Explain the difference between average speed and root-meansquare speed. Which is larger for a given gas sample at a fixed temperature?

The kinetic molecular theory (KMT) provides a conceptual model for understanding the behavior of gases at the molecular level. It's based upon several key postulates:

• Gases consist of many tiny particles (molecules) in constant, random motion.
• These particles are so small compared to the distances between them that the volume of the individual molecules can be assumed to be negligible.
• The collisions between gas particles and between particles and the container walls are perfectly elastic, meaning there is no net loss of kinetic energy during collisions.
• There are no forces of attraction or repulsion between the particles.
• The average kinetic energy of gas particles is directly proportional to the gas temperature in kelvins.

This last point relates directly to the concepts of average speed and root-mean-square speed. Temperature serves as a measure of the average kinetic energy of the particles, so as temperature increases, so does the kinetic energy and consequently the speed of the gas molecules. Understanding KMT is crucial for interpreting why certain properties of gases behave the way they do and for making sense of how average speed and RMS speed are derived and why they differ.

When we talk about the speed of gas molecules, we refer to how fast they travel in their random motion. The speeds of these molecules in a sample vary widely, as collisions can quickly change their direction and kinetic energy. There's not just one speed but a distribution of speeds, often depicted as a graph known as the Maxwell-Boltzmann distribution.

The average speed, as calculated in the exercise, provides a single value representative of the entire sample, despite the large range of individual speeds. It gives us a sense of the median rate at which particles move. However, because the distribution of speeds is skewed towards higher speeds, the RMS speed, which accounts for the 'weight' of these faster speeds by squaring them before averaging, offers a different perspective that more accurately reflects the energy of the particles. Better understanding individual molecule speeds can help us predict behaviors of gases, like diffusion rates or reaction kinetics, which are influenced by the molecular motion.

According to the kinetic molecular theory, there's a direct correlation between the temperature of a gas and the speeds of its molecules. Higher temperatures indicate higher average kinetic energies, leading to an increase in the speed of gas molecules.

Temperature is typically measured on the Kelvin scale in scientific contexts because it starts at absolute zero, the point where molecular motion ceases. Understanding this relationship can help explain the behavior of gases under different thermal conditions. For example, heating a gas will increase its temperature, making its molecules move faster, as reflected in both the average and RMS speeds. This increase in speed with temperature is also the reason why hot air balloons rise and why substances change their state of matter.

The exercise's comparison of average speed and RMS speed is a vivid demonstration of how temperature is fundamentally linked to molecular motion. The fact that RMS speed is consistently higher than the average speed at any given temperature reinforces the idea that temperature acts as a 'baseline' kinetic energy level for the particles in a gas.