On a single plot, qualitatively sketch the distribution of molecular speeds for \((\mathbf{a}) \operatorname{Kr}(g)\) at \(-50^{\circ} \mathrm{C},(\mathbf{b}) \operatorname{Kr}(g)\) at \(0^{\circ} \mathrm{C},(\mathbf{c}) \operatorname{Ar}(g)\) at \(0{ }^{\circ} \mathrm{C}\). [Section \(\left.10.7\right]\)

The Maxwell-Boltzmann distribution is a fundamental concept describing the statistical distribution of speeds among particles in a gas. It is vitally important in the field of thermodynamics and provides insights into how particles' speeds vary at different temperatures.

The distribution curve, when plotted, represents the probability of particles possessing a certain speed at a given temperature. It is characterized by its peak, where the most probable speed is located, and its tails, which indicate the presence of both slow and fast-moving particles. As the temperature of the gas increases, the peak shifts to higher speeds, illustrating that particles have greater kinetic energy and therefore move faster. Conversely, lower temperatures result in slower particle speeds, with the peak shifting to the left.

When we compare the distribution of molecular speeds at different conditions – say, for Krypton gas at two distinct temperatures – we can predict that the curve for the gas at the higher temperature will peak further to the right, signifying higher average speeds due to increased kinetic energy.

Kinetic Molecular Theory (KMT) provides insight into the behavior of particles in gases and underpins the Maxwell-Boltzmann distribution concept. This theory is based on several key assumptions: gases are composed of a large number of small particles that are far apart relative to their size, and these particles are in constant, random motion. The theory postulates that pressure is due to particle collisions with the container walls and that these collisions are perfectly elastic, meaning no kinetic energy is lost.

Under the KMT framework, the temperature of a gas is directly related to the average kinetic energy of its particles. When the temperature rises, the gas particles gain kinetic energy, leading to an increase in their speed, which is reflected in a widening and shifting of the Maxwell-Boltzmann distribution curve. Importantly, the theory also explains how at any given temperature, lighter gas molecules will travel faster on average than heavier ones, due to their lower mass and the influence of kinetic energy distribution.

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It has a significant role in determining the behavior of gas particles, as it directly correlates to the kinetic energy distribution among those particles. For instance, argon has a lower molar mass than krypton, meaning that, at a given temperature, argon's molecules will generally move faster than those of krypton.

The influence of molar mass becomes clear when considering its impact on gas behavior under the same environmental conditions. Lower molar mass gases will have a distribution curve that peaks at higher speeds and is narrower, indicating a higher average speed and a tighter speed range. On the other hand, gases with higher molar masses will exhibit a broader distribution, with a peak at slower speeds, as seen with krypton in the given exercise.