RMS velocity of molecules in a mixture

In summary, the conversation discusses the relationship between average kinetic energy and RMS velocity for a single monoatomic gas and a mixture of gases. The first equation states that for a single gas, the average kinetic energy is equal to half the mass times the square of the RMS velocity. The second equation, corrected by the participants, shows that the RMS velocity of the mixture is equal to the square root of the average of the squares of the individual RMS velocities. This leads to the conclusion that if the absolute temperature of the two gases is equal, the ratio of their RMS velocities will be 2.
  • #1
Saptarshi Sarkar
99
13
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.
 
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  • #2
Saptarshi Sarkar said:
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?
 
  • #3
haruspex said:
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##
 
  • #4
Saptarshi Sarkar said:
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?
 
  • #5
haruspex said:
Right, so consider one molecule of each. What is the RMS speed of the pair?

Should it be ##\frac 3 2v_{rms_1}##?
 
  • #6
Saptarshi Sarkar said:
Should it be ##\frac 3 2v_{rms_1}##?
How do you find the RMS of two numbers?
 
  • #7
haruspex said:
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##?
 
  • #8
Saptarshi Sarkar said:
By taking the root of the average of the square. So, will it be ##\sqrt \frac {5v_{rms_1}^2} 2##?
Yes.
 
  • Informative
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Related to RMS velocity of molecules in a mixture

1. What is RMS velocity?

RMS velocity, or root mean square velocity, is a measure of the average speed of molecules in a gas or liquid mixture. It takes into account the velocities of all the molecules present, and gives a more accurate representation of the overall speed than other measures such as average velocity.

2. How is RMS velocity calculated?

RMS velocity is calculated by taking the square root of the average of the squared velocities of all the molecules in the mixture. This can be expressed as:

V(rms) = √(1/N * Σv^2)

where N is the number of molecules and v is the velocity of each individual molecule.

3. What factors affect the RMS velocity of molecules in a mixture?

The RMS velocity of molecules in a mixture is affected by the temperature, mass of the molecules, and the nature of the gas or liquid. As temperature increases, the RMS velocity also increases. Heavier molecules will have a lower RMS velocity compared to lighter molecules at the same temperature. Additionally, the type of gas or liquid will also affect the RMS velocity as different molecules have different masses and properties.

4. Why is RMS velocity important?

RMS velocity is important because it helps us understand the behavior of gases and liquids at a molecular level. It is used in various equations and models to predict properties such as pressure, diffusion, and viscosity. It also helps in understanding the kinetic energy and heat transfer within a mixture.

5. How does the RMS velocity of molecules in a mixture relate to kinetic theory of gases?

The kinetic theory of gases states that the average kinetic energy of gas molecules is directly proportional to the temperature of the gas. RMS velocity is related to kinetic energy as it is the measure of the average speed of molecules. Therefore, the higher the RMS velocity, the higher the kinetic energy and temperature of the gas or liquid mixture.

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