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In this lesson, we will learn how to calculate the efficiency of heat engines and heat pumps from the temperature difference between their reservoirs.

Q1:

The temperature of a Carnot refrigerator’s cold reservoir is − 7 3 . 0 ∘ C . The hot reservoir has a temperature of 2 7 0 ∘ C . What is the coefficient of performance of the Carnot refrigerator?

Q2:

A Carnot engine operates between reservoirs at temperatures of 6 . 0 × 1 0 2 K and 3 . 0 × 1 0 2 K. If the engine absorbs 1 . 0 × 1 0 2 J per cycle at the hot reservoir, what is its work output per cycle?

Q3:

A Carnot engine working between two heat baths of temperatures 600 K and 273 K completes each cycle in 5.00 s. In each cycle, the engine absorbs 10.0 kJ of heat. Find the output power of the engine.

Q4:

A Carnot refrigerator works between 0 . 0 0 ∘ C and 3 0 . 0 0 ∘ C . The refrigerator is used to cool a bucket of water containing 1 . 0 0 × 1 0 − 2 m^{3} of water from a temperature of 3 0 . 0 0 ∘ C to a temperature of 5 . 0 0 ∘ C in 7 2 0 0 seconds. Find the total amount of work needed.

Q5:

A Carnot engine performs 100 J of work while rejecting 200 J of heat each cycle. After the temperature of the hot reservoir only is adjusted, it is found that the engine now does 130 J of work while discarding the same quantity of heat.

What is the initial efficiency of the engine?

What is the final efficiency of the engine?

What is the fractional change in the temperature of the hot reservoir?

Q6:

The Carnot cycle for a heat engine is represented by the temperature-entropy diagram shown.

How much heat is absorbed per cycle at the high-temperature reservoir?

How much heat is exhausted per cycle at the low-temperature reservoir?

How much work is done per cycle by the engine?

What is the efficiency of the engine?

Q7:

A Carnot engine employs 1.5 mol of nitrogen gas as a working substance, which is considered as an ideal diatomic gas with 𝛾 = 7 . 5 at the working temperatures of the engine. The Carnot cycle goes in the cycle 𝐴 𝐵 𝐶 𝐷 𝐴 with 𝐴 𝐵 being an isothermal expansion. The volume at points 𝐴 and 𝐶 of the cycle are 5 . 0 × 1 0 − 3 m^{3} and 0.15 L, respectively. The engine operates between two thermal baths of temperature 500 K and 300 K.

Find the values of volume at 𝐵 .

Find the values of volume at 𝐷 .

How much heat is absorbed by the gas in the 𝐴 𝐵 isothermal expansion?

How much work is done by the gas in the 𝐴 𝐵 isothermal expansion?

How much heat is given up by the gas in the 𝐶 𝐷 isothermal compression?

How much work is done by the gas in the 𝐶 𝐷 isothermal compression?

How much work is done by the gas in the 𝐵 𝐶 adiabatic expansion?

How much work is done by the gas in the 𝐷 𝐴 adiabatic compression?

Find the value of efficiency of the engine based on the net work and heat input.

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