Question

Determine the relative probability of a gas molecule being within a small range of speeds around \(2 {\text {rms }}\) to being in the same range of speeds around \(v {_\text {rms }}\).

Short answer

The relative probability of finding a gas molecule at a speed of 2 times the root mean square speed \(2v_{\text{rms}}\) is four times the probability of finding one at the root-mean-square speed \(v_{\text{rms}}\).

Step-by-step solution

Define the Probability Density Function

The function for the Maxwell-Boltzmann distribution is given by: \[ f(v) = \sqrt{ \frac{2}{\pi}} \left( \frac{v^2}{v_{\text{rms}}^3}\right) e^{-v^2/2v_{\text{rms}}^2}\] Where \(e\) is the base of the natural logarithm, and \(v_{\text{rms}}\) is the root mean square speed.

Determine Probability at \(v_{\text{rms}}\)

Substitute \(v = v_{\text{rms}}\) into the probability density function and calculate: \[f(v_{\text{rms}}) = \sqrt{ \frac{2}{\pi}} \left( \frac{v_{\text{rms}}^2}{v_{\text{rms}}^3}\right) e^{-v_{\text{rms}}^2/2v_{\text{rms}}^2} = \sqrt{ \frac{2}{\pi e}}\] Here \(e\) is the base of the natural logarithm.

Determine Probability at \(2 v_{\text{rms}}\)

Substitute \(v = 2 v_{\text{rms}}\) into the probability density function and calculate: \[f(2v_{\text{rms}}) = \sqrt{ \frac{2}{\pi}} \left( \frac{(2v_{\text{rms}})^2}{v_{\text{rms}}^3}\right) e^{-(2v_{\text{rms}})^2/2v_{\text{rms}}^2} = \frac{8}{\sqrt{\pi e^2}}\]

Find The Ratio

Determine the relative probability by taking the ratio of the two probabilities: \[\frac{f(2v_{\text{rms}})}{f(v_{\text{rms}})} = \frac{\frac{8}{\sqrt{\pi e^2}}}{\sqrt{ \frac{2}{\pi e}}} = \frac{8}{2} = 4\] This implies that the relative probability of finding a gas molecule at a speed of 2 times the root mean square speed is four times the probability of finding one at the root mean square speed.

Key concepts

Probability Density Function

The concept of a probability density function (PDF) is crucial in understanding the Maxwell-Boltzmann distribution applied to the velocities of gas molecules. The Maxwell-Boltzmann distribution describes how the speeds of particles are distributed in a gas that is in thermal equilibrium.
This function is characterized by a specific mathematical form:
\[ f(v) = \sqrt{ \frac{2}{\pi}} \left( \frac{v^2}{v_{\text{rms}}^3}\right) e^{-v^2/2v_{\text{rms}}^2} \]
Here, \(v\) represents the speed of gas molecules, while \(v_{\text{rms}}\) is the root mean square speed, and \(e\) is the base of the natural logarithm. The probability density function tells us the probability of finding particles with a specific speed \(v\).
The PDF is not the probability itself but relates to how probability is spread over different speeds. A higher value of the PDF means there is a greater likelihood of finding particles moving at speeds near that value.

Root Mean Square Speed

The root mean square (rms) speed is a measure of the average speed of particles in a gas. It provides insights into the kinetic energy of the molecules since temperature is related to kinetic energy.
The root mean square speed \(v_{\text{rms}}\) is calculated using the formula:
\[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]
where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a gas molecule.
This speed is significant in the Maxwell-Boltzmann distribution and serves as a key parameter in the PDF for gas speeds. Plugging \(v_{\text{rms}}\) into the probability density function lets us compare other speeds to this average measurement.
When you hear root mean square speed in this context, think of it as the 'average' speed of particles, considering their energetic and random motion.

Relative Probability

Relative probability in this context refers to the comparison of probabilities corresponding to different speeds of gas particles. Specifically, it means comparing the likelihood of finding particles at different speeds described by the PDF.
In the exercise, you explore the ratio of probabilities between velocities \(2v_{\text{rms}}\) and \(v_{\text{rms}}\).
Through the calculations, you find that:
- The probability density at speed \(v_{\text{rms}}\) is \(f(v_{\text{rms}}) = \sqrt{ \frac{2}{\pi e}}\)
- At double this speed, \(2v_{\text{rms}}\), the probability density is \(f(2v_{\text{rms}}) = \frac{8}{\sqrt{\pi e^2}}\)
- By examining the ratio:\[ \frac{f(2v_{\text{rms}})}{f(v_{\text{rms}})} = 4 \]
This ratio implies that the likelihood of finding a gas molecule moving at \(2v_{\text{rms}}\) is four times higher compared to a molecule moving at \(v_{\text{rms}}\). This understanding is pivotal in kinetic theory, helping to visualize how the speed of particles influences their distribution and behavior in a gas.