MCQ Detailed View

According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the absolute temperature of the system. This means that as temperature increases, the kinetic energy of the molecules increases in the same ratio. The mathematical relationship is expressed as:

Ek=32kBTE_{k} = \frac{3}{2} k_B TEk=23kBT

where EkE_{k}Ek is the average kinetic energy per molecule, kBk_BkB is the Boltzmann constant, and $T$ is the temperature in Kelvin.

In the given problem, the gas is initially at 300 K. The vessel is closed, so the number of molecules remains constant, and only the temperature changes. The gas is then heated to 600 K. Applying the proportionality:

E2E1=T2T1\frac{E_2}{E_1} = \frac{T_2}{T_1}E1E2=T1T2 E2E1=600300=2\frac{E_2}{E_1} = \frac{600}{300} = 2E1E2=300600=2

This calculation shows that the average kinetic energy of the molecules at 600 K is exactly double the kinetic energy at 300 K. The reason is that every doubling of absolute temperature results in a doubling of the molecular kinetic energy.

It is important to note that this relationship is only valid when temperature is expressed in the Kelvin scale. If Celsius were used, the calculation would give misleading results. For example, doubling from 27 °C to 327 °C does not represent doubling the absolute temperature, which is why Kelvin must be used in gas law and kinetic energy problems.

The direct proportionality between kinetic energy and absolute temperature helps explain many physical phenomena, such as the increase in gas pressure with heating, diffusion rates of gases, and the behavior of gases in thermodynamic processes. Thus, when a gas in a closed vessel is heated from 300 K to 600 K, the average kinetic energy of its molecules becomes double.