Why average translational KE per molecule of all gases is same at same temperaure?

Gas molecules move constantly and randomly throughout the volume of the container they occupy. When examining the gas molecules individually, we see that not all of the molecules of a particular gas at a given temperature move at exactly the same speed. This means that each molecule of a gas have slightly different kinetic energy. To calculate the average kinetic energy (eK) of a sample of a gas, we use an average speed of the gas, called the root mean square speed (urms).


eK=12mu2rmswith

eK is the kinetic energy measures in Joules
m is mass of a molecule of gas (kg)
urms is the root mean square speed (m/s)

The root mean square speed, urms, can be determined from the temperature and molar mass of a gas.urms=3RTM−−−−−√with

R the ideal gas constant (8.314 kg*m2/s2*mol*K)
T temperature (Kelvin)
M molar mass (kg/mol)

When examining the root mean square speed equation, we can see that the changes in temperature (T) and molar mass (M) affect the speed of the gas molecules. The speed of the molecules in a gas is proportional to the temperature and is inversely proportional to molar mass of the gas. In other words, as the temperature of a sample of gas is increased, the molecules speed up and the root mean square molecular speed increases as a result.



Figure 1. As the temperature of a gas sample increases from left to right, the molecular speed distribution increases indicate by the shift of the peaks form left to right from each graph.

Graham's Law states that the rate of effusion of two different gases at the same conditions are inversely proportional to the square roots of their molar masses as given by the following equation:RateofeffusionofARateofeffusionofB=(urms)A(urms)B=3RT/MA−−−−−−−−√3RT/MB−−−−−−−−√=MB−−−√MA−−−√In according with the Kinetic Molecular Theory, each gas molecule moves independently. However, the net rate at which gas molecules move depend on their average speed. By examining the equation above, we can conclude that the heavier the molar mass of the gas molecules slower the gas molecules move. And conversely, lighter the molar mass of the gas molecules the faster the gas molecules move.