Question

The compressibility factor for a real gas is expressed by $Z = 1 + \dfrac{{PB}}{{RT}}$ . The value of $B$ at $500{\text{ }}K$ and $600{\text{ }}bar$ is $0.0169{\text{ }}L/mol$ . Molar volume of the gas at $500{\text{ }}K$ and $600{\text{ }}bar$ is:
( $bar = 1{\text{ }}atm$ ) ( $R = 0.083{\text{ }}L - atm/mol - K$ )
\begin{array}{*{20}{l}} {A:0.01L} \\\ {B:9 \times {{10}^{ - 5}}L} \\\ {C:8.62 \times {{10}^{ - 2}}L} \\\ {D:1.65L} \end{array}

Answer 1

Hint : The compressibility factor (Z) is a beneficial thermodynamic property for changing the ideal gas law in order to account for behaviour of real gases. It refers to a measure of how much the thermodynamic properties of a real gas vary from that expected from an ideal gas. It may also be estimated as the ratio of the actual volume of a real gas to that volume as predicted by the ideal gas at the similar temperature and pressure as the actual volume.

Complete Step By Step Answer:
Compressibility factor (Z), which is generally defined as $Z = \dfrac{{PV}}{{RT}}$ (wherein P is pressure, V is the molar volume of gas, Z is compressibility factor, R is the universal gas constant and T is temperature), is always unity for an ideal gas. Though in the case of high-pressure region, the expression for the compressibility factor becomes $Z = 1 + \dfrac{{PB}}{{RT}}$ .
In the question, we are provided with the following data:
$B$ = $0.0169{\text{ }}L/mol$
$P = 600{\text{ }}bar$
$R = 0.083{\text{ }}L - atm/mol - K$
$T = 500{\text{ }}K$
Substituting these values, we will calculate the value of ‘Z’ as shown below:
$Z = 1 + \dfrac{{600 \times 0.0169}}{{0.083 \times 500}} = 1.247$
We know that $Z = \dfrac{{PV}}{{RT}}$
From this relation, we will calculate the value of molar volume as shown below:
$\begin{gathered} V = \dfrac{{1.247 \times 0.083 \times 500}}{{600}} \\\ = 0.0862 = 8.62 \times {10^{ - 2}}L \\\ \end{gathered}$
Hence, the correct answer is Option C.

Note :
The compressibility factor should not be confused with the coefficient of isothermal compressibility. In most engineering works, the compressibility factor is generally employed as a correction factor to ideal behaviour. Therefore, ${v_{real}} = Z{v_{id}}$ is employed to determine the actual volume, ${v_{real}}$ by multiplying the compressibility factor with the ideal gas volume, at the same temperature and pressure.