B.1 Thermal energy transfers

The question asks for the microscopic meaning of temperature, which is a fundamental definition in thermal physics.

Temperature is defined as a measure of the average random kinetic energy of the particles in a substance. This is directly expressed in the equation ${\bar{E}}_k=\frac{3}{2}{k}_{B}T$ where $T$ is the absolute temperature.

Temperature does not account for the total energy of a system. Internal energy includes both kinetic and potential energy of all particles. Temperature only reflects the average kinetic component.

The correct answer is 'The average random kinetic energy of the particles in a substance', which precisely matches the microscopic definition of temperature.

The question asks what temperature measures microscopically — we need to eliminate options that describe other thermal physics concepts.

This describes internal energy, not temperature. Internal energy sums both kinetic and intermolecular potential energy contributions across all particles, which is a different quantity.

This describes the concept of heat flow or thermal power, not temperature. Temperature is a state property, not a rate of transfer.

Collision frequency is related to particle density and speed, but it is not the definition of temperature. A gas could have many slow collisions at low temperature.

The remaining option, 'The average random kinetic energy of the particles in a substance', correctly defines temperature at the microscopic level.

The average random kinetic energy is given by ${\bar{E}}_k=\frac{3}{2}{k}_{B}T$ where $T$ must be in kelvin. Since ${\bar{E}}_k\propto T$, the ratio of kinetic energies equals the ratio of absolute temperatures.

Initial temperature: $T_1=27+273=300\text{ K}$. Final temperature: $T_2=327+273=600\text{ K}$.

$\frac{{\bar{E}}_{k2}}{{\bar{E}}_{k1}}=\frac{T_2}{T_1}=\frac{600}{300}=2$ The average kinetic energy doubles.

The average random kinetic energy doubles when the absolute temperature doubles. This illustrates why Celsius temperatures cannot be used directly in this equation.

The question asks for the factor of change in average kinetic energy. The key insight is that ${\bar{E}}_k\propto T$ (in kelvin), so we must convert temperatures first.

The temperature clearly increases, so from ${\bar{E}}_k=\frac{3}{2}{k}_{B}T$, the kinetic energy must also increase. This option is incorrect.

This would only be true if $T$ changed by a factor of 12. $T_1=300\text{ K}$ and $T_2=600\text{ K}$, giving a ratio of 2, not 12.

A factor of 4 would require $T_2=4\times T_1=1200\text{ K}$. However $T_2=600\text{ K}$, not 1200 K. This answer likely comes from incorrectly using Celsius: $327/27\approx 12$, not 4 either.

Since $T_2/T_1=600/300=2$, the kinetic energy doubles. The correct answer is 'It doubles'.