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 \( S_f = S_i \) when the partition is removed.

Step-by-step solution

Define the Initial and Final States

Initially, there are two separate chambers, each with volume V, containing equal numbers of gas molecules, N, at the same temperature. After removing the partition, the whole system has a volume of 2V with 2N gas molecules.

Calculate Initial Entropy Using Multiplicity

For an ideal gas, multiplicity \( \Omega \) is given by \( \Omega = \frac{V^N}{N!}\). Using Stirling's approximation, \( \ln N! \approx N \ln N - N \), the initial entropy \( S_i \) is: \( S_i = k_B \ln \( \frac{V^N}{N!}\) + k_B \ln \( \frac{V^N}{N!}\) = k_B \left( 2N \ln V - 2N \ln N + 2N \right) \).

Calculate Final Entropy Using Multiplicity

After removing the partition, the total number of gas molecules is now 2N in a volume of 2V. Thus, the multiplicity \( \Omega_f \) is given by \( \Omega_f = \frac{(2V)^{2N}}{(2N)!} \. \). Using Stirling's approximation, the final entropy \( S_f \) is: \( S_f = k_B \ln \( \frac{(2V)^{2N}}{(2N)!}\) = k_B \left( 2N \ln (2V) - 2N \ln (2N) + 2N \right) \).

Compare Initial and Final Entropies

Rewriting for clarity: \( S_i = k_B \left( 2N \ln V - 2N \ln N + 2N \right) \) \( S_f = k_B \left( 2N \ln V + 2N \ln 2 - 2N \ln N - 2N \ln 2 + 2N \right) = k_B \left( 2N \ln V - 2N \ln N + 2N \right) = S_i \). The terms involving \( \ln 2 \) cancel out, showing that \( S_f = S_i \), hence there is no change in entropy.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of molecules (N). It is expressed as PV = nRT, where:

• P is the pressure
• V is the volume
• n is the number of moles
• R is the gas constant
• T is the temperature

This law assumes that gas molecules do not interact with each other and that they occupy negligible space.
When we talk about ideal gases, we refer to a theoretical gas composed of many randomly moving point particles that only interact when they collide elastically. The Ideal Gas Law is useful in predicting the behavior of gases under different conditions, such as changes in temperature and volume.

Stirling's Approximation

Stirling's approximation is a mathematical formula used to estimate the factorial of a large number, N. It is given by:

• \ ln(N!) \approx N ln(N) - N

This is extremely useful in statistical mechanics and thermodynamics, particularly when dealing with large numbers of particles like in an ideal gas.
In the step-by-step solution, Stirling's approximation simplifies the calculation of entropy. Without Stirling's approximation, the factorial of a large number would be complex to handle.
Here's how it's applied: Suppose we have N gas molecules. The multiplicity (Ω), or the number of ways the molecules can be arranged, can be written as \( \Omega = \frac{V^N}{N!} \). Using Stirling's approximation for \( N! \), this becomes easier to compute, transforming to \( \Omega \approx \frac{V^N}{e^{N}N^N} \) which simplifies further entropy calculations.

Multiplicity in Thermodynamics

'Multiplicity' refers to the number of ways the state of a system can be realized. In thermodynamics, it is denoted as Ω.
For an ideal gas, the multiplicity is linked to the volume and number of gas molecules. For instance, with N molecules in volume V, Ω can be written as:

• \ \Omega = \frac{V^N}{N!}

This helps calculate the entropy, which is a measure of the disorder or randomness in the system. Entropy (S) is related to multiplicity by the formula:

• \ S = k_B \ ln(\Omega)

Where \( k_B \) is the Boltzmann constant.
In the context of the provided example, the multiplicities of the initial and final states are calculated to show that the entropy remains unchanged when the partition between two chambers of gas is removed. Multiplicity, in essence, quantifies the possible arrangements, leading to more profound insights into the system's entropy. When we realize these arrangements or microstates, we are better geared to predict the thermodynamic properties of the system, like energy and temperature.