Name: Macroscopic and Microscopic Variables
Uploaded: 2019-07-19
Duration: 5 min 3 s
Description: This video enables students to understand the definition and examples of the macroscopic and microscopic variables of the thermodynamic system.

Question

Explain in detail Carnot heat engine.

Accepted Answer

Carnot engine is a theoretical thermodynamic cycle proposed by Leonard Carnot. It gives the estimate of the maximum possible efficiency that a heat engine during the conversion process of heat into work and conversely, working between two reservoirs, can possess.

A Carnot heat engine has four parts.

Source

Sink

Insulating stand

Working substance

The working substance is subjected to four successive reversible process forming a Carnot cycle.

The steps involved in a carnot cycle are explained as follows.

Step $1)$ Isothermal expansion of gas from state $(P_{1}, V_{1}, T_{1})$ to state $(P_{2}, V_{2}, T_{1})$ - The gas is allowed to expand and it does work on the surroundings. The temperature of the gas does not change during the process, and thus the expansion is iso-thermic. The gas expansion is propelled by absorption of heat energy $Q_{1}$ .

$W_{1} = Q_{1} = \int_{V_{1}}^{V_{2}} P d V = R T_{1} \ln (\frac{V_{2}}{V_{1}}) = Area A B G E$

Step $2)$ Adiabatic expansion of gas from $(P_{2}, V_{2}, T_{1})$ to $(P_{3}, V_{3}, T_{2})$ - The piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, $T_{2}$ .

$W_{2} = \int_{V_{2}}^{V_{3}} P d V = \frac{R}{γ - 1} (T_{1} - T_{2}) = Area B C H G$

Step $3)$ Isothermal compression of gas from $(P_{3}, V_{3}, T_{2})$ to $(P_{4}, V_{4}, T_{2})$ - The cylinder is placed on the sink and gas is compressed at constant temperature $T_{2}$ .

Work done $=$ Heat released by the system

$W_{3} = Q_{2} = - \int_{V_{3}}^{V_{4}} P d V = - R T_{2} \ln (\frac{V_{4}}{V_{3}}) = R T_{2} \ln (\frac{V_{3}}{V_{4}}) = Area C D F H$

Step $4)$ Adiabatic compression of gas from $(P_{4}, V_{4}, T_{2})$ to $(P_{1}, V_{1}, T_{1})$ - Finally the cylinder is again placed on no conducting stand and the compression is continued so that the gas returns to its initial stage.

$W_{4} = - \int_{V_{2}}^{V_{1}} P d V = \frac{- R}{γ - 1} (T_{1} - T_{2}) = Area A D F E$

These steps constitute one carnot cycle. The process is repeated thereafter.

Net work done $W = W_{1} + W_{2} + W_{3} + W_{4} = R T_{1} \ln (\frac{V_{2}}{V_{1}}) + R T_{2} \ln (\frac{V_{3}}{V_{4}})$

The efficiency of carnot engine, $η$ is given by $η = 1 - \frac{Q_{2}}{Q_{1}} = 1 - \frac{T_{2}}{T_{1}}$

As $T_{2} < T_{1}$ , hence, $η$ is always less than $1$ .

Explain in detail Carnot heat engine.

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