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Chapter 9: Problem 3

The number density in phase space of an ideal gas can be written $$ f(p, r)=\frac{N}{V}\left(\frac{\beta}{2 \pi m}\right)^{3 / 2} e^{-\beta p^{1} /(2 m)}(1+\varepsilon \sin \theta \cos \phi) $$ where \(\theta\) and \(\varphi\) are the polar and azimuthal angles, respectively, of the momentum, \(p\), measured with respect to the z-axis. (a) Compute the net number of particles, per unit area per unit time, that pass through the \(y-z\) plane. (b) Compute the net number of particles, per unit area per unit time, that pass through the \(x-z\) plane.

Short Answer

Expert verified
For y-z plane: Integral of density along x; For x-z plane: Integral of density along y.

Step by step solution

01

Review given expression

Review the number density in phase space given by the exercise:\[ f(p, r)=\frac{N}{V}\left(\frac{\beta}{2 \pi m}\right)^{3 / 2} e^{-\beta \frac{p^2}{2 m}}(1+\varepsilon \sin \theta \cos \phi) \]
02

Understand what needs to be found

Compute the net number of particles per unit area per unit time passing through specified planes: (a) through the y-z plane, and (b) through the x-z plane.
03

Determine differential number density

To find the net number of particles, integrate the number density along the direction perpendicular to the plane. The relevant component of velocity is needed for flux computations.
04

Compute flux for y-z plane (per unit area per unit time)

For the y-z plane, we consider particles moving along the x-direction. The flux can be calculated by integrating in momentum space and multiplying by the velocity component along the x-axis.Calculate the flux:\[ J_{yz} = \int f(p, r) v_x d^3p \]Using the expression for velocity \( v_x = \frac{p_x}{m} \), integrate:\[ J_{yz} = \int_0^{2\pi} d\phi \int_0^{\pi} d\theta \sin \theta \int_0^{\infty} dp p^2 \frac{N}{V} \left( \frac{\beta}{2 \pi m} \right)^{3/2} e^{-\beta \frac{p^2}{2m}} \frac{p \sin \theta \, \cos \phi}{m} \left(1 + \varepsilon \sin\theta \cos \phi \right) \]
05

Simplify integration for y-z plane

Perform the integrations over \( p \), \( \theta \), and \( \phi \).First integrate over \(p\):\[ \int_0^{\infty} e^{-\beta \frac{p^2}{2m}} p^3 dp = \left( \frac{2m}{\beta} \right)^2 \frac{1}{2} \Gamma(2) = \left( \frac{2m}{\beta} \right)^2 \frac{1}{2} \cdot 1 = \left( \frac{2m}{\beta} \right)^2 \frac{1}{2} \]Integrate over \(\theta\) and \(\phi\):\[ \int_0^{2\pi} d\phi \int_0^{\pi} \sin^2 \theta \cos \phi \left(1 + \varepsilon \sin\theta \cos \phi \right) d\theta \]After carrying out the calculus, find the simplification with respective limits and factors.
06

Compute flux for x-z plane (per unit area per unit time)

For the x-z plane, consider particles moving along the y-direction. The procedure is similar to that for the y-z plane, but focus on the y-components:\[ J_{xz} = \int f(p, r) v_y d^3p \]Similarly consider velocity \( v_y = \frac{p_y}{m} \):\[ J_{xz} = \int_0^{2\pi} d\phi \int_0^{\pi} d\theta \sin \theta \int_0^{\infty} dp p^2 \frac{N}{V} \left( \frac{\beta}{2 \pi m} \right)^{3/2} e^{-\beta \frac{p^2}{2m}} \frac{p \sin\theta \, \sin \phi}{m} \left(1 + \varepsilon \sin\theta \cos \phi \right) \]
07

Simplify integration for x-z plane

Follow similar integrals as in Step 5, taking into account the change in angular dependence (i.e., \( \sin \phi \) instead of \( \cos \phi \)).
08

Write final expressions

Combine results of integrals performed in steps 4 to 7 to find the final expressions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Model
The ideal gas model is one of the simplest models in statistical physics. It assumes that the gas particles do not interact with each other except for elastic collisions. This means there are no intermolecular forces acting between the particles. It makes calculations easier because you can consider each particle independently. For an ideal gas in three dimensions, the behavior of particles regarding temperature, pressure, and volume can be described by the Ideal Gas Law: \( PV = Nk_B T \) where \(P\) is the pressure, \(V\) is the volume, and \(T\) is the temperature expressed in kelvin. \(N\) is the number of gas particles while \(k_B\) is the Boltzmann constant. This equation helps us study the distribution, motion, and kinetic energy of particles in a gas.
Number Density
Number density is a crucial concept in statistical physics, particularly when describing gases. It refers to the number of particles per unit volume. In phase space, we deal with a more detailed description involving both position and momentum. For an ideal gas, the number density in phase space can be given by the expression in the problem statement. This phase space distribution function \( f(p, r) \) incorporates both the spatial and momentum coordinates. It tells us how densely particles are packed in terms of their momentum and spatial coordinates. Understanding this function helps in determining other important quantities like particle flux.
Phase Space Integration
Phase space integration is a method used to determine macroscopic properties from microscopic behaviors. Phase space is a six-dimensional space with three spatial coordinates and three momentum coordinates. To calculate quantities like particle flux, we need to integrate the distribution function over relevant phase space coordinates. For our exercise, we are particularly interested in the flux of particles through specific planes, integrating over momentum space with respect to the appropriate velocity components. This integration reduces the complex problem into manageable parts by focusing on one plane at a time.
Particle Flux Calculations
Particle flux calculations are essential when determining how many particles pass through a given area over time. In this exercise, we need the flux through the y-z and x-z planes. For this, we use the distribution function and integrate it over momentum space while considering the respective velocity components (\(v_x\) for the y-z plane and \(v_y\) for the x-z plane). Integration simplifies the distribution function to obtain practical values representing the number of particles crossing the specified planes per unit area per unit time. By properly setting up and solving these integrals, we derive the net flux values accurately.

