For an ideal gas, which one of the following statements does not hold true according to the kinetic theory of gases?

Introduction / Context:

The kinetic theory of gases provides a microscopic model for the behaviour of ideal gases. It makes several simplifying assumptions about molecular motion, energy and interactions. Examinations often ask which statements are consistent with this model and which are not. This question targets your understanding of the distribution of molecular speeds and the basic assumptions regarding energy and forces in an ideal gas.

Given Data / Assumptions:

• An ideal gas is being considered.
• The kinetic theory assumptions apply.
• We must identify which statement does not correctly describe an ideal gas.

Concept / Approach:

In an ideal gas, molecules are assumed to be point particles that undergo perfectly elastic collisions and do not exert long range forces on each other. The potential energy of interaction is taken as zero. The molecules have a distribution of speeds described by the Maxwell Boltzmann distribution, even at a fixed temperature. Thus, while the average kinetic energy depends only on temperature, individual molecules have different speeds and kinetic energies at any instant. Therefore, any statement claiming that all molecules have the same speed is inconsistent with kinetic theory.

Step-by-Step Solution:

Verification / Alternative check:

Graphs of the Maxwell Boltzmann distribution show a continuous spread of molecular speeds rather than a single value. Even at constant temperature, there are molecules much faster or slower than the average. Furthermore, in the kinetic theory derivation of pressure, the assumption is not equal speeds but a large number of molecules with a distribution of velocities whose mean square speed relates to temperature.

Why Other Options Are Wrong:

Option B: The kinetic energies of all molecules are not the same is correct because energy varies with speed, and speeds are distributed.

Option C: The potential energy of the gas molecules is zero is an idealisation that the molecules do not have attractive or repulsive potential energy except during collisions, which is part of the ideal gas model.

Option D: There is no interactive force between the molecules beyond brief collisions is again an ideal gas assumption, ignoring intermolecular forces.

Common Pitfalls:

Some students misinterpret the statement that average kinetic energy is proportional to temperature and wrongly assume that every molecule must have exactly that kinetic energy at a given temperature. In reality, temperature relates to the average over many molecules. The key point is that ideal gas theory uses statistical distributions, not identical speeds for all molecules.

Final Answer:

The statement that does not hold true is The speed of all gas molecules is the same at a given temperature.