An ideal gas $A$ and a real gas $B$ have their volumes increased from $V$ to $2V$ under isothermal conditions. The increase in internal energy
[$AIPMT$ $1993$]

The mean free path of gas molecules depends on $(d =$ molecular diameter$)$

The value of critical temperature in terms of van der Waals' constants $a$ and $b$ is given by

Two ideal gas thermometers $A$ and $B$ use oxygen and hydrogen respectively. The following observations are made:

Temperature	Pressure thermometer $A$	Pressure thermometer $B$
Triple-point of water	$1.250 \times 10^{5} \; Pa$	$0.200 \times 10^{5} \; Pa$
Normal melting point of sulphur	$1.797 \times 10^{5} \; Pa$	$0.287 \times 10^{5} \; Pa$

$(a)$ What is the absolute temperature of the normal melting point of sulphur as read by thermometers $A$ and $B$?
$(b)$ What do you think is the reason behind the slight difference in answers of thermometers $A$ and $B$? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings?

Calculate the value of mean free path $(\lambda)$ for oxygen molecules at temperature $27^{\circ} C$ and pressure $1.01 \times 10^{5} Pa$. Assume the molecular diameter $d = 0.3 nm$ and the gas is ideal. Given Boltzmann constant $k = 1.38 \times 10^{-23} J K^{-1}$. (Result in $nm$)

Under which of the following conditions is the law $PV = RT$ obeyed most closely by a real gas?