Laws of Thermodynamics

There are two ways to give a gas energy: heat it, or push on it. The first law of thermodynamics is the energy account that keeps track of both.

Two ways to change internal energy#

By heating. Keep a piston fixed and make the cylinder walls hotter than the gas. Gas molecules bouncing off the faster-jiggling wall molecules gain energy on average, and share it: the internal energy rises.

By work. Keep the walls at the gas temperature but push the piston in. A molecule bouncing off an approaching piston rebounds faster (like a ball off a moving bat), so again the internal energy rises.

Usually both happen together: a gas in a can on a hot stove takes in heat and, by pushing its piston out, does work and loses energy.

The first law#

If heat ΔQ is given to the gas and it does work ΔW, its energy rises by ΔQ − ΔW. If none of this goes into bulk motion of the gas, it all becomes internal energy ΔU:

For an ideal monatomic gas, internal energy is just the translational KE of its molecules; in general it also includes rotational and vibrational energy and the potential energy of molecular forces.

Internal energy is a state function#

Take a system from state i to state f by different routes: ΔQ and ΔW will differ from route to route, but ΔQ − ΔW is always the same, ΔU = U_f − U_i. So there is an internal energy U with a definite value in each state. We write ΔU = U_f − U_i, but never “Q_f − Q_i” or “W_f − W_i”: heat and work belong to processes, not states.

A gas pushing a piston does work. On a pressure–volume graph that work has a simple picture: an area.

Work by an expanding gas#

Gas at pressure p pushes on a piston of area A with force pA. If the piston moves out a small distance Δx, the work done by the gas is (pA)Δx = pΔV. For a larger change the pressure may vary, so add up small steps:

Expanding (V₂ > V₁): W positive, the gas does work. Compressed: W negative, work is done on the gas.

Three standard processes#

Isothermal (constant T), ideal gas

With p = nRT/V:

Isobaric (constant p)

Isochoric (constant V)

The volume doesn’t change, so W = 0.

Work turns into internal energy easily: just rub. Getting work back out of internal energy needs a machine: a heat engine.

What a heat engine does#

A heat engine takes heat Q₁ from hot bodies, turns part of it into work W, and rejects the rest, Q₂, to colder bodies. Its working substance returns to its starting state each time: it runs in a cyclic process. Since ΔU = 0 over a cycle, the first law gives W = Q₁ − Q₂.

The steam engine#

A sturdy main cylinder holds a tight-fitting piston, whose rod leaves the cylinder. Beside it is the steam chest, sharing a wall pierced by three holes: the upper and lower holes open into the two ends of the cylinder, the middle one leads to a condenser. A box-shaped sliding valve (the D-valve, from its shape) always covers the middle hole and one of the other two.

The piston rod drives a crank and flywheel through a crank-rod. An eccentric disc on the same shaft drives the valve rod, so the valve moves up as the piston moves down. Steam from the boiler enters above the piston and drives it down while used steam below escapes through the lower and middle holes to the condenser; then the valve shifts and steam drives the piston up. Burning fuel to make steam, the whole machine typically delivers only 3–10% of the fuel’s energy as work.

Internal combustion engines#

These burn their fuel inside the cylinder itself, with no separate furnace.

Petrol (Otto) engine, 1876

Petrol and air are mixed in the carburettor. Two valves (inlet and outlet) and a spark plug sit at the top of the cylinder. Four strokes:

1. Charging: inlet open, piston moves down, drawing in the petrol–air mixture.
2. Compression: both valves shut, piston moves up, squeezing the mixture (it warms to about 500°C).
3. Working: a spark ignites it; the gas reaches about 2000°C and 15 atm and drives the piston down, turning the crankshaft.
4. Exhaust: outlet open, piston moves up, pushing out the burnt gases.

Diesel engine

No spark plug. Air alone is drawn in and compressed so strongly that it reaches about 1000°C and 36 atm. Diesel is then injected and ignites by itself, driving the working stroke.

A hot cup cools in a cold room; it never spontaneously warms up while the room cools. Energy would be conserved either way. Something else decides which way nature goes: the second law.

Processes that only go one way#

• Bodies at 100°C and 0°C in contact both end at 50°C. Two bodies at 50°C never become one at 100°C and one at 0°C.
• A sliding block stops, warming itself and the table. A warm block never cools down and starts sliding.
• Gas let into an empty chamber fills both halves. It never gathers back into one half by itself.

None of the reverse processes would break the first law. Another law must decide which processes actually happen.

The Kelvin–Planck statement#

An engine with Q₂ = 0 would be 100% efficient. It would need no cold body: it could take heat from one body at one temperature, even the surroundings, and turn it all into work. A ship could run on the warmth of the ocean, a scooter on the warmth of its own body. All attempts failed:

Reversible and irreversible processes#

Quasi-static processes

A gas is in thermodynamic equilibrium when a single p, V and T describe the whole of it. Heat it on a stove and different parts have different temperatures, so it isn’t. A quasi-static process is carried out so slowly that the system is always very nearly in equilibrium, so it can be drawn as a curve on a p–V diagram. Real processes only approach this.

Reversible processes

A process from i to f is reversible if it can be run backwards along the same curve, with every small part exactly undone: the heat taken in on the way out is returned, and the work done is done back on the system. That needs two things:

• it must be quasi-static, and
• it must be non-dissipative: no friction, viscosity or similar losses.

A cyclic process whose every part is reversible is a reversible cycle. From here on, processes are assumed reversible unless stated otherwise.

Entropy is the bookkeeping quantity that tells which way processes run. Energy is always conserved; entropy can only be created.

Defining entropy#

Like pressure or internal energy, entropy S has a definite value in each equilibrium state. Give a system at absolute temperature T a small amount of heat ΔQ:

In a reversible adiabatic process no heat flows, so the entropy stays the same.

Entropy and disorder#

Entropy measures disorder. If every molecule of a gas moved with the same velocity in the same direction, its entropy would be lower than in the real gas, where they move randomly every which way.

The second law in terms of entropy#

Entropy is not conserved. It can be created, and once created it can never be destroyed.

In 1824 Sadi Carnot imagined the perfect heat engine. It can’t be built, but it sets the best efficiency any engine between two temperatures can reach.

The Carnot cycle#

An ideal gas fills a cylinder with a diathermic (heat-conducting) base, adiabatic walls and an adiabatic piston. There are two large bodies at fixed temperatures, hot (T₁) and cold (T₂). Starting compressed at T₁ (state a):

1. a → b: on the hot body, expand isothermally at T₁; the gas does work and takes in heat Q₁.
2. b → c: on an insulating stand, expand adiabatically; the gas does work and cools to T₂.
3. c → d: on the cold body, compress isothermally at T₂; work is done on the gas and it rejects heat Q₂.
4. d → a: on the insulating stand, compress adiabatically back to T₁.

With a frictionless piston moved very slowly, the whole cycle is reversible.

Its efficiency#

On a temperature–entropy diagram the cycle is a rectangle. Along a → b the entropy rises from S₁ to S₂ while Q₁ flows in at T₁; the adiabats keep S fixed; along c → d it falls back while Q₂ leaves at T₂:

Carnot’s theorem

It follows from the second law, and caps every heat engine at η = 1 − T₂/T₁.

Refrigerators (heat pumps)#

Run the engine backwards: take heat Q₂ from a cold body, have work W done on the working substance, and deliver Q₁ = Q₂ + W to a hot body. Heat moves from cold to hot. Done reversibly between single temperatures, this is a Carnot refrigerator, which needs the least possible work: