Various Statements of the Second Law

Entropy. The second law introduces a new state function, entropy ($S$), as a measure of a system's molecular disorder (randomness). Its rigorous thermodynamic definition concerns the CHANGE in entropy resulting from a process:

Entropy statement of the second law: the entropy of an isolated system increases during a spontaneous process. For an irreversible (spontaneous) process, such as the spontaneous expansion of a gas, $\Delta S_{total}>0$, and more precisely

For a REVERSIBLE process, such as ice melting at its melting point under equilibrium conditions, $\Delta S_{system}=-\Delta S_{surrounding}$, so the two exactly cancel and $\Delta S_{total}=0$.

Kelvin-Planck statement: it is impossible to construct a machine that absorbs heat from a hot source and converts it COMPLETELY into work by a cyclic process, without transferring some part of that heat to a cold sink. This is exactly why even an ideal, perfectly frictionless engine can never convert 100% of the heat it absorbs into useful work. Carnot's analysis of heat engines showed that the MAXIMUM efficiency of a reversibly-operating heat engine depends only on the two temperatures it operates between:

where $q_h$ is heat absorbed from the hot reservoir and $q_c$ is heat rejected to the cold reservoir. For a reversible cyclic process, $\Delta S_{total}=\Delta S_{system}+\Delta S_{surroundings}=0$, so $\Delta S_{system}=-\Delta S_{surroundings}$, and expressing each in terms of $q/T$ gives $\dfrac{q_h}{T_h}=\dfrac{q_c}{T_c}$ (7.28); substituting this ratio back into (7.27) gives

Since $T_h\gg T_c$ in any real engine, $\eta$ is always less than 1 -- perfect (100%) conversion of heat to work is thermodynamically forbidden. Expressed as a percentage:

Worked example (Problem 7.10). An automobile engine burns petrol at $816^\circ$C, with surroundings at $21^\circ$C. Converting to Kelvin: $T_h=816+273=1089$ K, $T_c=21+273=294$ K. …