Velocity distribution functions, find rms velocity

[fahmed93]

My friends and I have been working on this for the last two hours and we're still on 1a. I'm desperate and I'm going anywhere for help. If anyone's taken/is good at modern physics then please help. We're still in a physics review thing so it's not as complicated. The question is "Using the velocity distribution function, find the formula for the rms velocity."

The starting equation is P(v) = (m/2piKT)^3/2 * 4piV^2 * e ^ (-mv^2/2kT)
where m is mass, k is the Boltzmann constant, v is velocity, and t is temperature.

The answer is SQRT(3RT/M)
where R is the universal gas constant, T is temperature and M is molar mass.

Some useful conversions.
K = R/Na
where Na is Avagodro's number

m*Na = M
m = mass, Na = Avagodro's, M = molar mass



and we figured out that you're supposed to integrate it in this equation.
SQRT(integral from 0 to infinity of (v^2 * P(v) dv))
but don't understand how to go from there to the final answer.
PLEASE HELP!

 

[Redbelly98]

.
Welcome to Physics Forums.

My friends and I have been working on this for the last two hours and we're still on 1a. I'm desperate and I'm going anywhere for help. If anyone's taken/is good at modern physics then please help. We're still in a physics review thing so it's not as complicated. The question is "Using the velocity distribution function, find the formula for the rms velocity."

The starting equation is P(v) = (m/2piKT)^3/2 * 4piV^2 * e ^ (-mv^2/2kT)
where m is mass, k is the Boltzmann constant, v is velocity, and t is temperature.

The answer is SQRT(3RT/M)
where R is the universal gas constant, T is temperature and M is molar mass.

Some useful conversions.
K = R/Na
where Na is Avagodro's number

m*Na = M
m = mass, Na = Avagodro's, M = molar mass



and we figured out that you're supposed to integrate it in this equation.
SQRT(integral from 0 to infinity of (v^2 * P(v) dv))
but don't understand how to go from there to the final answer.
PLEASE HELP!

That integral is, essentially

[some constants]·∫vⁿ e^-mv2/2kT dv, where n=___?​

integrated from 0 to ∞.

I suggest a change-of-variable, so that the exponential factor becomes simply e^-x. You should get an integral that can be looked up, or evaluated numerically.

 

[DEMANDRED]

I would like to add a question here because I am too embarassed to go to my teacher for help. I have a homework problem and I must be doing something wrong on a really basic level because I think I am getting really bad answers.

"Calculate the temperatures for which the molecules x,y,z equal the speed of sound in air 340 m/s"

OK. The question is a little vauge I think because a) molecules never equal speed they travel "WITH" speed and b) at any temperature could a certain molecule reach a certain speed. It is a randomized distribution i.e. Boltzmann. OK so smart ones out there, does my H.W. ask for most probable speed or rms speed. And how would i find both? This is how I tried and came up with the wrong answers.

Used maxwell Boltzmann distribution. Df/dx = 0 when Vp = SQRT(2RT/M) (so says wikipedia: please correct me if I am wrong my book is of no help). So I squared the Vp which i want to be 340 m/s divided through by 2R multiplied by M and calculated. But my answers were in the neighborhood of 3-40 degrees kelvin for hydrogen and helium and water (unless somehow i got units different than kelvin, this answer is rediculus...i think). Wikipedia says Vrms = SQRT(3/2) * Vp which is also low. Am I doing something wrong? Is the correct temperature really so low? 40 degrees kelvin or less. I feel stupid.

 

[DEMANDRED]

So I forgot to tell you, Vp is the most proable velocity.