Second Law of Thermodynamics, Clausius and Kelvin’s statements

The second law of thermodynamics states that the entropy of an isolated system never decreases; it either stays the same (in reversible processes) or increases (in irreversible processes). It places fundamental limits on the direction of natural processes and on the efficiency of heat engines and refrigerators. Two classical equivalent formulations are the the Kelvin–Planck (Kelvin)and Clausius statement statement.

(i) Kelvin–Planck Statement

It is not possible to design an engine which works in a cyclic process and converts all the heat extracted from a heat source into work so that the working substance may remain unaffected.

In other words, for the continuous production of work, a heat sink is necessary along with the heat source.

According to the original statement given by Kelvin, it is impossible to obtain a continuous supply of work by taking heat continuously from a heat source and lowering its temperature below the temperature of the surroundings.

The above statement can be understood from the fact that in a Carnot engine, when the temperature of the heat source is equal to that of the heat sink, the efficiency of the engine is zero, i.e. it is impossible to obtain work from it.

Complete conversion of heat extracted from the heat source into work means that no heat is delivered to the heat sink. This state can be obtained only by keeping the temperature of the sink at $0\,\mathrm{K}$, which is impossible to attain in practice.

Similarly, it is impossible to obtain work continuously from a heat source alone. Thus, it is necessary that the engine must have a heat sink and some amount of heat must be delivered to it.

\[ Q_1 = W + Q_2 \]

Hence, for a heat engine operating cyclically, the heat absorbed from the source is partly converted into work and the remaining heat is rejected to the sink.

\[ W = Q_1 - Q_2 \]

Therefore, the efficiency of a heat engine is

\[ \eta = \frac{W}{Q_1} = 1-\frac{Q_2}{Q_1} \]

For complete conversion of heat into work, $Q_2$ would have to be zero. According to the second law of thermodynamics, this is not possible for a cyclic heat engine operating between finite temperatures.

(ii) Clausius Statement

It is impossible to have a device which works in a cyclic process and, unaided by an external agency, transfers heat from a system at lower temperature to a system at a relatively higher temperature.

In other words, it is impossible for heat to flow from a lower-temperature system to a higher-temperature system by itself.

The above statement is based on the principle of a refrigerator in which the working substance extracts heat from a colder system and rejects heat to a hotter system.

For doing this, work has to be done by an external agency on the working substance.

\[ Q_1 = Q_2 + W \]

Here, $Q_2$ is the heat extracted from the low-temperature reservoir, $Q_1$ is the heat rejected to the high-temperature reservoir, and $W$ is the external work supplied to the refrigerator.

Key Point: Heat cannot flow spontaneously from a colder body to a hotter body. External work is required to transfer heat from a lower-temperature reservoir to a higher-temperature reservoir.

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