Second Law of Thermodynamics

Kelvin-Planck statement: it is impossible to construct an engine which, operating in a continuous cycle, will extract heat from a single reservoir and convert it entirely into work.

Explanation: no heat engine can be 100% efficient. Some heat must always be rejected to a lower-temperature sink to complete the cycle.

Clausius statement: it is impossible to construct a device operating in a cycle whose sole effect is to transfer heat from a colder body to a hotter body.

Explanation: heat flows naturally from hot to cold. To make it flow the other way, as in a refrigerator, external work must be supplied.

Source (hot reservoir): a body at a high temperature \(T_H\) with very large heat capacity, which supplies heat to the engine.

Working substance: the material (usually a gas or vapour) that undergoes the cycle, absorbing heat, doing work and rejecting heat.

Sink (cold reservoir): a body at a lower temperature \(T_L\) with very large heat capacity, to which the waste heat is rejected.

The efficiency \(\eta\) is the ratio of useful work output to the heat input. By conservation of energy \(W=Q_H-Q_L\), so

(i) Work done:

(ii) Efficiency:

For 100% efficiency the heat rejected \(Q_L\) would have to be zero, or the sink would have to be at absolute zero. Rejecting no heat violates the Kelvin-Planck statement of the second law, and absolute zero cannot be reached (third law). Every cycle must restore the working substance to its initial state, which requires rejecting some heat to the sink.

The Carnot cycle is an idealised, perfectly reversible cycle made of four processes:

1. Reversible isothermal expansion (A to B): the gas expands at the constant high temperature \(T_H\) and absorbs heat \(Q_H\).

2. Reversible adiabatic expansion (B to C): the gas expands further without heat exchange and does work while its temperature falls to \(T_L\).

3. Reversible isothermal compression (C to D): the gas is compressed at the constant low temperature \(T_L\) and rejects heat \(Q_L\).

4. Reversible adiabatic compression (D to A): the gas is compressed without heat exchange and its temperature rises back to \(T_H\).

The P-V diagram is a closed loop bounded by two isotherms and two steeper adiabats. The area enclosed by the loop equals the net work done in one cycle.

For 1 mole of ideal gas, the heat absorbed in the isothermal expansion A to B (volumes \(V_1\to V_2\)) and the heat rejected in the isothermal compression C to D (volumes \(V_3\to V_4\)) are

For the adiabatic processes B to C and D to A, \(TV^{\gamma-1}=\text{constant}\):

Hence the logarithms are equal and

Significance: the efficiency depends only on the temperatures (in kelvin), not on the design or the gas used. To increase efficiency, raise the source temperature \(T_H\) or lower the sink temperature \(T_L\).

P-V diagram of the Otto cycle: adiabatic compression, then an isochoric (vertical) rise in pressure as heat is added, then adiabatic expansion, and finally an isochoric (vertical) fall in pressure as heat is rejected.

P-V diagram of the Diesel cycle: adiabatic compression, then an isobaric (horizontal) expansion as heat is added, then adiabatic expansion, and finally an isochoric (vertical) fall in pressure as heat is rejected.

A refrigerator is a heat engine run in reverse. Instead of taking heat from a hot source to produce work, it uses external work \(W\) (from an electric compressor) to remove heat \(Q_L\) from a cold space (the inside of the refrigerator) and reject heat \(Q_H=Q_L+W\) to a warmer surrounding (the kitchen).

The coefficient of performance is the ratio of the heat removed from the cold space to the work supplied:

Entropy \(S\) is a state function of a system whose change is defined by

It measures the degree of randomness or disorder of the system, and also the part of the thermal energy that is unavailable for conversion into work. A highly ordered system such as a crystal has low entropy; a disordered system such as a hot gas has high entropy.

The second law states that for any spontaneous, irreversible process the entropy of an isolated system, and hence of the universe, increases. All real processes involve irreversibility such as friction, heat conduction and turbulence, so the total entropy keeps increasing; it stays constant only in an ideal reversible process.

Example 1: a glass that falls and shatters goes from an ordered to a disordered state. The pieces never spontaneously reassemble.

Example 2: an ice cube melting in a warm room. The ordered solid lattice becomes a disordered liquid, and heat flows from the warm air to the cold ice, both of which increase the total entropy.

Suppose a “super engine” more efficient than a Carnot engine existed between the same temperatures. For the same heat input it would produce more work. That extra work could drive a Carnot engine in reverse as a refrigerator. The combined system would transfer heat from the cold reservoir to the hot reservoir with no net external work, which violates the Clausius statement. So the assumption is false, and the reversible Carnot engine has the maximum possible efficiency.

Friction: friction between moving parts turns some useful work into waste heat.

Irreversibility: rapid processes cause turbulence and non-uniform pressure and temperature, so the infinitely slow, reversible conditions of the ideal cycle are not met.

Heat loss: real engines are not perfectly insulated and some heat escapes to the surroundings instead of doing work.

Finite temperature differences: heat transfer in a finite time needs a finite temperature difference, which generates entropy.

Check: \(\eta=1-\dfrac{Q_L}{Q_H}\Rightarrow0.25=1-\dfrac{Q_L}{600}\Rightarrow Q_L=450\ \mathrm{J}\).

(a) Efficiency:

(b) Heat absorbed: