Why speed of the molecules is bigger the less they weigh?

[aaaa202]

In statistical physics we have for an ideal gas about the average kinetic energy for its molecules:

E_kin=½kT

Now in my book this is derived using the ideal gas law as an experimental fact, but that does not really help you get a deeper understanding, does it? I'm assuming that this can be derived from statistical mechanics.

I wonna ask the following?
What is the intuition behind, that the speed of the molecules is bigger the less they weigh? This follows from the fact that every molecule regardless of mass, apparently on average have the same kinetic energy.

 

[James Leighe]

Kinetic energy is equal to half the mass times the velocity squared. Therefore, to have the same kinetic energy with something less massive, you need a greater velocity.

That is if I understood you properly.

 

[aaaa202]

well I was more interested in the deeper reason behind why <Ekin> is always 3/2kT, but maybe I should just wait with that till statistical mechanics.

 

[Philip Wood]

Sketch of an argument.
(1) You can show by a dynamics argument (See Jeans: Kinetic theory of gases) that a gas will exchange energy in collisions with its container walls, unless mean KE of gas molecules is the same as that of wall molecules. (2) But macroscopically it's temperature difference that controls heat transfer. (3) So two gases with the same mean KE have the same temperature. (4) But this doesn't show that mean KE is proportional to temperature. (5) Nothing can show this until we have defined a temperature scale. (6) the fundamental scale is the thermodynamic scale (of which the kelvin scale is the practical expression) which is defined in terms of heat taken in and given out in a Carnot cycle. (7) By taking an ideal gas through a Carnot cycle (in a thought-experiment), we can show that the kelvin temperature of the gas is proportional to PV. (8) Kinetic theory shows that PV is proportional to the mean KE of the molecules.

For a lot of elementary purposes, we can down the argument by defining an ideal gas scale of temperature such that T on this scale is proportional to the mean KE of the molecules. But at some stage, if you take Physics further, you're going to need to know how to establish the identity between the ideal gas scale and the thermodynamic scale.

 

[PJC01]

The reason why the speed of molecules is greater when they have a lower mass is due to the equation for average kinetic energy, Ekin=½kT. This equation shows that the average kinetic energy is directly proportional to the temperature (T) and inversely proportional to the mass (k). This means that as the mass of the molecule decreases, the average kinetic energy increases, resulting in higher speeds.

Intuitively, we can think of it like this: in a gas, the molecules are constantly moving and colliding with each other. The molecules with lower mass are able to move faster because they require less energy to move at the same speed as heavier molecules. This is because their smaller mass means they have less inertia, making it easier for them to accelerate.

Additionally, the average kinetic energy of a molecule is related to its temperature, which is a measure of the average kinetic energy of all the molecules in the gas. As the temperature increases, the average kinetic energy of the molecules increases, leading to higher speeds.

In summary, the speed of molecules is greater when they have a lower mass because of the relationship between temperature, average kinetic energy, and mass. This can be derived from statistical mechanics, providing a deeper understanding of the behavior of gases.