Question

Pulling out the partition between two chambers of a gas. A partition separates equal volumes containing equal numbers $N$ of ideal gas molecules of the same species at the same temperature. Using a simple lattice model for ideal gases, evaluate the relevant multiplicity terms to show that the entropy of the composite system does not change when the partition is removed. (Hint: use Stirling's approximation from Appendix B.)

Short answer

The entropy of the system does not change when the partition is removed due to equal opposing factors, resulting in $\mathrm{\Delta }S=0$.

Step-by-step solution

Understanding the Problem

First, identify what needs to be calculated: the entropy change when a partition is removed between two chambers of ideal gas. Both chambers have equal volumes with the same number of molecules and are at the same temperature.

Recall Entropy Formula

Entropy (S) is related to the multiplicity (Ω) of a system by the formula: $$S=k_B\ln \mathrm{\Omega }$$ where $k_B$ is the Boltzmann constant.

Lattice Model for Ideal Gas

Assume each gas molecule can be in either of the two volumes. Initially, the number of ways to distribute $N$ gas molecules in one chamber is given by the multiplicity: ${\mathrm{\Omega }}_1=\frac{V^N}{N!}$. Similarly, the multiplicity in the other chamber is ${\mathrm{\Omega }}_2=\frac{V^N}{N!}$.

Total Multiplicity Before Removing Partition

The total multiplicity of the system before removing the partition is the product of the individual multiplicities: $${\mathrm{\Omega }}_{initial}={\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2=\left(\frac{V^N}{N!}\right)\left(\frac{V^N}{N!}\right)={\left(\frac{V^N}{N!}\right)}^2$$.

Total Multiplicity After Removing Partition

Once the partition is removed, the gas molecules can be freely distributed in the combined volume. The new multiplicity is: $${\mathrm{\Omega }}_{final}=\frac{(2V{)}^{2N}}{(2N)!}$$.

Applying Stirling's Approximation

Using Stirling's approximation, $N!\approx \sqrt{2\pi N}(N/e{)}^N$, we approximate multiplicities before and after removing the partition: $${\mathrm{\Omega }}_{initial}={\left(\frac{V^N}{\sqrt{2\pi N}(N/e{)}^N}\right)}^2\approx \frac{V^{2N}}{(2\pi N){\left(N/e\right)}^{2N}}=\frac{(eV{)}^{2N}}{2\pi N^{2N+1}}$$ and $${\mathrm{\Omega }}_{final}\approx \frac{(2V{)}^{2N}}{\sqrt{2\pi (2N)}((2N)/e{)}^{2N}}\approx \frac{(2V{)}^{2N}}{\sqrt{4\pi N}(2N/e{)}^{2N}}\approx \frac{(2V{)}^{2N}}{\sqrt{4\pi N}{2}^{2N}{N}^{2N}/e^{2N}}=\frac{(2V{)}^{2N}}{2^{2N}\sqrt{4\pi N}{N}^{2N}/e^{2N}}=\frac{(e(2V){)}^{2N}}{4\pi \sqrt{N}{N}^{2N}}$$.

Finding Entropy Change

Determine the entropy change: $$\mathrm{\Delta }S=k_B\ln {\mathrm{\Omega }}_{final}-k_B\ln {\mathrm{\Omega }}_{initial}=k_B[(2N\ln 2V-2N\ln N-\frac{1}{2}\ln (4\pi N))-(2N\ln V-2N\ln N-\ln (2\pi N))]=k_B[2N\ln 2+\frac{1}{2}(\ln 4-\ln 2)]=k_B[2N\ln 2+\frac{1}{2}\ln 2]=k_{B}2N\ln 1=0$$.

Key concepts

partition removal in gases

When you have two chambers containing ideal gas separated by a partition, removing this partition means letting the gas molecules freely move throughout the combined volume. In our problem, both chambers have equal volumes and the same number of molecules at the same temperature. This setup makes the problem simpler to analyze.

The main question is how removing the partition affects the system's entropy, which depends on the distribution of the gas molecules. We use the concept of multiplicity, which is basically the number of ways the gas molecules can be arranged in the given space.

ideal gas laws

The ideal gas law is an important principle for understanding the behavior of gases. It states that the pressure, volume, and temperature of an ideal gas are related by the equation:

Stirling's approximation

Dealing with factorials of large numbers can be cumbersome. Stirling's approximation provides a way to simplify expressions involving factorials. It approximates the factorial of a large number N as:

multiplicity of states

The multiplicity of a system refers to the number of ways the system can be arranged. In our case, it means the number of ways to distribute the gas molecules in the available volume. The more ways to arrange the molecules, the higher the multiplicity.

Entropy is closely related to multiplicity: the higher the multiplicity, the higher the entropy. This is because systems tend to move towards more probable arrangements, which correspond to higher multiplicity.

For an ideal gas in a chamber, the multiplicity before removing the partition is restricted to the initial volumes. After removing the partition, the molecules enjoy more freedom. However, the calculation shows that this increased spatial freedom does not actually lead to a net change in entropy. This surprising result highlights the subtlety and power of using entropy and multiplicity concepts in thermodynamics.