Conditions for Real Gas to Behave Ideally

To determine the conditions under which a real gas behaves like an ideal gas, we can consider the assumptions of the Kinetic Molecular Theory of Gases that define ideal behavior:

1. The volume occupied by the individual gas molecules is negligible compared to the total volume of the gas container.
2. The intermolecular forces of attraction between the gas molecules are negligible.

For a real gas, the equation of state is given by the Van der Waals equation (for $1\text{ mole}$ of gas):
$\left(P + \frac{a}{V_{\text{m}}^2}\right)(V_{\text{m}} - b) = RT$

where:

• $V_{\text{m}}$ is the molar volume of the gas,
• $a$ accounts for the intermolecular attractive forces,
• $b$ accounts for the finite size (volume) occupied by the gas molecules.

For the real gas to obey the ideal gas equation $PV_{\text{m}} = RT$:

1. The term $\frac{a}{V_{\text{m}}^2}$ must be negligible compared to $P$, meaning the intermolecular forces are negligible.
2. The co-volume $b$ must be negligible compared to $V_{\text{m}}$, meaning the volume of the molecules is negligible.

Both of these conditions are satisfied when the molar volume $V_{\text{m}}$ is very large ($V_{\text{m}} \gg b$ and $\frac{a}{V_{\text{m}}^2} \to 0$).

According to the ideal gas law, the molar volume $V_{\text{m}}$ increases as:

• Pressure ($P$) becomes low: At low pressure, the gas expands, the molecules are far apart, and the intermolecular forces become negligible.
• Temperature ($T$) becomes high: At high temperature, the average kinetic energy of the molecules is very large compared to the intermolecular potential energy, and the gas expands to occupy a larger volume.

Thus, a real gas behaves like an ideal gas when its pressure is low and temperature is high.

Correct Option: D