Consider a sample of ideal gas molecules for the following question. a. How is the average kinetic energy of the gas molecules related to temperature? b. How is the average velocity of the gas molecules related to temperature? c. How is the average velocity of the gas molecules related to the molar mass of the gas at constant temperature?

The average kinetic energy of a gas molecule is given by the equipartition theorem, which states that for an ideal gas, the average kinetic energy per degree of freedom per molecule is proportional to the temperature of the gas. Mathematically, this is expressed as: \(K.E._{avg} = \frac{3}{2} kT\) where \(K.E._{avg}\) is the average kinetic energy, \(k\) is the Boltzmann constant, and \(T\) is the temperature of the gas in Kelvin. Thus, the average kinetic energy of the gas molecules is directly proportional to the temperature.

In order to find the relationship between the average velocity of the gas molecules and the temperature, we first need to find the root mean square (RMS) velocity, which is defined as: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) where \(v_{rms}\) is the root mean square velocity, \(m\) is the mass of a single gas molecule, and \(k\) and \(T\) are as defined earlier. The RMS velocity, however, is not a direct measure of the average velocity of the gas molecules. For an ideal gas, the average velocity is zero due to random motion of the gas molecules in all directions. But, the RMS velocity tells us about the magnitude of the velocities. The RMS velocity is directly related to the temperature as it obeys the same relationship as found earlier: \(v_{rms} \propto \sqrt{T}\) Thus, the magnitude of the average velocity of the gas molecules is related to the square root of the temperature.

To find the relationship between average velocity and molar mass at constant temperature, we can rewrite the equation for the RMS velocity in terms of the molar mass. Instead of using the mass of a single molecule, we can use the molar mass, \(M\), which is the mass of one mole of gas molecules: \(v_{rms} = \sqrt{\frac{3kT}{(M/N_A)}}\) where \(N_A\) is Avogadro's number (approximately \(6.022 \times 10^{23}\)) and \(M\) is the molar mass in kg/mol. We can now rewrite this equation in terms of the molar gas constant, \(R\), which is given by: \(R = kN_A\) Replacing \(kN_A\) with \(R\) in our previous equation, we get: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) At constant temperature, the term \(\frac{3R}{M}\) is constant. Therefore, the RMS velocity is inversely proportional to the square root of the molar mass: \(v_{rms} \propto \frac{1}{\sqrt{M}}\) In conclusion, at constant temperature, the magnitude of the average velocity of the gas molecules is inversely proportional to the square root of the molar mass of the gas.