Fraction of molecules have kinetic energy of 1/2mC^2

In summary, the task is to obtain an expression for the fraction of molecules having kinetic energy in the range e to e + ∆e, given a gas of molecules of mass m at equilibrium at temperature T. This can be done by integrating the speed distribution function from 0 to C, and then dividing 1/2mC2 by the total kinetic energy. To find the probability in the given energy range, you need to integrate the energy distribution from e to e + ∆e.
  • #1
igowithit
6
1

Homework Statement


Consider a gas of molecules of mass m at equilibrium at temperature T. Obtain an expression for the fraction of molecules having kinetic energy e = 1/2mC2 in the range e to e + de.

This is problem 5.3, page 48 of Vincenti and Kruger's Intro to Physical Gas Dynamics

Homework Equations


Maxwellian speed distribution

The Attempt at a Solution


I'm not sure where to start really, so I don't have much of a solution.

If I were finding the fraction of molecules with kinetic energy LESS than 1/2mC2, I would integrate the speed distribution function from 0 to 1/2mC2. But I'm not sure about finding the fraction at exactly 1/2mC2.

My first thoughts were to change the speed distribution to an energy distribution by knowing that c2 = 2e/m, and then integrating the kinetic energy distribution from 0 to inf to find the total kinetic energy. Then divide 1/2mC2 by that total to find a fraction. Don't feel like this is right.

Anyone care to nudge me in the right direction?
 
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  • #2
igowithit said:
Consider a gas of molecules of mass m at equilibrium at temperature T. Obtain an expression for the fraction of molecules having kinetic energy e = 1/2mC2 in the range e to e + de.
Hi igowithit! :welcome:

My reading of the task is that you are asked to:
obtain an expression for the fraction of molecules having kinetic energy in the range e to e + ∆e.

They have helpfully provided you with the formula to use, but wrote it where it has lead to confusion.
 
  • #3
igowithit said:
If I were finding the fraction of molecules with kinetic energy LESS than 1/2mC2, I would integrate the speed distribution function from 0 to 1/2mC2.
To elaborate on your reasoning here, what I believe you meant was:
$$P(E < \frac 12 mC^2) = P(v < C) = \int_0^C f_v(v)\,dv$$
You'd integrate the speed distribution from 0 to C.

But I'm not sure about finding the fraction at exactly 1/2mC2.
This fraction would be 0. The probability of picking one value out of the continuum of energies is 0. That's why the question is asking you to find the probability in a range of values, specifically you want to calculate
$$P(e<E<e+de) = \int_e^{e+de} f_E(E)\,dE.$$

My first thoughts were to change the speed distribution to an energy distribution by knowing that c2 = 2e/m, and then integrating the kinetic energy distribution from 0 to inf to find the total kinetic energy. Then divide 1/2mC2 by that total to find a fraction. Don't feel like this is right.

Anyone care to nudge me in the right direction?
 

1. What does it mean when a fraction of molecules have kinetic energy of 1/2mC^2?

When we say that a fraction of molecules have kinetic energy of 1/2mC^2, it means that a certain percentage of molecules in a given sample have a specific amount of kinetic energy, which is equal to half of the molecule's mass multiplied by the speed of light squared.

2. How is the fraction of molecules with kinetic energy of 1/2mC^2 calculated?

The fraction of molecules with kinetic energy of 1/2mC^2 is calculated by dividing the number of molecules with this specific kinetic energy by the total number of molecules in the sample. This calculation results in a percentage or fraction that represents the proportion of molecules with this energy level.

3. What factors affect the fraction of molecules with kinetic energy of 1/2mC^2?

The fraction of molecules with kinetic energy of 1/2mC^2 can be affected by various factors such as temperature, pressure, and the type of molecule. At higher temperatures, more molecules will have this energy level, while at lower temperatures, fewer molecules will have this energy level. Similarly, at higher pressures, there will be more molecules with this energy level compared to lower pressures. The type of molecule also plays a role, as heavier molecules will have a lower fraction with this energy level compared to lighter molecules.

4. Why is the fraction of molecules with kinetic energy of 1/2mC^2 important in science?

The fraction of molecules with kinetic energy of 1/2mC^2 is important in science because it helps us understand the behavior and properties of molecules in a given sample. This information is crucial in various fields such as chemistry, physics, and biology, where the kinetic energy of molecules plays a vital role in chemical reactions, energy transfer, and biological processes.

5. Can the fraction of molecules with kinetic energy of 1/2mC^2 change over time?

Yes, the fraction of molecules with kinetic energy of 1/2mC^2 can change over time. This change can occur due to various factors such as changes in temperature, pressure, or the addition/removal of molecules in the sample. For example, if the temperature of a sample increases, more molecules will have this energy level, resulting in a higher fraction. Similarly, if molecules are added to the sample, the fraction will increase, and if molecules are removed, the fraction will decrease.

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