A thin partition divides a container of volume V into two parts. One side contains nA moles of gas A in a fraction fA of the container; that is, VA = fAV. The other side contains nB moles of a different gas B at the same temperature in a fraction fB of the container. The partition is removed, allowing the gases to mix. Find an expression for the change of entropy. This is called the ,entropy of mixing.

Verified step by step guidance

1. Identify the initial conditions for each gas. For gas A, the initial volume is $V_A = f_A V$ and the number of moles is $n_A$. For gas B, the initial volume is $V_B = f_B V$ and the number of moles is $n_B$. Both gases are at the same temperature $T$.
2. Use the formula for the entropy change of an ideal gas when it expands or contracts isothermally: $\Delta S = nR \ln\left(\frac{V_{\text{final}}}{V_{\text{initial}}}\right)$, where $n$ is the number of moles of the gas, $R$ is the gas constant, and $V_{\text{initial}}$ and $V_{\text{final}}$ are the initial and final volumes of the gas, respectively.
3. Calculate the entropy change for gas A, $\Delta S_A$, as it expands from volume $V_A$ to the total volume $V$. Substitute into the entropy change formula: $\Delta S_A = n_A R \ln\left(\frac{V}{V_A}\right)$.
4. Calculate the entropy change for gas B, $\Delta S_B$, as it expands from volume $V_B$ to the total volume $V$. Substitute into the entropy change formula: $\Delta S_B = n_B R \ln\left(\frac{V}{V_B}\right)$.
5. Add the entropy changes of gas A and gas B to find the total entropy change due to mixing, $\Delta S_{\text{mixing}} = \Delta S_A + \Delta S_B$. This represents the increase in entropy as the gases mix and occupy the entire volume $V$.

Key Concepts

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the amount of energy in a physical system that is not available to do work. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and it tends to increase, reflecting the natural tendency of systems to evolve towards thermodynamic equilibrium.

Moles and Gas Laws

A mole is a unit in chemistry that represents a specific number of particles, typically atoms or molecules, equivalent to Avogadro's number (approximately 6.022 x 10²³). Understanding the behavior of gases, particularly through the ideal gas law (PV=nRT), is crucial for analyzing gas mixtures. This law relates pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas, providing a foundation for calculating properties of gas mixtures.

Entropy of Mixing

The entropy of mixing refers to the increase in entropy that occurs when two or more different gases are allowed to mix. This process results in a greater number of possible microstates, leading to increased disorder. The change in entropy during mixing can be calculated using the formula ΔS = -R(nA ln(fA) + nB ln(fB)), where R is the gas constant, and nA and nB are the moles of gases A and B, respectively, reflecting how the distribution of particles contributes to the overall entropy.