A Carnot heat engine operates between reservoirs at 182℃ and 0℃. If the engine extracts 25 J of energy from the hot reservoir per cycle, how many cycles will it take to lift a 10 kg mass a height of 10 m?

Verified step by step guidance

1. Step 1: Convert the temperatures of the hot and cold reservoirs from Celsius to Kelvin. Use the formula T(K) = T(°C) + 273.15. For the hot reservoir, T_hot = 182 + 273.15, and for the cold reservoir, T_cold = 0 + 273.15.
2. Step 2: Calculate the efficiency of the Carnot engine using the formula η = 1 - (T_cold / T_hot), where η is the efficiency, T_cold is the temperature of the cold reservoir, and T_hot is the temperature of the hot reservoir.
3. Step 3: Determine the work output per cycle of the engine. The work output (W) is given by W = η × Q_hot, where Q_hot is the energy extracted from the hot reservoir per cycle (25 J).
4. Step 4: Calculate the work required to lift the 10 kg mass a height of 10 m. Use the formula W_lift = m × g × h, where m is the mass (10 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height (10 m).
5. Step 5: Divide the total work required to lift the mass (W_lift) by the work output per cycle (W) to determine the number of cycles needed. Use the formula cycles = W_lift / W.

Key Concepts

Carnot Heat Engine

A Carnot heat engine is an idealized thermodynamic cycle that provides the maximum possible efficiency for a heat engine operating between two thermal reservoirs. It operates 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 given by the formula η = 1 - (T_c / T_h), where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively.

Work and Energy

In physics, work is defined as the energy transferred when a force is applied over a distance. The work done on an object can be calculated using the formula W = F × d, where F is the force and d is the distance moved in the direction of the force. In the context of lifting a mass, the work done is equal to the gravitational potential energy gained, which can be calculated using W = m × g × h, where m is mass, g is the acceleration due to gravity, and h is the height.

Efficiency and Energy Transfer

Efficiency in a heat engine is a measure of how well it converts the energy extracted from the hot reservoir into useful work. It is calculated as the ratio of the work output to the heat input from the hot reservoir. In this scenario, understanding the efficiency of the Carnot engine is crucial to determine how much work can be done per cycle, which directly influences how many cycles are needed to lift the specified mass.