The second law and reversibility
What the first law cannot tell us
A cup of hot chai left on the table cools down to room temperature. Have you ever seen a cup of chai at room temperature heat itself up by cooling the room? Never.
Yet the first law would allow it. Energy would be conserved either way. The first law tells us energy balances; it does not tell us which processes actually happen.
Here’s another. A book rests on a table. Could it cool itself down a little and use that internal energy to jump into the air? Energy would be conserved, but it never happens.
Nature follows a further rule: the second law of thermodynamics. It has two classic forms.
Picture a device that runs in a cycle. It takes heat Q₁ from a hot reservoir, does work W, and gives heat Q₂ to a cold reservoir. This is a heat engine, and its efficiency is η equals W over Q₁.
The Kelvin–Planck statement: no process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of the heat into work.
So every engine must throw away some heat Q₂. Its efficiency can never be one. A ship cannot run by simply cooling the ocean.
Now reverse the device. It takes heat from the cold side and dumps it on the hot side. That is a refrigerator, and it needs work W to run.
The Clausius statement: no process is possible whose sole result is the transfer of heat from a colder object to a hotter object. Your fridge works only because the compressor does work, paid for by the electricity bill.
The two statements are equivalent. If you could break one, you could build a machine that breaks the other.
Reversible and irreversible processes
Film a process and play the film backwards. If the backward film could also happen, with the system and its surroundings both returning exactly to where they started, the process is reversible.
Open a valve between a box of gas and an empty box. The gas rushes in and fills both. This is free expansion. The gas will never squeeze itself back into one box on its own. It is irreversible.
Heat flowing from a hot iron to the cooler cloth is irreversible. So is a block sliding to rest on a rough floor, where friction turns its kinetic energy into heat.
In fact, all spontaneous processes in nature are irreversible. There are two main reasons.
First, dissipative effects: friction, viscosity, electrical resistance. They waste ordered energy as disordered internal energy, and you cannot get it back for free.
Second, the process passes through non-equilibrium states, like the free expansion, or an explosive chemical reaction, where p and T are not even defined throughout the system.
A reversible process must therefore be quasi-static, and free of friction and other dissipation. For example, a very slow isothermal expansion of an ideal gas, with a frictionless piston, can be reversed step by step.
Reversible processes are an idealisation, but a very useful one. As we will see next, an engine built only from reversible steps gives the highest possible efficiency.
Try it: the second law auditor
- Design an engine: choose Q₁ and W. The first law fixes the heat Q₂ given to the cold reservoir.
- Push W up to Q₁, so Q₂ = 0. The auditor flags a Kelvin–Planck violation.
- Switch to Refrigerator and set the work to zero. That breaks the Clausius statement.
- The red zone on the gauge is the Carnot limit, which you meet in the next lesson.
Powered by the sea
An inventor claims a ship engine that takes 1000 J of heat from sea water in each cycle and turns all 1000 J into work, with no other effect. What is wrong with the claim?
- It violates the first law: energy is not conserved
- It violates the Kelvin–Planck statement of the second law
- It violates the zeroth law
- Nothing: sea water is a huge reservoir, so it is possible
Answer
B. Energy is conserved (1000 J in, 1000 J out), so the first law is fine. But the sole result is heat taken from one reservoir and turned completely into work, which is exactly what the Kelvin–Planck statement forbids. Any real engine must reject some heat to a colder body.
Engine efficiency
In each cycle, an engine takes 500 J of heat from a hot reservoir and gives 350 J of heat to a cold reservoir. What is the work done per cycle, and the efficiency?
- 150 J, 30%
- 150 J, 43%
- 350 J, 70%
- 850 J, 100%
Answer
A. Over a cycle ΔU = 0, so W = Q₁ − Q₂ = 500 − 350 = 150 J. Efficiency η = W/Q₁ = 150/500 = 0.30, or 30%. Dividing by Q₂ instead (150/350) gives the wrong 43%.
Summary
- Engine efficiency:
η = W/Q₁ = 1 − Q₂/Q₁ - Kelvin–Planck:
Q₂ = 0 is impossible ⇒ η < 1 - Clausius:
heat cannot flow cold → hot with W = 0
- The first law says energy is conserved; the second law says which energy-conserving processes can actually happen.
- Kelvin–Planck: no process can have as its sole result the absorption of heat from a reservoir and its complete conversion into work.
- Clausius: no process can have as its sole result the transfer of heat from a colder object to a hotter one. The two statements are equivalent.
- A reversible process can be retraced so that system and surroundings both return to their original states. It must be quasi-static and free of dissipation.
- All spontaneous processes (free expansion, heat flow across a temperature difference, friction) are irreversible.