There are \(n_{1}\) moles of an ideal monatomic gas at temperature \(T_{1}\) and pressure \(P\) in one compartment of an insulated container. In an adjoining compartment separated by an insulating partition are \(n_{2}\) moles of another ideal monatomic gas at temperature \(T_{2}\) and pressure \(P .\) When the partition is removed: (a) Show that the final pressure of the mixture is \(P\). (b) Calculate the entropy change when the gases are identical. (c) Calculate the entropy change when the gases are different.

In thermodynamics, an isentropic process is one where the entropy of the system remains constant. This typically occurs in idealized situations where there is no exchange of heat with the environment. During an isentropic process, the system is considered to be adiabatic (no heat transfer) and reversible. Key characteristics include:
- Constant entropy
- Reversible process
- Adiabatic conditions
In the exercise, since the container is insulated and the gas mixing does not involve heat exchange with the surroundings, we can see elements of isentropic behavior. However, because gases are allowed to mix freely, the process is not strictly isentropic.

Entropy is a measure of the disorder or randomness of a system. Mixing different gases increases the entropy because it increases the disorder of the system. For identical gases, mixing them at the same initial conditions results in no entropy change due to the lack of new disorder. But for different gases, the entropy change can be derived using the entropy formula for ideal gases:
\( \triangle S = nR \text{ln}\bigg(\frac{V_f}{V_i}\bigg) \)
Where:

• \(V_i \) is the initial volume of the gas, and
• \( V_f \) is the final volume after mixing.

For the given exercise where each gas in different compartments expands to the total volume, the change in entropy is:
\( \triangle S_1 = n_1 R \text{ln}\bigg(\frac{2V}{V}\bigg) = n_1 R \text{ln}(2) \)
\( \triangle S_2 = n_2 R \text{ln}\bigg(\frac{2V}{V}\bigg) = n_2 R \text{ln}(2) \)
Therefore, the total entropy change for different gases becomes:
\( \triangle S_{total} = (n_1 + n_2)R \text{ln}(2) \)

An isothermal process is one where the temperature of a system remains constant throughout the process. For an ideal gas, this means that any work done by or on the system is exactly balanced by heat exchange with the surroundings. Key properties include:
- Constant temperature
- Heat transfer equal to work done
In the context of the given exercise, the mixing of gases when they are identical occurs isothermally. This ensures that the temperature remains constant and the internal energy of the gases remains unchanged. This is why the entropy change for identical gases mixing under identical conditions results in no change in entropy.

The Ideal Gas Law is a mathematical relationship that describes the behavior of ideal gases and is given by:
\( PV = nRT \)
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 helps in understanding how different properties of the gas correlate. In the given exercise, since both compartments initially have the same pressure and temperature, when the partition is removed and the gases mix in an insulated container, the final pressure remains the same. This demonstrates the principles of the Ideal Gas Law, as the quantities remain balanced due to the initial conditions of the container.

Thermodynamic equilibrium refers to a state in which a system's macroscopic properties do not change over time. The system is in a balanced condition where mechanical, thermal, and chemical equilibria are achieved. For a system in thermodynamic equilibrium:
- Temperature is uniform throughout the system
- Pressure is uniform throughout the system
- No net chemical reactions occur
In the given exercise, after the partition is removed and gases are mixed, the system will eventually reach thermodynamic equilibrium, where the pressure and temperature become uniform throughout the container. This equilibrium occurs because the system naturally evolves towards a state of maximum entropy, where energy distribution is even.