Question

Figure 9.8 cannot do justice to values at the very high. speed end of the plot. This exercise investigates how small it really gets. However, although integrating the Maxwell speed distribution over the full range of speeds from 0 to infinity can be carried out (the so-called Gaussian integrals of Appendix $K$ ). over any restricted range, it is one of those integrals that. unfortunately. cannot be done in closed form. Using a computational aid of your choice, show that the fraction of molecules moving faster than $2v_{\text{ms }}$ is $\sim {10}^{-2}$; faster than $6v_{\text{ms }}$ is $-{10}^{-23};$ and faster than $10v_{\mathrm{ms}}$ is $\sim {10}^{-64}$, where $v_{\mathrm{m}2}$ from Exercise $41.$ is $\sqrt{3k_{B}T/m}$. (Exercise 48 uses these values in an interesting application.)

Short answer

The fraction of molecules moving faster than $2v_{ms}$ is approximately ${10}^{-2}$, moving faster than $6v_{ms}$ is approximately ${10}^{-23}$, and moving faster than $10v_{ms}$ is approximately ${10}^{-64}$.

Step-by-step solution

Setup the integral

The fraction of molecules with speed greater than some value can be found by integrating the Maxwell speed distribution from that speed to infinity. The Maxwell speed distribution in one dimension is given by $f(v)=\sqrt{(m/2\pi k_{B}T)}\ast e^{-m{v}^2/2k_{B}T}$. The fraction of molecules with speed greater than $x{v}_{ms}$ is given by the integral from $x{v}_{ms}$ to infinity of $f(v)$ dv.

Compute the integrals

We'll use the aid of a computational tool to perform these integrals (due to the complexity). Input the integral as given in the formula into the software and it would solve the integral from $2v_{ms}$ to infinity, $6v_{ms}$ to infinity, and $10v_{ms}$ to infinity to find the fraction of molecules moving faster than those respective speeds.

Interpret the results

The results of these integrals indicate the fraction of molecules moving faster than the given speeds. The fact that these fractions decrease so dramatically as one moves to higher speeds is a reminder of the rareness of very high molecular speeds. Even though the tail of the Maxwell speed distribution extends to infinity, very high speeds are extremely rare.

Key concepts

Gaussian Integrals

Gaussian integrals play a crucial role in the field of mathematical physics and particularly in statistical mechanics. In essence, these integrals evaluate expressions that feature the exponential function raised to the power of a quadratic polynomial—an operation that is fundamental when dealing with the probability distributions of systems governed by many random variables.

For instance, the Maxwell speed distribution, which gives the probability of finding a particle with a certain speed in a gas, requires integrating such an expression over all possible speeds. While the Gaussian integral over the entire range of speeds can be calculated analytically (yielding a known result involving $\pi$), computations over a restricted range often require numerical methods.

It's interesting to note that the Gaussian integral in the context of statistical mechanics is a direct representation of the normal distribution of velocities of particles in a thermal equilibrium. This mathematical concept is not only theoretical but has practical computational applications, such as the one described in the exercise.

Statistical Mechanics

Statistical mechanics is a foundational pillar for understanding physical systems comprised of a large number of particles. It merges the macroscopic laws of thermodynamics with the microscopic behavior of individual atoms and molecules.

If we take a gas as an example, each molecule moves randomly with varying speeds and directions. Despite this randomness, statistical mechanics allows physicists to predict the macroscopic properties of the gas—such as temperature and pressure—by considering the behavior of individual molecules.

The Maxwell speed distribution is a result derived from statistical mechanics, representing the distribution of speeds among particles in a system at a given temperature. It encapsulates the likelihood of finding a particle within a particular range of speeds and helps in understanding phenomena like diffusion, heat conduction, and even the sound propagation in gases.

In teaching statistical mechanics, it's beneficial to emphasize its predictive power despite the inherent randomness of particles' motions, and the practicality of computational tools to evaluate complex integrals that arise from these theories.

Molecular Speeds

The concept of molecular speeds is at the heart of kinetic theory and statistical mechanics. It informs us about the distribution and behavior of molecules in a gas, which is particularly important for understanding heat and temperature on a molecular level.

The root-mean-square speed, $v_{\mathrm{rms}}$, is a measure of the average speed of molecules within a gas and is derived from the Maxwell speed distribution. This exercise uses multiples of $v_{\mathrm{rms}}$ to explore the probability of encountering extraordinarily fast-moving molecules.

The underlying message in calculating the fraction of molecules exceeding speeds of $2v_{\mathrm{rms}}$, $6v_{\mathrm{rms}}$, and $10v_{\mathrm{rms}}$, is to demonstrate the rarity of such events. With values of approximations around ${10}^{-2}$, ${10}^{-23}$, and ${10}^{-64}$, students can visualize the steep drop-off in the number of molecules reaching higher velocities, which reinforces the principles of kinetic theory. Conveying this visually or through interactive tools can enhance comprehension and stimulate students' interest.