Question

Compressibility factor of ${{H}_{2}}$ behaving as real gas is
A) 1
B) \[\left( 1-\dfrac{a}{RTV} \right)\]
C) \[\left( 1+\dfrac{Pb}{RT} \right)\]
D) $\dfrac{RTV}{(1-a)}$

Answer 1

Hint: The answer to this question includes calculation using the Vander-Waals equation which is given by the formula,$\left( P+\dfrac{a}{{{V}^{2}}} \right)\times \left( V-b \right)=RT$ for 1 mole of gas.

Complete step by step solution:
We have learnt from the previous lessons in chemistry that hydrogen behaves as real gases at higher pressure. Thus, at this time the Van Der-Waals constant ‘a’ is negligible and hence the formula for one mole of a gas that is n=1, reduces as shown below,
$\left( P+\dfrac{a}{{{V}^{2}}} \right)\times \left( V-b \right)=RT$ \[\]
\[P+\dfrac{a}{{{V}^{2}}}\] $\approx P$ [At higher pressure]
Therefore, the equation of Van Der-Waals reduces to,$P\times \left( V-b \right)=RT$
Rearranging this equation we get as,
\[\dfrac{P\left( V-b \right)}{RT}=1\]
Solving the brackets of above equation we have,
$\dfrac{PV}{RT}-\dfrac{Pb}{RT}=1$ …….(1)
Since, the compressibility factor is given by the term ‘Z’ which is defined by $Z=\dfrac{PV}{RT}$and substituting this equation in the above equation (1) we get the simplified equation as,
\[Z-\dfrac{Pb}{RT}=1\]
By rearranging this equation we get,
$Z=1+\dfrac{Pb}{RT}$

Thus, the correct option is option C).

Additional information:
-The compressibility factor refers to the thermodynamic property for a real gas to deviate from those expected of an ideal gas. This factor also modifies the ideal gas law to account for behaviour of real gases. For an ideal gas the compressibility factor always has the value of unity.
-At higher temperature, the compressibility factor will be greater than unity. This is because the kinetic energy of the molecules increases and it is due to this that they tend to move apart. Therefore, the actual volume of gas will be comparably more than that of the ideal volume.

Note: The Vander-Waals equation for ideas gas has formula $PV=nRT$and while solving the equation , the real gas equation has Van Der-Waals constants and the correct formula is to be used based on what question is being asked.