Physics 2nd Paper · Classes 11-12 · Chapter 1 · 1.9

Carnot Cycle.

Picture an engine with no friction, no heat leaking anywhere, and a source and a sink that never warm up or cool down. Nobody can build it, yet once you see its four-step turn, the later heat-engine calculations get much easier. You need not memorise the steps, only a rhythm.

1. An imaginary ideal engine

The French scientist Sadi Carnot designed an ideal engine free of every fault. It cannot be built in practice; it is an imaginary engine. This Carnot engine completes its work in four stages.

The principle fits in one breath. Through reversible processes the working substance takes heat from the source. By one isothermal expansion, one adiabatic expansion, one isothermal compression and one adiabatic compression, part of that heat is turned into work. The rest is rejected to the sink, and the substance comes back to its initial state.

2. The parts of the engine

In cylinder A the three walls are fully insulating and the base is fully conducting. An ideal gas, the working substance, is shut inside. The insulating piston P slides without friction.

The heat source M is a hot body of very high heat capacity at absolute temperature T₁, and its temperature always stays fixed. The heat sink N is a similar cold body at T₂, with T₂ ≪ T₁.

The stand S is a fully heat-insulating platform. With the cylinder sitting on it, in the adiabatic steps, no heat is exchanged.

The Carnot cycle is reversible, for three reasons. (1) There is no friction between piston and cylinder. (2) The processes take place very slowly. (3) Ideal insulators and conductors are used, and the source and sink have such high heat capacity that the isothermal steps run at constant temperature.

3. What makes it a cycle

A cycle in which an ideal gas, as the working substance, goes through one isothermal expansion, one adiabatic expansion, one isothermal compression and one adiabatic compression and returns to its initial state is called the Carnot cycle.

The diagram that shows this action and the work done is called the indicator diagram.

4. Four steps, from A back to A

Step 1, A→B, isothermal expansion. The cylinder sits on the heat source and the gas stays at T₁. It takes in heat Q₁ from the source. The work done is W₁ = ∫PdV = the area ABba.

Step 2, B→C, adiabatic expansion. The cylinder is on the insulating stand. The gas expands and its temperature falls from T₁ to T₂. The work is W₂ = ∫PdV = the area BCcb.

Step 3, C→D, isothermal compression. The cylinder is on the heat sink. The gas is compressed, its pressure rises, and it gives out heat Q₂ to the sink at T₂. The work is W₃ = the area CDdc.

Step 4, D→A, adiabatic compression. Back on the insulating stand, work is done on the gas and its temperature rises to T₁. Pressure and volume return to P₁ and V₁, and the cycle is back at its start. The work is W₄ = the area DAad.

Watch the four steps run together: the indicator diagram on the left, the cylinder on the right.

Now put the steps in order yourself.

Keep the sign convention in mind: work done by the gas is positive, work done on the gas is negative. So W₁ and W₂ are positive, while W₃ and W₄ are negative. Go round the table.

5. The net work, and one calculation

The total work in the whole cycle is W = W₁ + W₂ − W₃ − W₄ = the area ABCD.

So on the indicator diagram, the area enclosed by the two isothermal lines and the two adiabatic lines is the work done by the working substance.

Now put a number in. When 1 mol of ideal gas at T₁ = 600 K doubles its volume in step 1, W₁ = Q₁ = nRT₁ ln 2 = 8.31 × 600 × 0.693 ≈ 3456 J. Here ln is the natural logarithm. Change n, the temperature and the volume ratio and watch.

6. Check yourself

Three questions in the exam's style. Start with the first.

For the second, think of the steps.

For the last, recall the whole picture of the cycle.

The whole note on eight cards.

Source: Physics 2nd Paper, Classes 11-12 (NCTB), Chapter 1 “Thermodynamics”, section 1.9.