A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work W is performed by the engine?

Verified step by step guidance

1. Understand the concept of a Carnot heat engine: A Carnot engine is an idealized heat engine that operates between two thermal reservoirs, converting heat into work with maximum efficiency. The efficiency is determined by the temperatures of the hot and cold reservoirs.
2. Identify the key information: The heat rejected by the engine melts 0.0400 kg of ice. The latent heat of fusion for ice is approximately 334,000 J/kg. This means the heat rejected, Q_c, can be calculated using the formula Q_c = m * L_f, where m is the mass of ice and L_f is the latent heat of fusion.
3. Calculate the heat rejected: Use the formula Q_c = m * L_f to find the heat rejected by the engine. Substitute m = 0.0400 kg and L_f = 334,000 J/kg into the equation to find Q_c.
4. Apply the Carnot efficiency formula: The efficiency of a Carnot engine is given by η = 1 - (T_c / T_h), where T_c is the temperature of the cold reservoir and T_h is the temperature of the hot reservoir. Use this formula to find the efficiency η.
5. Relate efficiency to work done: The efficiency η is also defined as η = W / Q_h, where W is the work done by the engine and Q_h is the heat absorbed from the hot reservoir. Rearrange this formula to solve for W, using the relationship Q_h = Q_c / (1 - η) to find the heat absorbed from the hot reservoir.

Key Concepts

Carnot Heat Engine

A Carnot heat engine is an idealized engine that operates on the Carnot cycle, which is the most efficient cycle possible for a heat engine. It involves two isothermal processes and two adiabatic processes. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs, making it a useful model for understanding the limits of real-world engines.

Latent Heat of Fusion

The latent heat of fusion is the amount of heat required to change a unit mass of a substance from solid to liquid at constant temperature and pressure. For ice, this value is approximately 334,000 J/kg. In the context of the problem, it is used to calculate the heat rejected by the engine, which is responsible for melting the ice.

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. In the context of a heat engine, it implies that the work done by the engine is equal to the heat absorbed from the hot reservoir minus the heat rejected to the cold reservoir. This principle is crucial for calculating the work performed by the engine in the given problem.