Does Halving Molecules Double RMS Speed?

[sArGe99]

Does the rms speed of a gas change when the no of molecules are lowered, isothermally?
I know that at same temp, avg kinetic energy remains the same.
m₁ (v₁)² = m₂ (v₂)²

If m2 = m1/2, v₂² = 2 v₁²
Is this correct?

 

[kuruman]

The number of molecules is N as in pV = NkT. The symbol m stands for the mass of a molecule, not the number of molecules.

 

[kerimek]

I can confirm that the statement "at the same temperature, the average kinetic energy remains the same" is correct. This is known as the kinetic theory of gases, which states that the average kinetic energy of gas molecules is directly proportional to the temperature of the gas.

However, when it comes to the question of whether halving the number of molecules in a gas will double the root mean square (RMS) speed, the answer is not as straightforward. The RMS speed is a measure of the average speed of gas molecules, and it is calculated by taking the square root of the average of the squared speeds of all the molecules in the gas.

According to the ideal gas law, at a constant temperature, the number of molecules in a gas is inversely proportional to its volume. This means that if the number of molecules is halved, the volume of the gas will double. This change in volume will also affect the RMS speed of the gas molecules.

In an ideal gas, the RMS speed is directly proportional to the square root of the absolute temperature. This means that if the number of molecules is halved, the RMS speed will decrease by a factor of √2. In other words, halving the number of molecules will not double the RMS speed, but it will decrease it.

In conclusion, while the average kinetic energy of gas molecules remains the same at a constant temperature, halving the number of molecules in a gas will not double the RMS speed, but it will decrease it by a factor of √2 due to the change in volume.