# Why is the speed of ideal gases mass dependent while the kinetic energy is not?

The rms speed of an ideal gas is $v_{\text{rms}}$ = $\sqrt{\frac{3RT}{M} }$ and the kinetic energy is $E_\text{k} = \frac32RT$. From this, it is concluded that the speed is mass dependent, while the kinetic energy isn't.

This doesn't make sense to me. I know the speed and kinetic energy for things other than ideal gases are surely mass dependent. The heavier an object is, the slower it is and will have a bigger kinetic energy. Why isn't this true for ideal gases?

## 3 Answers

As the mass increases, the velocity decreases, yes that is true. But how does the kinetic energy remain the same?

The increment in the mass is balanced by the decrement in the velocity. In other words, the velocity changes as the mass changes in such a manner that the kinetic energy of the molecules of the gas will remain to be same no matter what. It's something like the multiplication of two variables always giving the same constant value. Keeping in mind that this is true in an ideal environment.

• I think that the answer should be pointed out that this assumes ideal gas behavior.
– MaxW
Feb 26, 2018 at 19:24
• I added that last bit. Thank you @MaxW. Since his question was "Why isn't this true for ideal gases?" I assumed I was writing for ideal conditions. Feb 26, 2018 at 19:32
• I didn't claim that the kinetic energy doesn't change but that is is independent on the gas mass. as the equation suggests Ek = 1.5RT the only thing that influences the kinetic energy is the temperature. Feb 26, 2018 at 20:27
• @ILoveIL Yes, only temperature influences kinetic energy. What is the issue? Feb 26, 2018 at 20:39
• @ILoveIL Two different gases in two separate vessels at the same temperature will have same kinetic energy but not the same Vrms. The difference in masses compensates for different Vrms values. Could you reframe your question again? So that I can understand your problem and help you solve it. Feb 26, 2018 at 20:46

We start with two samples of pure gas, one containing molecules of A (the larger molecule) and one containing molecules of B (the smaller molecule) with the same average speed (it will turn out that gas A has the higher temperature). If you mix these samples, their kinetic energy (per degree of translation freedom) will be different at first. However, they will quickly achieve thermal equilibrium. This means that the energy transfer in collisions between molecules of A and molecules of B average out. In a collision between a bat (molecule A) and a baseball (molecule B) coming at each other at the same speed, there is a kinetic energy transfer from bat to ball (the bat slows own a bit, the ball speeds up - if we assume an elastic collision).

Here is a simulation that shows one diatomic molecule and many monoatomic molecules. The diatomic is a bit slower on average.

• imo, the confusion has nothing to do with ideal gases at all. For any object, the velocity and kinetic energy are NOT always dependent on mass.. It depends on what is kept constant in the comparison. If I accelerate a proton and electron through a p.d, their k.e is independent of mass whereas velocity is not. If I put a proton and an electron in a g-field, their velocities are independent of mass, whereas k.e is not. Jan 12, 2021 at 17:21

The answer can be $$Speed \propto1/√m$$ And$$K.E. = 1/2mv^2$$. Finally m will cut, and therefore K.E. does not depend on mass.

• This directly follows from the formula. I believe the OP was asking for an intuitive approach rather than a formula-based approach. May 15, 2018 at 9:37