Problem — True or false on gas properties

Problem

True or false?

• (a) In an ideal gas the molecules interact with each other at a distance via elastic forces.
• (b) The mean translational kinetic energy of a molecule depends only on temperature.
• (c) All molecules of a gas at the same $T$ have the same speed.
• (d) The pressure of a gas is due to the molecules colliding with the walls.
• (e) Absolute zero corresponds to zero mean speed of the molecules.

Solution

(a) False. In the ideal-gas model the molecules are point-like and do not interact at a distance: they exchange energy and momentum only in (instantaneous) collisions with the walls and with each other. Forces at a distance appear only in real gases (Van der Waals).

(b) True. The mean translational kinetic energy is $\bar E = \tfrac{3}{2}k_B T$: it depends only on the absolute temperature, not on mass or species.

(c) False. At a given $T$ the speeds are distributed according to Maxwell-Boltzmann: slow and very fast molecules coexist. The temperature fixes only the mean value (of $v^2$), not a single value.

(d) True. Macroscopic pressure is the average of the countless molecular collisions on the walls: each collision transfers momentum, and the average force per unit area is the pressure.

(e) True. In the classical model, absolute zero ($T=0$ K) corresponds to the vanishing of thermal agitation, i.e. zero (mean) speed. It is the limit in which $\bar E = \tfrac{3}{2}k_B T \to 0$. $\ev{\text{F, T, F, T, T}}$

Links

Topics: Kinetic theory of gases Concepts: Temperature · Pressure Skills: Micro-macro interpretation