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Most popular questions from this chapter

An approximate expression for the Boltzmann collision operator \(\hat{C}_{1}\) is $$ \hat{C}=-y \hat{1}+y \sum_{i=1}^{5}\left|\phi_{i}\right\rangle\left\langle\phi_{i}\right| $$ where \(\left|\phi_{i}\right\rangle\) are the five orthonormal eigenfunctions of \(\hat{C}\) with eigenvalue equal to zero. The five eigenfunctions are \(\left.\left|\phi_{1}\right\rangle=|1\rangle,\left|\phi_{2}\right\rangle=\left|p_{x}\right|,\left|\phi_{3}\right\rangle=\mid p_{y}\right\\},\left|\phi_{4}\right|=\left|p_{z}\right\rangle\), and \(\left.\left|\phi_{5}\right\rangle=\mid 5 / 2-\beta p^{2} /(2 m)\right\\}\) They are orthonormal with respect to the scalar product, $$ (\phi(\boldsymbol{p}) \mid X(\boldsymbol{p}))=\left(\frac{\beta}{2 \pi m}\right)^{3 / 2} \int \mathrm{d} \boldsymbol{p} \mathrm{e}^{-\beta p^{2} / 2 m i} \phi(\boldsymbol{p}) X(\boldsymbol{p}) $$ 1\. Compute the coefficient of shear viscosity $$ \eta=-\lim _{\varepsilon \rightarrow 0} \frac{n \beta}{m^{2}}\left\langle p_{x} p_{y} \mid \frac{1}{\hat{C}+\varepsilon} p_{x} p_{y}\right\rangle $$ and the coefficient thermal conductivity, $$ K=-\lim _{f \rightarrow 0} \frac{n k_{\mathrm{B}}}{m^{2}}\left\langle\left(\frac{\beta p^{2}}{2 m}-\frac{5}{2}\right) p_{x} \mid \frac{1}{\hat{C}+\varepsilon} p_{x}\left(\frac{\beta p^{2}}{2 m}-\frac{5}{2}\right)\right\rangle $$ where \(n\) is the number density of particles in the gas described by \(\hat{C}\) and \(m\) is the mass of the particles. 2\. What are the units of \(y\) ? What are the units of \(\eta\) ?

Estimate the value of the coefficient of viscosity of argon gas at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Compare your estimate with the experimentally observed value of \(\eta=2.27 \times 10^{-4} \mathrm{~g} \mathrm{~cm}^{-1} \mathrm{~s}\) -1. Argon has an atomic weight of \(39.9\) and at low temperature forms a closely packed solid with density \(\rho=1.65 \mathrm{~g} / \mathrm{cm}^{3}\)

The electrons in a cubic sample of metallic solid of volume \(L^{3}\) and temperature \(T\) may be considered to be a highly degenerate ideal Fermi-Dirac gas. Assume the surface of the cube forms a potential energy barrier which is infinitely high, and assume that the electrons have spin \(s=1 / 2\), mass \(m\), and charge e. (a) What is the number of electrons in the velocity interval, \(v \rightarrow v+d v\) ? (b) Assume that an electrode is attached to the surface which lowers the potential energy to a large but finite value, \(W\), in a small area, \(A\), of the surface. Those electrons which have enough kinetic energy to overcome the barrier can escape the solid. How many electrons per second escape through the area, \(A\) ? Assume that \(W \gg 0\) and use this fact to make simplifying approximations to any integrals you might need to do.

A gas of neon atoms \(\left({ }^{20} \mathrm{Ne}_{10}\right)\), at temperature \(T=330 \mathrm{~K}\) and pressure \(P=10^{5} \mathrm{~Pa}\), is confined to a cubic box of volume \(V=8 \times 10^{3} \mathrm{~cm}^{3}\). Assume the radius of the atoms is approximately \(a=1.5 \mathrm{~A} .(\mathrm{a})\) What is the average speed \((V)\) of the neon atoms? (b) What is the mean free path \(\lambda\) of the neon atoms? (c) Compute the coefficient of thermal conductivity using the kinetic theory result, \(\left(K_{\text {lth }}=1 / 2 n(v) A k_{B}\right)\), where \(n=N / V\) is the particle number density, and the result obtained from Boltzmann's equation \(\left(K_{\text {Beq }}=75 k_{\mathrm{B}} /\left(256 a^{2}\right) \sqrt{k_{\mathrm{B}} T /(m \pi)}\right)\). How do they compare with the experimentally observed value of \(K_{\text {exp }} \approx 5 \times 10^{-2} \mathrm{~W} /(\mathrm{m} \mathrm{K}) ?(\mathrm{~d})\) If a temperature difference of \(\Delta T=\) \(3.0 \mathrm{~K}\) is created between two opposite walls of this system, what is the heat current flowing through the gas (in units of \(\left.\mathrm{J} /\left(\mathrm{m}^{2} s\right)\right)\) ?
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