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. Calculate the heat Q_c rejected to the cold reservoir. Use the latent heat of fusion for ice, L_f, which is approximately 334 kJ/kg. The formula to find the heat rejected is Q_c = m imes L_f, where m is the mass of ice melted.
2. Identify the temperatures of the hot and cold reservoirs. The temperature of the boiling water (hot reservoir) is approximately 100°C or 373 K, and the temperature of the ice-water mixture (cold reservoir) is 0°C or 273 K.
3. Use the Carnot efficiency formula, \\eta = 1 - \frac{T_c}{T_h}, where T_c is the temperature of the cold reservoir and T_h is the temperature of the hot reservoir. Convert temperatures to Kelvin for this calculation.
4. Calculate the work done by the engine using the formula W = Q_h imes \\eta, where Q_h is the heat absorbed from the hot reservoir. Since it's a Carnot engine, Q_h can be related to Q_c by Q_h = \frac{Q_c}{1 - \\eta}.
5. Substitute the values obtained from the previous steps into the work formula to find the work done by the engine during the 5 minutes of operation.

Key Concepts

Carnot Heat Engine

A Carnot heat engine is an idealized thermodynamic cycle that operates between two heat reservoirs, achieving maximum efficiency. It is based on the principles of reversible processes and is defined by the temperatures of the hot and cold reservoirs. The efficiency of a Carnot engine is determined by the ratio of the temperature difference between the reservoirs to the temperature of the hot reservoir.

Heat Transfer and Latent Heat

Heat transfer refers to the movement of thermal energy from one object or substance to another, which can occur through conduction, convection, or radiation. In the context of melting ice, latent heat is the energy required to change a substance from solid to liquid at its melting point without changing its temperature. For ice, this latent heat is crucial for calculating the amount of heat absorbed during the melting process.

Work Done by the Engine

The work done by a heat engine is the energy transferred from the engine to perform mechanical work, calculated as the difference between the heat absorbed from the hot reservoir and the heat rejected to the cold reservoir. In the case of the Carnot engine, this work can be determined using the first law of thermodynamics, which states that the change in internal energy of the system is equal to the heat added minus the work done by the system.