Regarding root mean square velocity

In summary, the equation c(rms) = √(3RT/M) works for problems when the number of moles is not 1, but the average speed is an intensive parameter, and it does not depend on the volume of gas taken into consideration.
  • #1
fireflies
210
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I am stuck with this question: why can't we use the equation

c(rms)=√(3RT/M) for problems when no of moles is not 1?

Because I think the mole will be deduced as

c(rms)=√(3nRT/nM) =√(3RT/M)

For instance I found this problem, given atmospheric pressure is 10^5 Pa, molecular mass of
Nitrogen is 28, and at STP the density is 1.25 kg/m^3. What is the c(rms) at 100°C?

If the problem is solved with c(rms)=√(3P/rho) the answer is 576.35 which is the actual answer, and when taken the above equation the answer is 18.22

Then what's wrong with my concept?
 
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  • #2
The formula is OK. Posiibly something wrong in the way you use it. Impossible to say if you don't show your actual calculation.
The number of moles has nothing to do with that formula. The average speed is an intensive parameter, it does not depend on the volume of gas taken into consideration.
 
  • #3
Can you see that, for an ideal gas, RT/M = P/rho, using the ideal gas equation and the definition of density?
 
  • #4
Khashishi said:
Can you see that, for an ideal gas, RT/M = P/rho, using the ideal gas equation and the definition of density?
 
  • #5
Sorry, I had been trying to upload the calculation but due to some problems it was not uploading. So, here I'm typing the calculations:

Calculation 1 (using the first equation)

Crms=√(3RT/M) = √(3*8.314*373/28) = 18.228 (approx)

Calculation 2

we know PV=nRT,
or, PV= (m/M)RT
or, P=(m/V)*R*T/M
∴ P=ρRT/M

i.e. P/(ρ*T)=constant

so in this case we get P/(ρT)=P'/(ρ'T')
or, P'/ρ' = (PT')/(ρT)

Now, crms = √(3p'/ρ') = √{(3*10^5*373)/(1.25*273)} = 572.63(approx)

The later one is the solution and anser given. But it doesn't match with the answer of the previous one. Why?
 
  • #6
Your units in the first calculation are not consistent.
If you use R=8.31 this is in J/mol*K. Then you have to take the molar mass per mol too. It is 28g/mol which is .028 kg/mol. You need to use kilograms to be consistent with the Joules in the unit for R. So you are off by a factor of sqrt(1000) or about 32. Which will give you the same value as the second calculation (around 500 m/s).
 
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Likes fireflies
  • #7
Yes, now the answer comes. Thanks for helping
 
  • #8
fireflies said:
Yes, now the answer comes. Thanks for helping
You are welcome.:smile:
 

FAQ: Regarding root mean square velocity

1. What is root mean square velocity?

Root mean square velocity, also known as RMS velocity, is a measure of the average speed of particles in a gas or liquid. It takes into account both the magnitude and direction of the individual velocities of the particles, making it a more accurate representation of the overall velocity of the system.

2. How is root mean square velocity calculated?

The root mean square velocity is calculated by taking the square root of the average of the squared velocities of all the particles in a system. This is represented by the equation Vrms = √(V1² + V2² + ... + Vn²)/n, where n is the total number of particles in the system.

3. What is the significance of root mean square velocity in thermodynamics?

In thermodynamics, root mean square velocity is used to calculate important properties such as the kinetic energy and pressure of a gas. It is also a key factor in the Maxwell-Boltzmann distribution, which describes the distribution of particle velocities in an ideal gas.

4. How does root mean square velocity relate to temperature?

Root mean square velocity is directly proportional to the square root of temperature. This means that as temperature increases, the average speed of particles also increases. This relationship is described by the equation Vrms = √(3kT/m), where k is the Boltzmann constant and m is the mass of the particles.

5. What is the difference between root mean square velocity and average velocity?

The main difference between root mean square velocity and average velocity is that RMS velocity takes into account the direction of the individual velocities, while average velocity only considers the magnitude. This means that RMS velocity is a more accurate measure of the overall speed of particles in a system.

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