91Ó°ÊÓ

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of an ideal gas. This law is expressed by the formula:

• \( PV = nRT \)

Where:

• \( P \) is the pressure of the gas
• \( V \) is the volume of the gas
• \( n \) is the number of moles of gas
• \( R \) is the ideal gas constant, approximately 8.314 J/(mol·K)
• \( T \) is the absolute temperature in Kelvin

This law assumes that the gas particles are small and that they do not interact with each other. For many gases under standard conditions, this approximation works very well.
In the example given, you can see how initial conditions, such as temperature, volume, and moles, are vital. They help us determine changes in the gas behavior when conditions change, like in processes where volume changes but temperature remains the same.

An isothermal process is a change in a system where the temperature remains constant. This means that, while the system might expand or contract, the thermal energy is regulated so that the temperature doesn't change.
In mathematical terms, during an isothermal process, we can use the formula:

• \( Q = W \)

Here:

• \( Q \) is the heat added to the system
• \( W \) is the work done by the system

For an ideal gas undergoing an isothermal expansion, the change in internal energy \( \Delta U \) is zero. Because \( T \) is constant, any work done is balanced by the heat added or removed.
In the provided exercise, you calculate the entropy change using the formula for an isothermal process. The fact that temperature remains constant, even as volume changes, is crucial in determining how entropy changes without a change in internal energy.

An irreversible process, as opposed to a reversible one, is a process where the system cannot return to its initial state without leaving changes in its surroundings. There is an increase in entropy and energy loss is inevitable.
Key indicators of irreversibility include:

• Free expansion
• Mixing of different gases
• Uncontrolled chemical reactions

These processes are not at equilibrium at all points; they happen spontaneously and cannot be reversed step-by-step without external intervention.
In the given problem, the expansion of helium gas from the tank into the jar is deemed irreversible. This is because it is a spontaneous process into a vacuum where balance with the surroundings isn’t maintained. Hence, the process results in an increase in entropy.

Free expansion refers to the process where a gas expands into an available space without performing external work and without the addition or removal of heat. This is also known as Joule expansion.
This process is characterized by:

• Sudden and rapid expansion
• Unresisted movement of gas particles
• No energy exchange with the surroundings through work

During free expansion, ideal gases show no change in internal energy nor temperature, as the expansion involves no work nor heat exchange. This is different from other processes where heat or work could alter these conditions.
In the task presented, when helium gas leaks into the evacuated jar, it undergoes free expansion. The entropy change is calculated because the gas spreads into a larger volume, although temperature is held constant, signifying an isothermal free expansion.