- #1
arccosinus
- 1
- 0
Hello, I have a question regarding the Maxwell-Boltzmann distribution.
As you know, the distribution basically looks like
n(v) = constant \cdot v^2e^{-\frac{mv^2}{2kT}}
, where v is the speed, m is the particle mass, is k the Boltzmann constant and T is the absolute temperature.
Now, one can calculate the mean value of the squared velocity <v^2> by evaluating the integral
\frac{\int_0^\infty v^2 n(v) dv}{\int_0^\infty n(v) dv} = \frac{3kT}{m}
From here, we can calculate the average (translational-)kinetic energy of particles,
<\frac{mv^2}{2}> = \frac{m}{2} <v^2>=\frac{3kT}{2}
Here comes the question, say we have air of temperature T. How can one easily show that the fraction of (for example) oxygen molecules with kinetic energy greater than \frac{3kT}{2} is less than 50%?
My idea is basically to create a new distribution, say n_1(v^2) by substituting v with v^2 in the original distribution, integrating from 0 to \frac{3kT}{m} and then from \frac{3kT}{m} to infinity. One can then compare these integrals and conclude whether the fraction of molecules with kinetic energy greater than \frac{3kT}{2} is less than 50%. However, this approach seems extremely superflous and I am not even sure the integrals converge (even though I think so). There has to be an easier argument! Can anyone help please?
As you know, the distribution basically looks like
n(v) = constant \cdot v^2e^{-\frac{mv^2}{2kT}}
, where v is the speed, m is the particle mass, is k the Boltzmann constant and T is the absolute temperature.
Now, one can calculate the mean value of the squared velocity <v^2> by evaluating the integral
\frac{\int_0^\infty v^2 n(v) dv}{\int_0^\infty n(v) dv} = \frac{3kT}{m}
From here, we can calculate the average (translational-)kinetic energy of particles,
<\frac{mv^2}{2}> = \frac{m}{2} <v^2>=\frac{3kT}{2}
Here comes the question, say we have air of temperature T. How can one easily show that the fraction of (for example) oxygen molecules with kinetic energy greater than \frac{3kT}{2} is less than 50%?
My idea is basically to create a new distribution, say n_1(v^2) by substituting v with v^2 in the original distribution, integrating from 0 to \frac{3kT}{m} and then from \frac{3kT}{m} to infinity. One can then compare these integrals and conclude whether the fraction of molecules with kinetic energy greater than \frac{3kT}{2} is less than 50%. However, this approach seems extremely superflous and I am not even sure the integrals converge (even though I think so). There has to be an easier argument! Can anyone help please?