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The average kinetic energy of a gas is directly related to its temperature. When you heat a gas, you're essentially giving its molecules more energy. Imagine these molecules as tiny balls bouncing around.
The faster they move, the more energy they have.
In the kinetic theory of gases, this energy is calculated as:

• Average kinetic energy per molecule = \(\frac{3}{2}kT\)

where \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature in Kelvin.So, when the temperature of the nitrogen gas in the exercise increases from \(293.15\,\text{K}\) to \(523.15\,\text{K}\), the average kinetic energy of all the gas molecules increases. It means every molecule, on average, is moving energetically than before.

The root-mean-square speed (RMS speed) is a measure of how fast molecules in a gas are moving on average. It's a bit like taking the average speed of all these bouncy balls we mentioned earlier.
But instead of just averaging, we consider their speed squared, take an average, and then the square root of that number.
The formula for RMS speed is:

• \(v_{rms} = \sqrt{\frac{3RT}{M}}\)

where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.As the temperature of the gas increases, so does the RMS speed. So, when the temperature in our scenario jumps, these tiny balls move even faster, hence the RMS speed increases too.

The temperature of a gas is a key player in the kinetic theory. It's not just a measurement of heat but it tells us how energetic the molecules in the gas are.
In the formula for average kinetic energy and RMS speed, absolute temperature (measured in Kelvin) is the variable the others are directly dependent on.
When you convert from Celsius to Kelvin for calculations, for instance, you make sure that zero temperature truly represents zero motion. With the N2 gas at higher temperatures in the given problem, the molecules were energized to interact more strongly with their surroundings. This results in not just faster motion but more frequent collisions with the container's walls. That's why gas temperature is such a vital concept—it connects directly to a gas's energy dynamics.