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Chapter 2: Problem 9

A gas of \(N\) identical particles is free to move among \(M\) distinguishable lattice sites on a lattice with volume \(V\), such that each lattice site can have at \(N \ll M\) most one particle at any time. The density of lattice sites is \(\mu=M / V\). Assume that and that all configurations of the lattice have the same energy. (a) Compute the entropy of the gas. (b) Find the equation of state of the gas. (Note: the pressure of an ideal gas is an example of an entropic force.)

Short Answer

Expert verified
Entropy: \( S = Nk_B \left( \ln\left( \frac{M}{N} \right) + 1 \right) \). Equation of state: \( P = \frac{Nk_B}{V} \).

Step by step solution

01

Define Variables and Given Information

We are given a gas of \(N\) identical particles moving among \(M\) distinguishable lattice sites, with volume \(V\). Each lattice site can hold at most one particle. The density of lattice sites is \(\mu = \frac{M}{V}\). We need to calculate the entropy of the gas and find the equation of state.
02

Determine the Number of Microstates

The total number of ways to distribute \(N\) particles among \(M\) sites (where each site can hold at most one particle) is given by the binomial coefficient: \[ \Omega = \binom{M}{N} = \frac{M!}{N!(M-N)!} \]
03

Use Stirling's Approximation

For large \(M\) and \(N\), we can use Stirling's approximation \( \ln(N!) \approx N \ln(N) - N \) to simplify the expression for \(\Omega\): \[ \ln(\Omega) \approx M \ln(M) - M - N \ln(N) + N - (M-N) \ln(M-N) + (M-N) \]
04

Simplify the Expression for Entropy

Simplify the expression using the fact that \(N \ll M\): \[ \ln(\Omega) \approx M \ln(M) - (M-N) \ln(M) - N \ln(N) + N \] \[ \ln(\Omega) \approx N \ln\left(\frac{M}{N}\right) \] Thus, the entropy \(S\) is: \[ S = k_B \ln(\Omega) = Nk_B \ln\left(\frac{Me}{N}\right) \]
05

Find Entropy Expression

Thus, the entropy of the gas is: \[ S = Nk_B \ln\left(\frac{M}{N}\right) + Nk_B \]
06

Compute the Equation of State

The equation of state relates pressure \(P\) to other thermodynamic quantities. Here, the pressure can be seen as an entropic force. Using the thermodynamic relation: \[ P = \left( \frac{\partial S}{\partial V} \right)_N \] Substitute the expression for entropy into this relation: \[ S = Nk_B \ln\left(\frac{M}{N}\right) \] Recall \(\mu = \frac{M}{V}\), hence \(M = \mu V\): \[ S = Nk_B \ln\left(\frac{\mu V}{N}\right) + Nk_B \] Differentiate with respect to volume \(V\): \[ \frac{\partial S}{\partial V} = Nk_B \frac{1}{V} \] Thus, the pressure \(P\) is: \[ P = Nk_B \frac{1}{V} = \frac{Nk_B T}{V} \]
07

Summarize Results

The entropy of the gas is: \[ S = Nk_B \left( \ln\left( \frac{M}{N} \right) + 1 \right) \] The equation of state for the gas is: \[ P = \frac{Nk_B}{V} \]

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Key Concepts

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

Entropy
Entropy is a measure of the disorder or randomness in a system. In statistical mechanics, it quantifies the number of possible microscopic configurations (microstates) that correspond to a macroscopic state. This can be expressed using the Boltzmann formula: \(S = k_B \text{ln}(\text{Ω})\), where \(S\) represents entropy, \(k_B\) is the Boltzmann constant, and \(Ω\) is the number of microstates. In our exercise, we derived the entropy of the gas as \(S = Nk_B \text{ln} \frac{M}{N} + Nk_B\). This formula comes from considering the different ways we can arrange \(N\) particles in \(M\) distinguishable lattice sites, emphasizing the relationship between microstates and entropy.
Microstates
Microstates refer to the different possible arrangements of particles in a system that result in the same macroscopic properties. Each microstate represents a unique way the particles can be distributed within the system. In the context of our gas, each microstate is a different configuration of how the \(N\) particles are distributed among the \(M\) lattice sites. The total number of such microstates is given by the combinatorial expression \(Ω = \binom{M}{N}\), which accounts for all possible distributions. This number is crucial for determining the entropy, as shown in the entropy expression we derived.
Equation of State
An equation of state is a mathematical equation that describes the relationship between different state variables, such as pressure (P), temperature (T), volume (V), and number of particles (N) in a thermodynamic system. In our exercise, we derived an equation of state for the gas, represented as \(P = \frac{Nk_B T}{V}\). This equation connects the pressure of the gas to its volume, temperature, and the number of particles, illustrating how changes in one variable affect the others. It’s an important tool for understanding the behavior of gases, providing a quantitative relationship between macroscopic properties.
Ideal Gas
An ideal gas is a theoretical gas composed of many randomly moving particles that do not interact except when they collide elastically. The ideal gas law, \(PV = Nk_BT\), is a simple but remarkably accurate equation of state for many gases under various conditions. In our case, the equation of state we derived, \(P = \frac{Nk_B T}{V}\), is consistent with the ideal gas law, assuming constant temperature. This illustrates that our lattice gas behaves similarly to an ideal gas, where the pressure depends linearly on the number of particles and temperature, and inversely on volume. Understanding these ideal behaviors helps in analyzing real gases by providing foundational principles.

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Most popular questions from this chapter

Find the number of permutations of the letters in the word, MONOTONOUS. In how many ways are four O's together? In how many ways are (only) 3 O's together?

A fair six-sided die is thrown \(N\) times and the result of each throw is recorded. (a) If the die is thrown \(N=12\) times, what is the probability that odd numbers occur three times? If it is thrown \(N=120\) times, what is the probability that odd numbers occur 30 times? Use the binomial distribution. (b) Compute the same quantities as in part (a) but use the Gaussian distribution. (Note: For part (a) compute your answers to four places.) (c) Plot the binomial and Gaussian distributions for \(N=2\) and \(N=12\).

An Einstein solid (in 3D space) has 100 lattice sites and 300 phonons, each with energy \(\hbar \omega=0.01 \mathrm{eV}\). (a) What is the entropy of the solid (give a number)? (b) What is the temperature of the, solid (give a number)?

Consider a lattice with \(N\) spin- 1 atoms with magnetic moment \(\mu\). Each atom can be in one of three spin states, \(S_{z}=-1,0,+1\). Let \(n_{-1}, n_{0}\), and \(n_{1}\) denote the respective number of atoms in each of those spin states. Find the total entropy and the configuration which maximizes the total entropy. What is the maximum entropy? (Assume that no magnetic field is present, so all atoms have the same energy. Also assume that atoms on different lattice sites cannot be exchanged, so they are distinguishable.)

A system consists of \(N\) noninteracting, distinguishable two-level atoms. Each atom can exist in one of two energy states, \(E_{0}=0\) or \(E_{1}=\varepsilon\). The number of atoms in energy level, \(E_{0}\), is \(n_{0}\) and the number of atoms in energy level, \(E_{1}\), is \(n_{1}\). The internal energy of this system is \(U=n_{0} E_{0}+n_{1} E_{1}\). (a) Compute the multiplicity of microscopic states. (b) Compute the entropy of this system as a function of internal energy. (c) Compute the temperature of this system. Under what conditions can it be negative? (d) Compute the heat capacity for a fixed number of atoms, \(N\).
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