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. Boltzman. 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 boltzman 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.

 

[rwisz]

Dear friend,

I understand that you and your friends are working on a problem related to velocity distribution functions and finding the formula for rms velocity. It's great to see your enthusiasm towards understanding modern physics concepts. Let me try to provide some guidance that may help you in solving this problem.

Firstly, let's define what rms velocity is. RMS stands for root mean square, and it is a statistical measure of the magnitude of a set of values. In this case, it represents the average velocity of particles in a gas. To find the formula for rms velocity, we need to use the velocity distribution function, which describes the probability of a particle having a certain velocity.

The starting equation, P(v), that you have provided is the velocity distribution function for a gas particle. To find the rms velocity, we need to integrate this function over all possible velocities and take the square root of the result. This is what the equation you have mentioned, SQRT(integral from 0 to infinity of (v^2 * P(v) dv)), represents.

To solve this integral, we can use the substitution method, where we let u = mv^2/2kT. This will simplify the integral to SQRT((2kT/m) * integral from 0 to infinity of (ue^(-u) du)). This integral can be solved using integration by parts, and the final result will be SQRT(3RT/M), where R is the universal gas constant, T is temperature, and M is the molar mass of the gas.

I hope this explanation helps you in understanding how to solve this problem. Remember to always break down complex equations and use appropriate substitutions to make the problem more manageable. Keep up the good work and don't hesitate to ask for help when needed. Best of luck!