RMS velocity of molecules in a mixture

[Saptarshi Sarkar]

Homework Statement

A vessel at temperature T contains equal number of molecules of two gases whose masses are m and 4m respectively. What is the RMS velocity of molecules in the mixture?

Relevant Equations

$v_{rms} = \frac {\sqrt {3k_bT}} m$

I know that for a single monoatomic gas with RMS velocity $v_{rms}$ , $$\frac 1 2mv_{rms}^2 = \frac 3 2k_bT$$ where $m$ is mass of a single molecule, $k_b$ is Boltzmann constant and $T$ is temperature of the gas.

For a mixture of gas, I know that the average kinetic energy after mixing the gases will be equal to the sum of average kinetic energy of the two constituent gases before mixing.

$\frac 1 2mv_{rms_1}^2 + \frac 1 24mv_{rms_2}^2 = \frac 1 25mv_{rms}^2$

But, I have no idea how to use this to find the RMS velocity of the mix.

 

[haruspex]

the average kinetic energy after mixing the gases will be equal to the sum of average kinetic energy of the two constituent gases before mixing.

I don’t think you mean that.
It might be the average of the averages, which fits better with the equation you wrote, but not the sum.
What does your first equation tell you about the relationship between the two individual rms speeds?

 

[Saptarshi Sarkar]

I don’t think you mean that.
It might be the average of the averages, which fits better with the equation you wrote, but not the sum.
What does your first equation tell you about the relationship between the two individual rms speeds?

I did a mistake and wrote the formula wrong, it should be $v_{rms} = \sqrt \frac {3k_bT} m$

From this I can say that if the absolute temperature of the two gases be equal, then $\frac {v_{rms_1}} {v_{rms_2}} = \sqrt \frac {4m} m = 2$

 

[haruspex]

I did a mistake and wrote the formula wrong, it should be $v_{rms} = \sqrt \frac {3k_bT} m$

From this I can say that if the absolute temperature of the two gases be equal, then $\frac {v_{rms_1}} {v_{rms_2}} = \sqrt \frac {4m} m = 2$

Right, so consider one molecule of each. What is the RMS speed of the pair?

 

[Saptarshi Sarkar]

Right, so consider one molecule of each. What is the RMS speed of the pair?

Should it be $\frac 3 2v_{rms_1}$?

 

[haruspex]

Should it be $\frac 3 2v_{rms_1}$?

How do you find the RMS of two numbers?

 

[Saptarshi Sarkar]

How do you find the RMS of two numbers?

By taking the root of the average of the square. So, will it be $\sqrt \frac {5v_{rms_1}^2} 2$?

 

[haruspex]

By taking the root of the average of the square. So, will it be $\sqrt \frac {5v_{rms_1}^2} 2$?

Yes.