Problem 6 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The internal energy change when a system goes from state \(A\) to \(B\) is \(40 \mathrm{~kJ} /\) mol If the system goes from \(A\) to \(B\) by a reversible path and returns to state \(\mathrm{A}\) by an irreversible path, what would be the net change in internal energy? (a) \(40 \mathrm{~kJ}\) (b) \(>40 \mathrm{~kJ}\) (c) \(\leq 40 \mathrm{~kJ}\) (d) Zero

Step-by-Step Solution

Answer

The net change in internal energy is (d) Zero.

Step 1: Understanding the First Law of Thermodynamics

Recall that the First Law of Thermodynamics states that the change in internal energy of a system is the same for any path between two states; it is a state function and is independent of path.

Step 2: Analyzing the Reversible and Irreversible Paths

Given that the system goes from state A to B by a reversible path and then returns to state A by an irreversible path, the net change in internal energy must be the same as the internal energy change from A to B, but with the opposite sign when returning from B to A.

Step 3: Calculating the Net Change in Internal Energy

Since the internal energy change for the pathway from state A to B is +40 kJ/mol and the change from B back to A should be -40 kJ/mol, the net change over the complete cycle is zero.

Key Concepts

Internal Energy

The internal energy of a system refers to the total energy contained within it, which includes kinetic energy due to the motion of particles and potential energy from forces between particles. In thermodynamics, it's denoted as 'U' and measured in units of joules (J) or kilojoules (kJ). It's a key concept in the study of energy transformations in physics and chemistry.

Understanding the internal energy is crucial because it's one of the state properties that determine the state of a system, along with pressure, volume, and temperature. Changes in internal energy can occur due to heat transfer, work done by or on the system, or a combination of both. These changes are always linked to the way in which a system interacts with its surroundings.

Reversible Path

In thermodynamics, a reversible path is a theoretical phenomena where changes occur infinitesimally slowly, allowing the system to remain in equilibrium with its surroundings at all times. It is an idealized process that can never be fully achieved in practice but is a useful model for understanding the maximum efficiency possible in energy transformations.

A reversible path can serve as a standard to compare real-world processes. For instance, when a gas expands or compresses reversibly, it does so in such a way that the process could be reversed by an infinitesimally small change in conditions, and no energy would be lost to friction or turbulence. Consequently, in a reversible path, one can retrieve all the work done or heat transferred between a system and its environment.

Irreversible Path

Opposite to a reversible path, an irreversible path includes processes that occur with finite differences causing the system to stray from equilibrium; such events introduce inefficiencies and dissipate energy, commonly in the form of heat. Real-world processes such as natural heat flow, friction, and sudden expansion of gases are irreversible because once they occur, they cannot be undone without leaving a net change in the universe.

For example, when a gas expands into a vacuum (free expansion), it does so rapidly and without doing external work. This process is irreversible because it cannot be reversed simply by infinitesimally altering the conditions. As a result, irreversible processes create entropy, reflecting a degree of irrecoverable energy dispersal in a system.

State Function

A state function is a property of a system that depends only on its current state, not on the path taken to reach that state. The key characteristic of a state function is that its value is fixed for each state and does not depend on how the system arrived at that state. Examples of state functions include internal energy (U), pressure (P), volume (V), and temperature (T).

In the context of the original exercise, the internal energy change when the system moves from state A to state B is independent of whether the path between them is reversible or irreversible. The focus is on the states themselves. If the system starts and ends at the same state, such as in a cycle where it returns to state A from state B via any path, the change in a state function (like internal energy) will always be zero because state functions are path-independent.