Question

The compressibility factor for a real gas at high pressure is:
A.$1 + \dfrac{{RT}}{{Pb}}$
B.1
C.$1 + \dfrac{{Pb}}{{RT}}$
D.$1 - \dfrac{{Pb}}{{RT}}$

Answer 1

Compressibility factor of a gas is the measure of the deviation of the gas from ideal to real behavior. It is denoted by ‘Z’ and is given by the formula,
$\Rightarrow$ $Z = \dfrac{{PV}}{{RT}}$
And for real gases the gas equation with pressure and volume correction terms called the Van der Waals equation is,
$\Rightarrow$ $(P + \dfrac{a}{{{V^2}}})(V - b) = RT$

Complete step by step answer:
Van der Waals equation for real gases is,
$\Rightarrow$ $(P + \dfrac{a}{{{V^2}}})(V - b) = RT$
The term $\dfrac{a}{{{V^2}}}$ is for pressure correction and $b$ is for volume correction. At very high pressure, P is so large that the term $\dfrac{a}{{{V^2}}}$ holds negligible importance and therefore, it can be ignored.
So, the Van der Waals equation can be written as,
$P(V - b) = RT$
$\Rightarrow PV - Pb = RT$
Dividing both sides by RT,
$\Rightarrow \dfrac{{PV - Pb}}{{RT}} = \dfrac{{RT}}{{RT}}$
$\Rightarrow \dfrac{{PV}}{{RT}} - \dfrac{{Pb}}{{RT}} = 1$
$\Rightarrow \dfrac{{PV}}{{RT}} = 1 + \dfrac{{Pb}}{{RT}}$ …(eq. $1$)
The compressibility factor is defined as the ratio of the actual molar volume of a gas to the calculated or theoretical molar volume of the gas under identical conditions. Or it can also be defined as the ratio of volume occupied by a real gas to the volume occupied by an ideal gas under identical conditions. So, $Z = \dfrac{{{V_{real}}}}{{{V_{ideal}}}}$.
For one mole of an ideal gas, $P{V_{ideal}} = RT$
Substituting the value of ${V_{ideal}}$ in the above equation,
$\Rightarrow$ $\dfrac{{P{V_{real}}}}{Z} = RT$
$\Rightarrow Z = \dfrac{{P{V_{real}}}}{{RT}}$ …(eq. $2$)
Substituting the value of Z from eq. $2$ into eq. $1$.
$\Rightarrow$ $Z = 1 + \dfrac{{Pb}}{{RT}}$.

Therefore, C is the correct answer.

Note:
When gases deviate from ideal to real behavior, the pressure and volume corrections terms are added because unlike ideal gases, molecules of real gases interact with each other which causes the pressure on the gas to increase. Due to this the pressure P changes to $P + \dfrac{{a{n^2}}}{{{V^2}}}$.
Also, the gas does not occupy the entire volume of the container as the gas molecules occupy some volume too. Thus, the actual volume occupied by the gas is slightly less and is given by, $V - nb$.