Problem 152 - A Complete Resource Book in Chemistry for JEE Main

Question

An ideal gas is allowed to expand both reversibly and irreversibly in an isolated system. If \(\mathrm{T}_{i}\) is the initial temperature and \(\mathrm{T}_{\mathrm{f}}\) is the final temperature, which one of the following statements is correct? [2006] (a) \(\left(\mathrm{T}_{f}\right)_{\text {irrev }}>\left(\mathrm{T}_{\mathrm{F}}\right)_{\mathrm{rev}}\) (b) \(\mathrm{T}_{\mathrm{f}}>\mathrm{T}_{i}\) for reversible process but \(\mathrm{T}_{\mathrm{f}}=\mathrm{T}_{1}\) for irreversible process (c) \(\left(\mathrm{T}_{\mathrm{f}}\right)_{\text {irrev }}=\left(\mathrm{T}_{\mathrm{F}}\right)_{\mathrm{rev}}\) (d) \(\mathrm{T}_{\mathrm{f}}=\mathrm{T}_{i}\) for both reversible and irreversible processes

Step-by-Step Solution

Answer

(d) T_{ ext{f}} = T_{ ext{i}} for both reversible and irreversible processes.

Step 1: Understanding Ideal Gas Expansion

In an isolated system, the total internal energy is conserved. An ideal gas expanding in such a system does not exchange heat with its surroundings.

Step 2: Identifying Temperature Changes

For an isolated system during gas expansion, since there is no heat exchange, the internal energy change is related only to work done by or on the system. For both reversible and irreversible processes, the energy remains constant due to isolation.

Step 3: Consider Reversible Expansion

In a reversible expansion of an ideal gas, the process occurs very slowly, allowing the system to adjust and reach equilibrium at each stage, thus conserving internal energy. Hence, the final temperature remains equal to the initial temperature ( T_{ ext{f}} = T_{ ext{i}} ).

Step 4: Consider Irreversible Expansion

In irreversible expansion, the process is quick, leading to a loss in work and hence entropy increases. However, since the system is isolated and no heat exchange occurs, the internal energy and therefore the temperature remains unchanged. Thus, T_{ ext{f}} = T_{ ext{i}} .

Step 5: Conclusion of Temperature Changes

From the analysis, for both reversible and irreversible expansions, the final temperature is the same as the initial temperature in an isolated system. Thus, T_{ ext{f}} = T_{ ext{i}} for both scenarios.

Key Concepts

Reversible Process

When dealing with a reversible process, it's essential to think of it as a very controlled and slow expansion of the gas. In such a process, every infinitesimal step is in equilibrium. This balance means the process can be reversed without leaving any impact on the surroundings. Imagine it like a bike slowly rolling down a gentle hill, where at any point, you can decide to roll back up effortlessly.

• A reversible expansion implies maximum efficiency because no energy is wasted.
• However, this is more of an ideal concept and doesn't occur in real-world scenarios.
• This careful alignment means, for an isolated system, the final temperature will precisely equal the initial temperature.

So, the energy isn't lost or gained, and that's why the temperature remains stable from start to finish.

Irreversible Process

Irreversible processes are quite the opposite of reversible ones. They happen spontaneously and cannot be undone without a change in the environment. Think of dropping a glass and watching it shatter—it won't unshatter by itself. Irreversible processes are quicker, often characterized by rapid changes, and don't maintain equilibrium.

• In real life, irreversible processes are common, as they tend to be faster.
• During irreversible expansion in an isolated system, while you might think energy is "lost," it's actually distributed differently due to the speed of the change.
• However, like reversible processes in isolated systems, the temperature ultimately remains unchanged because no energy is exchanged with the surroundings.

This quick expansion results in increased entropy, but due to system isolation, the internal energy, and thus the temperature, stays constant.

Internal Energy Conservation

Internal energy is a crucial concept when studying ideal gas expansion, especially within an isolated system. An isolated system is closed off from the environment, meaning it cannot exchange heat or matter with its surroundings. Think of it like a sealed thermos bottle.

• In such systems, the law of energy conservation is strictly followed—energy within the system remains constant.
• For ideal gases, internal energy is directly related to temperature.
• Since no external energy is introduced or lost, any expansion, whether reversible or irreversible, must result in the internal energy remaining the same.

Thus, even if the processes occur differently, both start and end on an equal footing in terms of temperature. This conservation is why temperature doesn't change after either type of process in an isolated setup.