Question

The compressibility factor Z for real gas at high temperature is:
$\left( A \right)1 \\\ \left( B \right)1 + \dfrac{{Pb}}{{RT}} \\\ \left( C \right)1 - \dfrac{{Pb}}{{RT}} \\\ \left( D \right)1 + \dfrac{{RT}}{{Pb}} \\\ \\\$

Answer 1

As per the Boltzmann distribution at high temperature particles move apart and hence, real gas tends to behave as ideal gas. The compressibility factor is greater than unity. At higher temperature the kinetic energy of a particle increases as the particle moves apart.

Complete step-by-step answer: “Compressibility factor is the ratio of the actual volume of the real gas to the predicted by the ideal gas”. The deviation from the ideal nature can be measured by compressibility factor Z and which is mathematically written as given below.
$Z = \dfrac{{pV}}{{nRT}} - - - - - \left( 1 \right)$
Where, R = Gas constant and T = temperature.
For the real gas equation 1 can be rewrite as: this below equation is known as van der Waals equation.
$\left( {p + \dfrac{{a{n^2}}}{{{V^2}}}} \right)\left( {V - nb} \right) = nRT - - - - \left( 2 \right)$
Where, p= pressure, V= Volume, nb = total volume occupied, n = number of moles,
Where, Constant a and b are called van der waals constants.
a = it is the measure of magnitude of intermolecular attractive forces within the gas
b= is the constant, having taken into account the corrections for pressure and volume.
At high temperature value of Volume is high and hence, $\dfrac{{a{n^2}}}{{{V^2}}}$ can be neglected and number of moles $\left( n \right)$is 1. Hence, equation (2) can be reduced to
$\left( p \right)\left( {V - b} \right) = RT - - - - \left( 3 \right)$
$\Rightarrow pV - pb = RT - - - - \left( 4 \right)$
$\Rightarrow pV = RT + pb$
Now, Divide above equation by RT:
$\Rightarrow \dfrac{{pV}}{{RT}} = 1 + \dfrac{{pb}}{{RT}}$
Hence, $Z = 1 + \dfrac{{pb}}{{RT}}$ $Z = 1 + \dfrac{{pb}}{{RT}}$

Therefore, Option $\left( B \right)$ is the correct answer for the given question. At high temperature the value of compressibility factor Z is more than one.

Note: The behaviour of Real gas depends on the conditions such as temperature and pressure. At high temperature and low pressure, the shift in behaviour of real gas is observed. The real gas behaves as ideal gas at high temperature due to increase in intermolecular distance. The intermolecular forces of attraction are negligible. The compressibility factor is greater than unity.