The Carnot Cycle: Videos & Practice Problems

A heat engine operating on the basis of the Carnot cycle is designed by a mechanical engineering student. The student uses an electrical resistor operating at 150 °C as the hot thermal reservoir and a large block of ice at 0 °C as the cold thermal reservoir. The heat that comes out of the engine melts the ice at a rate of 100 g/min. What is the work produced by the engine in 10 minutes?

An ideal and reversible heat engine absorbs 7000 J of heat and operates between two thermal pools. The high-temperature source is at 420 °C, and the sink is at 300 °C. Find the efficiency (e) of this engine.

An engine operating on the basis of the Carnot cycle is used for a certain function. The engine runs between two thermal sources with temperatures of 600 K and 400 K. In each cycle, 2500 J is transferred from the higher temperature source. Find the amount of work (W) that can be obtained from the engine per cycle.

In each cycle, a Carnot engine withdraws 3500 J of heat from a high-temperature source at 450 K and delivers an amount of heat (Q) to a heat sink at a temperature of 320 K. In each cycle, how many joules are delivered to the heat sink?

Evaluate the heat engine shown in the figure and determine whether it follows (i) the first law or (ii) the second law of thermodynamics.

Given a Carnot engine with a hot reservoir temperature of 600 K and an efficiency of 45%, find the temperature of the cold reservoir, expressed in degrees Celsius

Consider a heat engine that operates on a Carnot cycle. The engine's thermal efficiency is 35% when the temperature of the hot reservoir is 450℃. To increase the efficiency to 65%, determine the necessary reduction in the temperature of the cold reservoir.

A rocket engine combusts fuel at a temperature of 3,000°C. It is cooled by liquid hydrogen at a temperature of -250°C. The rocket lifts a payload of 5000 kg to an altitude of 100 km. How much heat energy is transferred to the engine by burning fuel if the engine is 50% as efficient as a Carnot engine? Assume that the specific heat capacity of the rocket exhaust gases is 1.0 kJ/(kg•K) and that the gravitational acceleration is 9.8 m/s².

A Carnot engine works between a high-temperature reservoir at 700 K and a low-temperature reservoir at 300 K. If the engine is 100% efficient and extracts 50 J of energy from the hot reservoir per cycle, determine the number of cycles needed to displace a 20 kg mass by 12 m.

Picture a single mole of a diatomic gas undergoing various stages of a Carnot cycle, as shown in the figure. It is initially at a temperature of 400°C. The pressure experienced by the gas at this stage is 10 atm (point a). At the end of the isothermal expansion, its volume triples (point b). Afterward, the gas temperature falls to 250 °C (point c). Calculate the pressure and volume at different points of the cycle (points a, b, c, and d).

A single mole of a monatomic gas is subjected to a Carnot cycle, with the hot reservoir at 450°C and the cold reservoir at 300°C. The initial pressure is set at 10 atm, and the volume triples during the isothermal expansion phase. Calculate the efficiency of this cycle.

A 1.0 mole of helium gas powers a Carnot engine. In the isothermal expansion phase, the gas volume doubles. During the subsequent adiabatic expansion, the volume expands to 5.5 times its original size. Each cycle of the engine delivers a work output of 1000 J. Determine the temperatures of the two reservoirs that the engine operates between.

A heat engine operates between two thermal reservoirs at different temperatures and follows the Carnot cycle, undergoing four processes: two isothermal and two adiabatic. The engine operation is represented on a Pressure - Volume (PV) diagram.

i. What is the work done by the engine during one complete Carnot cycle using the PV diagram?

ii. What is the work done in any other reversible thermodynamic cycle that can be represented on a PV diagram?

Consider a Carnot cycle using a diatomic gas. The volumes at states 1 and 2 are 6.0 L and 12.0 L, respectively. The high temperature is 550°C, and the low temperature is 300°C. The gas used in the cycle is 0.75 moles of a diatomic gas. Calculate the pressures at states 1 and 2. Hint: For a diatomic gas, the adiabatic index $\gamma$ is 1.4.

A heat engine operates using one mole of a monatomic gas in a thermodynamic cycle. The engine absorbs heat at 500 K and releases it at 300 K. Starting at state 1, the gas has a pressure of 2.5 atm and a volume of 16.4 liters. During isothermal expansion to state 2, the volume doubles to 32.8 liters, and the pressure drops to 1.25 atm. During an adiabatic expansion to state 3, the volume increases to 66.7 liters, and the pressure drops to 0.37 atm. Isothermal compression to state 4 reduces the volume to 33.4 liters, and the pressure increase to 0.74 atm. Adiabatic compression returns the system to state 1. Calculate the heat transferred (Q), work done (W), and change in internal energy (ΔE_int) for each segment: 1-2, 2-3 processes only.