Question

A system is maintained at thermal equilibrium (at the same temperature) with its surroundings and has an enthalpy of \(50 \mathrm{~kJ}\). It has \(100 \mathrm{~kJ}\) of heat transferred to it, which causes it to expand against a constant pressure of \(1 \mathrm{~atm}\). It is then compressed back to its initial volume. All steps are at constant temperature. What is its final enthalpy?

Short answer

The final enthalpy is 50 kJ.

Step-by-step solution

Understand the Problem

The system is initially at thermal equilibrium with its surroundings, meaning its temperature is constant throughout the process. We need to determine the final enthalpy of the system after it undergoes expansion and then compression at constant temperature.

Recognize Key Thermodynamic Concepts

Enthalpy (\( H \)) is defined by the relationship \( H = U + PV \), where \( U \) is internal energy, \( P \) is pressure, and \( V \) is volume. Since the system starts and ends at the same state, the enthalpy change over the complete cycle should be zero at constant temperature as it returns to its initial state.

Analyze Initial to Final State

The system gains \(100 \, \text{kJ}\) of heat and expands against a constant pressure of \(1 \, \text{atm}\). However, once it's compressed back to its original volume and thermal equilibrium is maintained, the initial and final enthalpies should be equal at constant temperature.

Apply Thermodynamic Principles

In an isothermal process where the system returns to its original state, the enthalpy is a state function. Therefore, any reversible process that returns the system to its original conditions will also return the enthalpy to its original value.

Conclusion - Calculate Final Enthalpy

Given the process is reversible and the system returns to its original state at constant temperature, the final enthalpy must equal the initial enthalpy. Therefore, the final enthalpy is the same as it was initially: \(50 \, \text{kJ}\).

Key concepts

Thermal Equilibrium

Thermal equilibrium is a crucial concept in the study of thermodynamics. When a system is in thermal equilibrium with its surroundings, it means that there is no net flow of thermal energy between the two. Essentially, the system and its surroundings have reached the same temperature. This equilibrium state allows us to predict the behavior of the system under changes.
When a system is at thermal equilibrium:

• The temperature remains constant, implying that any heat added to the system does not change its temperature.
• The system exists in a stable state where no spontaneous heat transfer occurs.

In the provided exercise, the system starts in thermal equilibrium with its surroundings, ensuring that as heat is added or removed, the overall temperature remains constant throughout the process. This concept is vital for describing the behavior of the system when it undergoes changes in enthalpy or volume, as a constant temperature often simplifies the analysis of thermodynamic problems.

Isothermal Process

An isothermal process is one where the temperature remains constant throughout the entire process. This is a common assumption in thermodynamics because it substantially simplifies the calculations involved. During an isothermal process, even though energy in the form of heat may be transferred to or from the system, the internal energy does not change.
Here are some key points about isothermal processes:

• The system does work while maintaining constant temperature, often involving heat transfer to compensate for work done.
• In ideal conditions, especially for gases, the pressure-volume relationship is expressed through Boyle's Law: \[ PV = ext{constant} \]
• Since the temperature remains constant, any change in pressure and volume will not affect the internal energy, meaning ∆U = 0.

In the context of the exercise, the entire process—expansion and compression—occurs isothermally. This means that as the system expands and does work against the surrounding pressure, it must absorb heat to keep the temperature constant. Similarly, when compressed back to its initial state, heat is released yet the temperature does not deviate. This constancy in temperature ensures that the enthalpy, a state function, remains unchanged once the system completes the cycle.

Thermodynamic Cycle

A thermodynamic cycle consists of a series of processes that begin and end at the same state. These cycles are fundamental in understanding and designing engines and refrigerators. In a cycle, the system undergoes various processes but returns to its starting point, which means any state function, like enthalpy, remains unchanged over one complete cycle.
Some important characteristics of thermodynamic cycles include:

• After completing a cycle, properties such as pressure, volume, and enthalpy return to their initial values.
• For an ideal gas undergoing an isothermal process, the work done by or on the system can be directly equated to the heat exchanged.
• The net work done during one complete cycle is the area enclosed by the cycle on a PV diagram.

In the exercise, we have a thermodynamic cycle starting at an initial enthalpy of 50 kJ and returning to the same state. Since enthalpy is a state function, its value is the same at the start and end of the cycle, meaning that the final enthalpy of the system is 50 kJ. This fundamental property of cycles in thermodynamics assures us that provided the process is reversible, no energy is lost or gained concerning enthalpy.