When does a gas deviate the most from its ideal behavior?

For an ideal gas we use the familiar equation $$PV = nRT$$, which is based on two crucial assumptions: (i) the molecules themselves occupy a negligible volume, and (ii) there are no intermolecular attractive or repulsive forces. Whenever these two assumptions begin to fail, the gas shows non-ideal or real-gas behaviour.

To see how these assumptions break down, we recall the van der Waals equation, which is the first correction applied to the ideal-gas law:

$$\left(P + \frac{a n^2}{V^2}\right)\,(V - nb) = nRT.$$

Here, $$\frac{a n^2}{V^2}$$ compensates for attractive forces between molecules, and the term $$nb$$ corrects for the finite molecular volume. Let us now discuss how pressure and temperature influence these two corrections one by one.

First, look at the attractive-force correction $$\frac{a n^2}{V^2}$$. The magnitude of this term increases when the distance between the molecules becomes smaller, i.e. when the volume $$V$$ becomes small. A small volume is usually produced by a high pressure. Furthermore, low temperature decreases the kinetic energy of the molecules, allowing attractive forces to act for a longer time. Thus at a low temperature the attractive-force correction grows more important. So, attractions become most significant at high pressure and low temperature.

Now consider the volume correction $$nb$$. This is a direct measure of how much space the molecules themselves occupy. When the external pressure is high, the total volume $$V$$ of the sample becomes comparable to $$nb$$, and the assumption of “negligible molecular volume” fails badly. Hence, again, a high pressure exaggerates this deviation. The role of temperature in $$nb$$ is indirect, but at a low temperature the gas cannot easily overcome the limit set by its own molecular size, so the deviation persists.

Combining both effects, we see that both corrections of the van der Waals equation become simultaneously the largest under one common set of conditions:

$$\text{High pressure} \; \& \; \text{Low temperature}.$$

Under such conditions (high $$P$$, low $$T$$) the ideal-gas equation $$PV = nRT$$ gives the poorest prediction, whereas the real-gas equation must be used. Conversely, at low pressure and high temperature the corrections tend to vanish and the gas becomes almost ideal.

Therefore, a gas deviates most from ideal behaviour at high pressure and low temperature.

Hence, the correct answer is Option D.