A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heatconducting piston that divides the cylinder into two parts. Initially, the piston is clamped in the center with I liter of air at \(300 \mathrm{~K}\) and \(2 \mathrm{~atm}\) pressure on one side and 1 liter of air at \(300 \mathrm{~K}\) at 1 atm pressure on the other side. The piston is released and reaches equilibrium in pressure and temperature at a new position. Compute the final pressure and temperature and increase of entropy if air is assumed to be the ideal gas. What irreversible process has taken place?

The Ideal Gas Law is crucial in understanding how gases behave under various conditions. It is given by the formula:
\[ PV = nRT \]
where:

• \(P\) is the pressure
• \(V\) is volume
• n is the number of moles of the gas
• \(R\) is the gas constant
• \(T\) is the temperature in Kelvin

This formula relates the state variables of a gas, allowing us to calculate unknowns when some quantities are given.
In the given problem, we used the Ideal Gas Law to compute the number of moles of air in the cylinder before and after the piston movement. For example, for the high-pressure side, the moles were calculated as:
\[ n_1 = \frac{P_1 \cdot V_1}{R \cdot T_1} \]
Plugging in the values, we get:
\[ n_1 = \frac{2 \text{ atm} \cdot 1 \text{ liter}}{0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \cdot 300 \text{ K}} = 0.0812 \text{ moles} \]
Understanding and applying the Ideal Gas Law helps predict how gases in different compartments will interact and what changes will occur when conditions alter.

An isothermal process is a thermodynamic process during which the temperature remains constant. Since temperature doesn’t change, the internal energy of an ideal gas does not change either in this type of process.
For isothermal processes, the key takeaway is:

• Energy transfer is only in the form of work done by or on the system
• No change in the internal energy of the gas since the temperature is constant

In our example, the temperature of the air on both sides of the piston remains constant at 300 K throughout the process. Consequently, when the piston is released and the system reaches equilibrium, the final temperature will still be 300 K. This is critical for ensuring that calculations remain accurate when applying the Ideal Gas Law and understanding subsequent processes like entropy change.

Irreversible processes in thermodynamics are those that cannot return both the system and the surroundings to their original states. In irreversible processes, there is an increase in entropy.
The problem features an irreversible process: the free piston movement within an insulated system. Characteristics of such a process include:

• No external work is done on or by the system; the only internal factors drive changes
• It is a spontaneous process, as a system tends to move towards equilibrium naturally
• Entropy increases

In our case, when the piston is released, there is a free expansion of gas. The initial constraint is gone, and the air expands or compresses to reach a uniform pressure of equilibrium. This expansion is not reversible without additional external work, making the process typically irreversible in thermodynamic terms.

Entropy is a measure of disorder or randomness in a system and increases in irreversible processes.
To calculate the change in entropy for the problem at hand, we use the formula:
\[ \Delta S = nR\ln \frac{V_f}{V_i} \]
Entropy change is calculated separately for each side of the piston and then added together for the total change.
For the high-pressure side:
\[ \Delta S_1 = n_1 R \ln \frac{V_f}{V_i} = 0.0812 \times 0.0821 \times \ln 2 \]
For the low-pressure side:
\[ \Delta S_2 = n_2 R \ln \frac{V_f}{V_i} = 0.0406 \times 0.0821 \times \ln 2 \]
Total entropy change is:
\[ \Delta S = \Delta S_1 + \Delta S_2 = 0.0587 \times \ln 2 \]
This results in an overall increase in entropy, signifying a more disordered system post-process. Understanding entropy change is essential for grasping how systems evolve towards equilibrium and the nature of energy dispersal in thermodynamic processes.