac{a}{\mathrm{Rb}}$
and at this temperature net attractive and repulsive forces cancel out so for any T<$T_b$ graph will go below Z = 1, at T = $T_b$graph will match Z=1 line for a presure range and for T >$T_b$ graph will go above the line Z=1
Now critical temperature is $\frac{8a}{27\mathrm{Rb}}$ which is less than $T_b$ so option (c) is also correct