The molecules of a given mass of a gas have RMS speed of \(200 \mathrm{~m} / \mathrm{s}\) at \(300 \mathrm{~K}\) and \(1,00,000\) bar pressure. When the absolute temperature is doubled and the pressure is halved, the RMS speed of molecules will become (a) \(200 \mathrm{~m} / \mathrm{s}\) (b) \(400 \mathrm{~m} / \mathrm{s}\) (c) \(100 \mathrm{~m} / \mathrm{s}\) (d) \(200 \sqrt{2} \mathrm{~m} / \mathrm{s}\)

The kinetic theory of gases is a fundamental principle that helps us understand the behavior of gases at the molecular level. This theory is grounded on the assumption that gases consist of many small particles which are in constant, random motion.

According to this theory, the pressure exerted by a gas is a result of collisions between the gas molecules and the walls of their container. The temperature of the gas reflects the average kinetic energy of these molecules. Therefore, when we say that a gas is at a higher temperature, it implies that, on average, its molecules are moving faster. The RMS (root-mean-square) speed is especially important because it relates to the speed at which these molecules travel. It is the square root of the average of the squares of the velocities of each molecule, providing an effective measure of their speed.

In our case study, the RMS speed was initially given, along with the gas's temperature and pressure. To determine how changes in conditions affect the RMS speed, the kinetic theory of gases provides the necessary concepts, as it suggests that such speed depends on the temperature and is independent of pressure.

The Boltzmann constant, symbolized as 'k', is a crucial bridge connecting the macroscopic and microscopic worlds, linking variables that describe gases at a scale perceptible to humans, like temperature, to the energy levels observable only at the molecular or atomic scale.

This constant appears in many fundamental equations in physics and chemistry, including the one for RMS speed of gas molecules: \( v = \sqrt{\frac{3kT}{m}} \). Here, 'T' stands for absolute temperature, and 'm' is the mass of an individual gas molecule. With the Boltzmann constant playing a role in this relationship, it enables us to calculate the RMS speed, providing us a way to translate the average kinetic energy of gas molecules into a temperature we can measure.

Understanding the role of the Boltzmann constant is vital as it allows us to make predictions about the behavior of gas molecules. For example, if we know the absolute temperature and the mass of the gas molecules, we can estimate their average speed.

Absolute temperature, usually measured in kelvins (K), is a scale that relates directly to the energy contained within particles. It’s absolute because it starts from absolute zero, where theoretically, molecular motion ceases.

In relation to gas behavior, temperature plays a key role; it determines the kinetic energy of the gas molecules, which in turn affects their movement speed. The RMS speed of gas molecules increases with the square root of absolute temperature, illustrating that a higher temperature means faster moving molecules.

The exercise highlighted a scenario where the gas temperature is doubled, leading to an increase in RMS speed by a factor of the square root of two. It's essential to note that this increase in speed is driven solely by the temperature change and not influenced by the halving of the pressure, affirming that in kinetic theory, the RMS speed depends on temperature and not pressure for a given mass of gas.