Maxwell Kinetic Energy Distribution

[TeslaPow]

I want to calculate the kinetic energy distribution amongst let's say nitrogen molecules by using M.K.E.D, but not sure where to start.
I posted a picturefrom my physics book where the formula is shown, there was no example in the book.

As for g(K), is K the same as the kinetic energy formula Ktot = N[1/2m(v`^2)] = 3/2NKbT or 3/2KbT ?

 

[mjc123]

K there is the kinetic energy of a molecule, as you can tell by the scale of 10^-21 J. You can use the molar kinetic energy if you replace k_B in the formula by R = N_Ak_B. Or total kinetic energy if you replace k_B with nR.

 

[TeslaPow]

So how do I proceed from here on how to calculate the kinect energy of a molecule between K and K+dK ?

 

[mjc123]

K = 1/2*mv². You can't use 3/2k_BT because that's the mean kinetic energy of all the molecules in the sample.

 

[vanhees71]

The probability to find a molecule with kinetic energy between $K$ and $K+\mathrm{d} K$ is $g(K) \mathrm{d} K$. So what's the average kinetic energy of a molecule when you have a gas with temperature $T$?

 

[TeslaPow]

The probability to find a molecule with kinetic energy between $K$ and $K+\mathrm{d} K$ is $g(K) \mathrm{d} K$. So what's the average kinetic energy of a molecule when you have a gas with temperature $T$?

Exactly, but there's a derivation that needs to be done, and from previous experience the solution to this can be looked up at an integration table. Don't know what the solution is.

 

[mjc123]

Solution to what? It's not clear what you're after. You first asked about calculating the kinetic energy distribution. You have it, it's g(K). Then you ask how to calculate the ke of a molecule between K and K+dK. It's K. You don't calculate it, it's just a value of ke. There is a distribution of ke values in any gas sample. K isn't a function of the properties of the gas or conditions. g(K) is. K is just a variable.

 

[TeslaPow]

Solution to what? It's not clear what you're after. You first asked about calculating the kinetic energy distribution. You have it, it's g(K). Then you ask how to calculate the ke of a molecule between K and K+dK. It's K. You don't calculate it, it's just a value of ke. There is a distribution of ke values in any gas sample. K isn't a function of the properties of the gas or conditions. g(K) is. K is just a variable.

Here's an earlier thread I posted from the distribution of speeds,

https://www.physicsforums.com/threads/maxwell-boltzmann-distribution.757539/post-4772356

Is the procedure similiar? Thanks for the engagement

 

[mjc123]

STATE VERY SPECIFICALLY WHAT PROBLEM YOU'RE TRYING TO SOLVE. You've been asked to do that and haven't.

The thread you refer to suggests you're looking for how to integrate g(K). But read your earlier posts, that's not what you've asked for. You asked for the kinetic energy of a molecule between K and K+dK. That's just K. Did you mean you wanted the fraction of molecules with kinetic energy between K and K+dK? That's just g(K)dK. Do you want to find the fraction of molecules between two values K1 and K2? Integrate g(K)dK from K1 to K2.

Note that as KE = 1/2 mv², the distribution of speeds is different from the distribution of energies. You used an error function before, but not here. g(K) has the form of a gamma distribution. The general form of a gamma distribution is
f(x; α,β) = x^α-1/β^α*e^x/β/Γ(α)
Here α = 3/2 and β = kT

 

[TeslaPow]

Thanks for your advice and patience, I want to find the fraction of molecules between two values:

So I just fill in the missing parameters in equation 8-14, by using E = 1/2*m*v^2 ?
What are the values of K1 (E1) and K2 (E2) in the definite integral?

 

[mjc123]

There are no missing parameters in equation 8-14.
The values of E1 and E2 in the definite integral are 0 and ∞. You are also given the answer. Are you saying you want to integrate between two limits not equal to 0 and ∞?
This, however, gives you the average KE over that range. But you said you want to find the fraction of molecules between two values. For that you need to integrate equation 8-13 between E1 and E2 (and divide by N).

 

[TeslaPow]

There are no missing parameters in equation 8-14.
The values of E1 and E2 in the definite integral are 0 and ∞. You are also given the answer. Are you saying you want to integrate between two limits not equal to 0 and ∞?
This, however, gives you the average KE over that range. But you said you want to find the fraction of molecules between two values. For that you need to integrate equation 8-13 between E1 and E2 (and divide by N).

So by looking at integral tables, integral-tables.com, which one is the right to use?

 

[mjc123]

I'm not familiar with that website, but you can do it e.g. in Excel, using the gamma.dist function. For example, if you want the fraction of molecules with KE between K1 and K2, you would use the formula
=gamma.dist(K2, alpha, beta, true) - gamma.dist(K1, alpha, beta, true)
where K1 and K2 are your limiting energy values, K2 > K1
alpha = 1.5
beta = the value of k_BT
"true" is the value of the variable CUMULATIVE

 

[TeslaPow]

Alright, then I have an alternative to double check my answer. Thanks for your help.

 

[TeslaPow]

Is there somebody who can help me with the integration for the maxwell kinetic energy equation in 8-13 with the fraction of molecules with KE between K1 and K2?

 

[mjc123]

I have told you how to do it. What's your difficulty?

As your problem seems to be more mathematical (how to do an integration) than physical (understanding Maxwell's distribution), I suggest you post a question in the maths forum; you may get more appropriate help.

 

[DrClaude]

Is there somebody who can help me with the integration for the maxwell kinetic energy equation in 8-13 with the fraction of molecules with KE between K1 and K2?

There is no analytical formulation for usch a definite integral. @mjc123 has explained how to do it with Excel. You can also do it with, e.g., WolframAlpha. Note that you need numerical values for K1 and K2.

 

[TeslaPow]

I need the solution on how to solve this without a calculator, so where should I start?

https://towardsdatascience.com/gamma-function-intuition-derivation-and-examples-5e5f72517dee

Is this it?

 

[jambaugh]

There is an anti-derivative to the Maxwell distribution function, i.e. a cumulative distribution function, but it is expressed in terms of the error function, another integral. But technically logarithms are defined as integrals so that, at least, is a better format.

Using the kinetic energy form the differential probability density is:
$$d\rho = g(K)dK = \frac{2}{\sqrt{\pi}} E_T^{-3/2}K^{1/2}e^{-K/E_T} dK$$
where $E_T = k_B T$ is the thermal energy factor.

Converting to unitless kinetic energy factor $u= K/E_T$ gives:
$$d\rho = \frac{2}{\sqrt{\pi}} u^{1/2} e^{-u}du$$

Integrating once by parts and performing another substitution, $u=t^2$ on the remaining integral term will yield a Gaussian integral equivalent to the error function. So i.t.o. the error function you should get:
$$\rho - \rho_0 = \text{erf}(\sqrt{u}) - \frac{2}{\sqrt{\pi}}u^{1/2}e^{-u}$$

You should double-check my work by differentiating this (looking up erf) and verify it is the correct anti-derivative, and also checking my earlier substitutions.