Two different gases are at the same temperature, and both have low enough densities that they behave like ideal gases. Do their molecules have the same thermal speeds? Explain.

The kinetic theory of gases provides an explanation for the behavior of gas molecules based on their motion and how they interact at the microscopic level. It assumes that the gas comprises small particles in constant random motion and that these particles collude elastically with each other and the walls of the container. One fundamental implication of this theory is that the temperature of a gas is directly proportional to the average kinetic energy of its molecules.

Therefore, when we talk about thermal speeds, which refer to how fast gas molecules are moving on average, they're intrinsically linked to the kinetic energy of these molecules. As the temperature increases, so does the thermal speed of gas molecules. This is why, even at the same temperature, gases with different molar masses exhibit different speeds— since kinetic energy depends on both mass and velocity.

The ideal gas law is a fundamental equation in thermodynamics and physical chemistry that describes the state of an ideal gas. It is expressed as \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvins.

Assuming low density and high temperature conditions, real gases approximate the behavior of ideal gases, and the ideal gas law serves as an excellent predictive tool. When used in conjunction with the kinetic theory, the ideal gas law can help us understand why the root mean square speed of gas particles is based on temperature and the individual gas's molar mass. It shows us that the state of an ideal gas is not just about pressure and volume, but also about the microscopic motion of molecules as characterized by temperature.

Root mean square speed (\(v_{rms}\)) is a measure of the average speed of particles in a gas, which factors in the various velocities of all the particles. The formula itself, \(v_{rms} = \sqrt{\frac{3kT}{m}}\), showcases the direct relationship between the average speed and the temperature (\(T\)) of the gas, as well as its inverse relationship with the molar mass (\(m\)) of the gas particles.

What's important to consider here is that at a given temperature, all gases have the same average kinetic energy, but this does not mean that all gas molecules move at the same speed. Lighter molecules move faster than heavier ones because for two particles to have the same kinetic energy, the lighter must compensate with higher velocity. As such, understanding \(v_{rms}\) offers critical insight into how molecular properties influence the behavorial patterns of gases, and relates to why two different gases at the same temperature can exhibit different thermal speeds.