21.6 : Statements of the Second Law of Thermodynamics

The second law of thermodynamics can be stated in several different ways, and all of them can be shown to imply the others. The Clausius’ statement of the second law of thermodynamics is based on the irreversibility of spontaneous heat flow. It states that heat will not flow from the colder body to the hotter body unless some other process is involved. Additionally, as per the Kelvin’s statement, it is impossible to convert the heat from a single source into work without any other effect. This statement describes an unattainable “perfect engine”. The Kelvin statement is a manifestation of a well-known engineering problem. Despite advancing technology, we are not able to build a heat engine that is 100% efficient. The first law does not exclude the possibility of constructing a perfect engine, but the second law forbids it.

The Kelvin statement can be shown to be equivalent to the Clausius statement. To prove this, we show that, if one statement is false, it necessarily follows that the other statement is also false. Let us first assume that the Clausius statement is false, meaning that the perfect refrigerator does exist. Now, consider a real heat engine working in the same temperature range. Suppose, if these two devices are combined, then no net heat transfer occurs to or from the cold reservoir, but still, the work is done on some external body. The combination of a perfect refrigerator and a real heat engine is itself an ideal heat engine, thereby contradicting the Kelvin statement. Thus, if the Clausius statement is false, the Kelvin statement must also be wrong.