Question

The formula for compressibility of a gas(z) is
A) $\dfrac{{PV}}{{nRT}}$
B) $\dfrac{1}{P}\dfrac{{dV}}{{dP}}$
C) $V\dfrac{{dP}}{{dV}}$
D) $\dfrac{1}{V}\dfrac{{dV}}{{dP}}$

Answer 1

The value of the compressibility factor of a gas is used to classify it as a less compressible gas or a more compressible gas. Thus the compressibility of a gas refers to the ratio of the volume of the real gas to the volume of the ideal gas. The volume of the ideal gas can be obtained from the ideal gas equation.

Formulas used:
-The compressibility of a gas is given by, $z = \dfrac{{{V_{real}}}}{{{V_{ideal}}}}$ where ${V_{real}}$ is the volume of the real gas and ${V_{ideal}}$ is the volume of the ideal gas.
-The ideal gas equation is given by, $PV = nRT$ where $P$ is the pressure of the ideal gas, $V$ is the volume of the ideal gas, $n$ is the number of moles present in the sample of the ideal gas, $R$ is the universal gas constant and $T$ is the temperature of the ideal gas.

Complete step by step solution:
Step 1: Based on the definition of the compressibility of a gas express the general formula for the compressibility of the gas.
The compressibility of a gas can be considered as the amount of deviation of the real gas from an ideal gas. Compressibility factor is denoted by $z$ and it will be the ratio of the volume of the real gas to that of an ideal gas.
i.e., $z = \dfrac{V}{{{V_{ideal}}}}$ -------- (1) where $V$ is the volume of the real gas and ${V_{ideal}}$ is the volume of the ideal gas.
Step 2: Express the relation for the volume of an ideal gas from the ideal gas equation.
The ideal gas equation is expressed as $P{V_{ideal}} = nRT$ where $P$ is the pressure of the ideal gas, ${V_{ideal}}$ is the volume of the ideal gas, $n$ is the number of moles present in the sample of the ideal gas, $R$ is the universal gas constant and $T$ is the temperature of the ideal gas.
From the above equation, we have ${V_{ideal}} = \dfrac{{nRT}}{P}$ ------- (2)
Now substituting equation (2) in (1) we get, $z = \dfrac{V}{{\left( {\dfrac{{nRT}}{P}} \right)}} = \dfrac{{PV}}{{nRT}}$
Thus the formula for the compressibility factor of a gas is obtained as $z = \dfrac{{PV}}{{nRT}}$ .

So the correct option is A.

Note: For an ideal gas, the value of compressibility $z$ will be one as the volume of the real gas and the ideal gas will be the same. However, for a real gas, the value of compressibility will not be one i.e., $z \ne 1$ . If the value of $z$ of some gas is such that $z < 1$ , then that gas is said to be more compressible whereas if $z > 1$ then the gas will be less compressible. At low pressures, all gases are considered to be ideal gases.