Question

How much heat is added to the system when \(2.00 \mathrm{~kJ}\) of work is performed by an ideal gas in an isothermal process? Give a reason for your answer.

Short answer

Answer: In an isothermal process, the heat added to the system (Q) is equal to the work done by the system (W). So, when an ideal gas performs 2.00 kJ of work, 2.00 kJ of heat is added to the system.

Step-by-step solution

Identify the work done by the system

We are given that 2.00 kJ of work is performed by the ideal gas in the isothermal process. It's important to remember that this value is positive because work is done by the system.

Apply the equation for an isothermal process

As determined in the analysis, for an isothermal process, the heat added to the system (Q) is equal to the work done by the system (W). So, \(Q = W\).

Calculate the heat added to the system

Using the equation from step 2 and the given work value from step 1, we can calculate the heat added to the system: \(Q = W\) \(Q = 2.00 \mathrm{~kJ}\) So, 2.00 kJ of heat is added to the system when 2.00 kJ of work is performed by the ideal gas in an isothermal process.

Provide reasoning

The reasoning behind the solution is the first law of thermodynamics and the fact that in an isothermal process, the internal energy of an ideal gas remains constant, which is related to the constant temperature. This relationship, combined with the first law of thermodynamics, leads us to conclude that the heat added to the system is equal to the work done by the system in an isothermal process.

Key concepts

First Law of Thermodynamics

Understanding the first law of thermodynamics is pivotal when evaluating energy changes within a physical system. Simply put, this law states that energy cannot be created or destroyed in an isolated system; it can only change forms. The law is usually formulated as
\( \text{Change in internal energy} = \text{Heat added to the system} - \text{Work done by the system} \), or
\( \triangle U = Q - W \).
In the context of our exercise, when an ideal gas expands isothermally—that is, at a constant temperature—the work done by the gas indeed results in an equal amount of heat being added to maintain its internal energy constant. This ties back to the conservation principle of energy, underlining that the 2.00 kJ of work performed must be matched by 2.00 kJ of heat added to the system.

Ideal Gas Laws

The behavior of ideal gases is described aptly by the ideal gas law, expressed as
\( PV = nRT \), where
P is the pressure,
V is the volume,
n is the number of moles,
R is the ideal gas constant, and
T is the temperature in Kelvin. For an isothermal process, as presented in the exercise, the temperature
T remains unchanged, and thus the product of pressure and volume must stay constant too. During such a process, an ideal gas complies by assuming the work-done is entirely compensated by the external heat supplied, adhering to the ideal gas law principles that underlie such conclusion.

Work-Energy Principle

The work-energy principle connects the work done by or on a system to its change in energy. For a mechanical system, work done on an object generally results in an increase in kinetic energy, referred to as the work-kinetic energy theorem. In thermodynamics, particularly for gases, this principle helps us understand how the work done during expansion or compression affects internal energy and by extension, the temperature of the gas. Since an isothermal process maintains temperature, and thus internal energy constant, all work done must be balanced by the heat transfer—a manifestation of the work-energy principle at the molecular level.

Heat Transfer Physics

The physics of heat transfer deals with the energy movement from one place to another or between a system and its surroundings. It occurs in three forms: conduction, convection, and radiation. In the case of the isothermal process described in the exercise, we are primarily concerned with heat transfer as a means to compensate for the work done by the system. While the system performs work, it loses energy, and to stay isothermal, it must gain an equivalent amount of heat. This points to a direct heat transfer into the gas, matching the work done to fulfill the requirement of constant internal energy. Therefore, heat transfer here acts to counterbalance the energy changes due to work, ensuring the process adheres to isothermal conditions.