Question

At the critical point for ${{H}_{2}}$ gas, value of Z=3/8, then the value of Z for the similar conditions of $C{{O}_{2}}$,${{O}_{2}}$ at their respective critical points will be :
A) Greater than 3/8
B) Smaller than 3/8
C) Equal to 3/8
D) Nothing can be said.

Answer 1

Hint: The answer here is based on the compressibility factor Z calculated for 1 mole which is given by the formula $Z=\dfrac{PV}{RT}$ and the pressure and volume along with temperature are the values measure at critical points according to Van Der Waals liquefaction of real gases.

Complete answer:
We have studied in our lower classes of chemistry that deals with the concept of the compressibility factor and also the formulas for volume and temperature and also the pressure at critical points according to the Van Der Waals liquefaction formula for real gases.
Let us now revise those formulas which help us to deduce the correct answer.
- Compressibility factor according to thermodynamics means a correction factor which describes the deviation of real gas from the ideal gas behaviour or in simple words, it is the ratio of molar volume of an ideal gas at same temperature and pressure.
Now, according to the Van Der Waals liquefaction of real gases at critical points, we can denote volume, pressure and temperature as${{V}_{c}},{{P}_{c}}$and${{T}_{c}}$ respectively.
Now, ${{V}_{c}}$is given by ${{V}_{c}}$=3b where b is the Van Der Waals constant
Similarly, ${{T}_{c}}$=$\dfrac{8a}{27Rb}$ and ${{P}_{c}}$=$\dfrac{a}{27{{b}^{2}}}$
Substituting this in the compressibility factor equation that is,$Z=\dfrac{PV}{RT}$ we have
\[Z=\dfrac{\left( \dfrac{a}{27{{b}^{2}}} \right)\times 3b}{\left( \dfrac{8a}{27Rb} \right)\times R}=\dfrac{3}{8}\]
Therefore, we can say that even if the individual gases have different values of a and b but at critical conditions it has value 3/8.

Thus, the correct answer is option C) Equal to 3/8.

Note:
Note that compressibility factor is also known as gas deviation factor or compression factor and do not confuse the compressibility factor with that of compressibility (coefficient of compressibility or isothermal compressibility) where compressibility measures relative volume change of solid or fluid in response to a pressure change.