Problem 9 - A Complete Resource Book in Chemistry for JEE Main

Question

If a gas at constant temperature and pressure expands, then its (a) internal energy decreases (b) entropy increases and then decreases (c) internal energy increases (d) internal energy remains constant

Step-by-Step Solution

Answer

The correct answer is (d) internal energy remains constant.

Step 1: Understanding the Ideal Gas Behavior

For an ideal gas undergoing an isothermal (constant temperature) process, the internal energy depends solely on temperature. Since the temperature is constant, the change in internal energy, according to the first law of thermodynamics, is zero.

Step 2: Evaluating Changes in Entropy

When a gas expands at constant temperature, its entropy must increase because the gas molecules have more volume to occupy, increasing the disorder or randomness of the system.

Step 3: Revisiting Internal Energy

Since the internal energy is not directly affected by volume changes in an ideal gas at constant temperature, it remains unchanged.

Key Concepts

Isothermal Process

In an isothermal process, the temperature remains constant throughout the change, which is a core concept for gases, especially ideal gases. When a gas expands isothermally, it does so at a steady temperature.
This means there's a delicate balance between heat exchange with surroundings and work done by the gas. Even though the system might exchange heat with its surroundings, it doesn’t lead to a temperature change.

• Heat absorbed is equal to work done.
• Because temperature remains constant, the internal energy of an ideal gas doesn't change.

Consequently, formulas such as the ideal gas law can simplify the analysis of such processes, with pressure and volume varying inversely as explained by the equation: P imes V = n imes R imes T, where n is the number of moles, R is the universal gas constant, and T is temperature.
This balancing act is crucial in understanding energy dynamics in physical systems.

First Law of Thermodynamics

The first law of thermodynamics is essentially a principle of energy conservation. It is often summarized as, "energy cannot be created or destroyed, only transformed from one form to another."
In the context of an isothermal process for an ideal gas, the first law can be expressed as: ΔU = Q - W, where

• ΔU is the change in internal energy.
• Q is the heat added to the system.
• W is the work done by the system.

Since the temperature is constant, ΔU = 0 for an ideal gas, implying that the heat added to the system is equal to the work done by the system: Q = W.
This interplay between heat and work showcases how the first law governs energy exchange, particularly in efficient processes like those observed in engines and thermodynamic cycles.

Entropy

Entropy is a measure of disorder or randomness in a system, and it plays a significant role in thermodynamic processes. During an isothermal expansion, the entropy of the ideal gas increases. This increase is due to the particles having more volume to spread out and move around, which increases randomness.
Entropy is closely linked to the second law of thermodynamics, which states that entropy of an isolated system always tends to increase over time. This principle helps explain the natural tendency towards equilibrium.

• Entropy change can be calculated using the equation: ΔS = rac{Q}{T}, where ΔS is the change in entropy, Q is the heat exchanged, and T is the temperature.
• In any real process, some energy will become unavailable for doing useful work, accounting for the natural tendency toward increasing disorder.

Understanding entropy gives insight into why certain processes occur naturally and how energy distribution affects them.