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.