In a Carnot cycle, the isothermal expansion of an ideal gas takes place at \(412 \mathrm{~K}\) and the isothermal compression at \(297 \mathrm{~K}\). During the expansion, \(2090 \mathrm{~J}\) of heat energy are transferred to the gas. Determine \((a)\) the work performed by the gas during the isothermal expansion, ( \(b\) ) the heat rejected from the gas during the isothermal compression, and \((c)\) the work done on the gas during the isothermal compression.

An isothermal process is a thermodynamic procedure during which the temperature of a system remains constant. Imagine heating a balloon gently and evenly such that it expands but its temperature stays the same; this is a classic example of isothermal expansion. In such a process, the system interacts with its surroundings in a way that the internal energy change is balanced by a transfer of energy as work, resulting in no temperature change.

In the context of the Carnot cycle, which is an idealized thermodynamic cycle, isothermal expansion means the gas expands and performs work while staying at a constant high temperature, such as the given example at 412 K. During this phase, any heat energy added to the gas is fully converted into work done by the gas as it expands. Similarly, during isothermal compression, the gas is compressed at a constant lower temperature, here 297 K, and if performed ideally, the work done on the gas is completely removed as heat.

The first law of thermodynamics is a version of the law of conservation of energy tailored for thermodynamic systems. It's elegantly simple: the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the work done by the system on its surroundings. Expressed mathematically, it's written as \( \Delta U = Q - W \).

In our isothermal expansion of the gas, the idea is that the entirety of the heat energy \( Q \) absorbed (\( 2090 \, \text{J} \) from the problem statement), converts into work done \( W \) by the expanding gas. There is no change in internal energy because the temperature remains constant. This relationship highlights a fundamental principle in energy exchange within thermodynamic systems and underpins the concepts integral to understanding the Carnot cycle.

Heat transfer is a phenomenon where thermal energy is exchanged due to a temperature difference. It's the movement of energy from a hotter object to a cooler one. There are three modes of heat transfer: conduction, convection, and radiation. However, in thermodynamics, we're often concerned with the quantity of heat transferred rather than the mechanism.

During the Carnot cycle's isothermal steps, heat transfer is crucial. As stated in the example problem, \( 2090 \, \text{J} \) of heat is transferred to the gas during expansion, and a different amount of heat is rejected during compression. The ability to precisely quantify this transfer allows us to use the first law of thermodynamics to determine other properties of the system, such as work done.

Work in thermodynamics is a measure of energy transfer when a force moves its point of application in the direction of the force, often resulting in volume changes of a gas. It's calculated as the product of force and displacement. When a gas expands isothermally, like in our example, it does work on its surroundings by pushing against a force, often attributed to atmospheric pressure or the resistance of a piston in an engine.

In our exercise, the work done by the gas during expansion is equated to the heat absorbed (\(2090 \, \text{J}\)) since it's an isothermal process. Conversely, work done on the gas during compression is needed to decrease its volume, which results in the removal of heat from the system.

Thermodynamic efficiency is a dimensionless measure of how well a thermodynamic system converts the heat input into useful work output. It's a ratio that typically ranges from 0 to 1, where 1 would represent 100% conversion of heat into work and is practically unattainable due to inherent losses, like friction. The theoretical limit of efficiency is determined by the Carnot efficiency, dictated by the temperatures of the heat reservoirs.

The formula for the Carnot efficiency is \(\eta = 1 - \frac{T_2}{T_1}\), where \(\eta\) is the efficiency, \(T_1\) is the temperature during isothermal expansion, and \(T_2\) is the temperature during isothermal compression. Using this formula, in conjunction with the amount of heat put into and rejected from the system, allows us to calculate the cycle's efficiency and thereby its effectiveness in performing work.