Question

At critical temperature, pressure and volume. The compressibility factor $(Z)$ is:
$(1)\dfrac{8}{3}$
$(2)\dfrac{5}{3}$
$(3)\dfrac{3}{5}$
$(4)\dfrac{3}{8}$

Answer 1

This problem will be solved by using the formula for compressibility factor. As a special case, all the values of temperature, pressure and volume will be replaced by the values of these quantities which are present in the critical condition.

Complete step by step solution:
Compressibility factor is defined as the correction factor which is used to describe the deviation of a real gas from hypothetical ideal gas behaviour.
This is a very useful thermodynamic property which is used to modify the ideal gas law to compensate for the behaviour of real gases. It can also be defined as the measure of how much the thermodynamic properties of a real gas change from those which were expected of an ideal gas.
The formula for the compressibility factor of any gas is given by,
$Z = \dfrac{{PV}}{{nRT}}$
We know that at the critical temperature,
${T_c} = \dfrac{{8a}}{{27bR}}.......(1)$
Similarly, at critical pressure,
${P_c} = \dfrac{a}{{27{b^2}}}.....(2)$
Also, at critical volume,
${V_c} = 3b......(3)$
So, at the critical condition,
$Z = \dfrac{{{P_c}{V_c}}}{{R{T_c}}}$
On putting the value of equation (1), (2) and (3) in the formula of compressibility factor for te critical condition, we get,
$Z = \dfrac{a}{{27{b^2}}} \times \dfrac{{3b}}{R} \times \dfrac{{27bR}}{{8a}}$
On cancelling the terms in the numerator and the denominator, we get,
$Z = \dfrac{3}{8}$
So, the final answer is $(4)\dfrac{3}{8}$.

Note:
It is important to note that the compressibility factor must not be compared to compressibility. These two terms are completely different from each other. Compressibility is basically defined as the measure of the change in volume which is a result of the external pressure applied to the surface of a particular object. On the other hand, the compressibility factor is defined as the correction factor which is used to describe the deviation of a real gas from hypothetical ideal gas behaviour.