An ideal gas is taken through a Carnot cycle. The isothermal...

An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at $250^{\circ} \mathrm{C}$ , and the isothermal compression takes place at $50.0^{\circ} \mathrm{C}$ . The gas takes in 1200 $\mathrm{J}$ of energy from the hot reservoir during the isothermal ex- pansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle.

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the process. In the context of the Carnot cycle, isothermal expansion occurs when the system absorbs heat at a constant high temperature, and isothermal compression occurs when heat is expelled at a constant low temperature. These processes involve heat transfer while maintaining constant internal energy for an ideal gas.

Carnot Cycle

The Carnot cycle is an idealized thermodynamic cycle that represents the most efficient possible heat engine operating between two temperature reservoirs. It consists of two isothermal processes where heat is absorbed and rejected, and two adiabatic processes where the system is insulated and changes temperature without heat transfer. This cycle provides a benchmark for the maximum efficiency attainable by any heat engine operating between given temperature limits.

Thermal Efficiency

Thermal efficiency is a measure of a heat engine’s ability to convert heat from a high-temperature reservoir into work. For a Carnot engine, efficiency is defined as 1 minus the ratio of the absolute temperature of the cold reservoir to that of the hot reservoir (? = 1 - Tc/Th). This concept highlights the fundamental limitation imposed by the second law of thermodynamics on the performance of all heat engines.

Energy Balance in Thermodynamic Cycles

The energy balance in a thermodynamic cycle is based on the first law of thermodynamics, which states that the net heat absorbed by the system over one complete cycle is equal to the net work done by the system. In a reversible cycle like the Carnot cycle, the work output is determined by the difference between the energy absorbed from the hot reservoir and the energy expelled to the cold reservoir. This concept is critical for calculating the net work and understanding energy conservation in cyclic processes.

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