A Carnot engine is operated between two heat reservoirs at temperatures of 520 K and 300 K. (a) If the engine receives 6.45 kJ of heat energy from the reservoir at 520 K in each cycle, how many joules per cycle does it discard to the reservoir at 300 K?

Verified step by step guidance

1. Identify the temperatures of the hot and cold reservoirs. The hot reservoir temperature (T1) is 520 K, and the cold reservoir temperature (T2) is 300 K.
2. Calculate the efficiency of the Carnot engine using the formula: \( \eta = 1 - \frac{T2}{T1} \), where \( \eta \) is the efficiency, \( T1 \) is the temperature of the hot reservoir, and \( T2 \) is the temperature of the cold reservoir.
3. Determine the amount of heat energy converted to work by the engine. Since the engine receives 6.45 kJ of heat from the hot reservoir, multiply this value by the efficiency calculated in the previous step to find the work done per cycle.
4. Calculate the heat discarded to the cold reservoir. Subtract the work done from the total heat received from the hot reservoir: \( Q_{cold} = Q_{hot} - W \), where \( Q_{cold} \) is the heat discarded, \( Q_{hot} \) is the heat received, and \( W \) is the work done.
5. Convert the result to joules if necessary, as the heat discarded will be in the same units as the heat received (kJ in this case).

Key Concepts

Carnot Engine

A Carnot engine is an idealized heat engine that operates on the Carnot cycle, which is the most efficient cycle possible between two heat reservoirs. It is defined by its two isothermal processes (heat absorption and rejection) and two adiabatic processes. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs, given by the formula: efficiency = 1 - (T_cold / T_hot).

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or system to another due to a temperature difference. In the context of the Carnot engine, heat is absorbed from the hot reservoir and discarded to the cold reservoir. The amount of heat transferred is crucial for calculating the work done by the engine and the efficiency of the cycle.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another. In the case of the Carnot engine, the energy input as heat from the hot reservoir is converted into work done by the engine, with the remainder being expelled as heat to the cold reservoir. This principle is essential for determining the heat discarded in each cycle.