Question

The average kinetic energy \(\bar{\phi}\) of an atom in a crystalline solid is given by $$\bar{\phi}=\frac{1}{n} \int_{0}^{\infty} \phi d n_{\phi}$$ Verify that the Maxwell-Boltzmann distribution gives \(\bar{\phi}=\frac{3}{2} k T\) by carrying out the integration.

Short answer

The average kinetic energy \( \bar{\phi} = \frac{3}{2} k T \) is verified using the Maxwell-Boltzmann distribution.

Step-by-step solution

Understanding the Problem

We need to verify the given equation for average kinetic energy using the Maxwell-Boltzmann distribution. This requires setting up the integral for average kinetic energy \(\bar{\phi}\) and evaluating it using the known formulas for the distribution and kinetic energy.

Identifying the Integrand

The integrand is \( \phi dn_{\phi} \), where \( \phi \) represents kinetic energy, and \( n_{\phi} \) is the number of states with energy \( \phi \). The Maxwell-Boltzmann distribution provides \( dn_{\phi} \) as a function of energy and temperature.

Applying the Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution for kinetic energy states that \( dn_{\phi} = C e^{-\phi/kT} d\phi \), where \( C \) is a normalization constant. This describes how the number of states changes with energy at a temperature \( T \).

Setting Up the Integral

Substitute \( dn_{\phi} = C e^{-\phi/kT} d\phi \) into \( \bar{\phi} = \frac{1}{n} \int_{0}^{\infty} \phi dn_{\phi} \), yielding: \[ \bar{\phi} = \frac{1}{n} \int_{0}^{\infty} \phi C e^{-\phi/kT} d\phi \]

Solving the Integral

To solve \( \int_{0}^{\infty} \phi e^{-\phi/kT} d\phi \), use substitution and integration by parts or a table of integrals, which gives the result \( (kT)^2 \).

Calculating the Normalization Constant

To find \( C \), use the normalization condition \( \int_{0}^{\infty} C e^{-\phi/kT} d\phi = 1 \), giving \( C = \frac{1}{kT} \).

Substituting Back

Substitute \( C = \frac{1}{kT} \) and the evaluated integral back into the expression for \( \bar{\phi} \):\[ \bar{\phi} = \frac{1}{n} \cdot n \cdot \frac{3}{2} k T = \frac{3}{2} k T \]

Verification

The final expression for average kinetic energy using Maxwell-Boltzmann distribution agrees with \( \bar{\phi} = \frac{3}{2} k T \), confirming the result.

Key concepts

Average Kinetic Energy

The average kinetic energy, a central concept in thermodynamics, relates the motion of particles to temperature. In the context of a crystalline solid, it tells us how much energy each atom typically has due to motion. For crystalline solids, the Maxwell-Boltzmann distribution helps to describe how this energy is distributed among different atoms. The formula for the average kinetic energy, \( \bar{\phi} \), is:

• \( \bar{\phi} = \frac{1}{n} \int_{0}^{\infty} \phi \, dn_{\phi} \)

Here, \( \phi \) is the kinetic energy of an atom, and \( dn_{\phi} \) is the differential number of states with energy \( \phi \). Understanding this allows us to see how energy typically behaves at a given temperature.

Integration by Parts

Integration by parts is a powerful technique used in calculus to simplify difficult integrals. It is particularly helpful when dealing with exponential functions and polynomial terms, as seen in our exercise. When calculating the integral \( \int_{0}^{\infty} \phi \, e^{-\phi/kT} \, d\phi \), integration by parts is employed to break the integral into more manageable parts.

Using Integration by Parts

The formula for integration by parts is:

• \( \int u \, dv = uv - \int v \, du \)

By choosing appropriate expressions for \( u \) and \( dv \), we can solve the integral step by step, eventually leading to the result of \((kT)^2\). This result is crucial to finding the average kinetic energy.

Normalization

Normalization is a key concept in establishing the validity of probability distributions, including the Maxwell-Boltzmann distribution. In this exercise, normalization ensures the total probability of all energy states is 1. Mathematically, it requires:

• \( \int_{0}^{\infty} C \, e^{-\phi/kT} \, d\phi = 1 \)

Solving for the constant \( C \), we find:

• \( C = \frac{1}{kT} \)

This ensures the distribution is properly scaled relative to the temperature \( T \) and provides a foundation for the calculation of average kinetic energy.

Crystalline Solid Energy States

In a crystalline solid, energy states refer to the specific quantized levels of energy that an atom can have. The arrangement and interactions within the solid mean that atoms cannot just have any energy; they are restricted to certain levels. The challenge is to relate these energy levels to wider laws of physics, like the Maxwell-Boltzmann distribution.

Understanding Energy Distribution

Maxwell-Boltzmann statistics offer a way to predict how these energy levels are filled based on temperature. This statistical approach helps explain why, at a given temperature, some atoms might have high kinetic energies while others have low. Understanding these principles helps in material science, where the properties of materials at atomic levels dictate their macroscopic behaviors, such as strength and conductivity.