Not much chemistry is performed without making products. Combining these results leads to the conclusion that all reversible engines working between the same two reservoirs have the same efficiency.When two chemicals are mixed together do they react? This is an existential question for a synthetic chemist. If both engines are then reversed, the same reasoning implies that the efficiency of D cannot be greater than the efficiency of E. If they are coupled as shown in (Figure)(b), the efficiency of E cannot be greater than the efficiency of D, or the second law would be violated. Suppose that D and E are both reversible engines. Now it is quite easy to demonstrate that the efficiencies of all reversible engines operating between the same reservoirs are equal. The original assumption must therefore be wrong, and it is impossible to construct an irreversible engine such that E is more efficient than the reversible engine D. Since and the net result of each cycle is equivalent to a spontaneous transfer of heat from the cold reservoir to the hot reservoir, a process the second law does not allow.
Suppose the cycle of D is reversed so that it operates as a refrigerator, and the two engines are coupled such that the work output of E is used to drive D, as shown in (Figure)(b). (b) The coupled engines, with D working in reverse. (a) Two uncoupled engines D and E working between the same reservoirs. Thus, if the Clausius statement is false, the Kelvin statement must also be false. Since, the combination of a perfect refrigerator and a real heat engine is itself a perfect heat engine, thereby contradicting the Kelvin statement. The net heat removed from the hot reservoir is, no net heat transfer occurs to or from the cold reservoir, and work W is done on some external body. Suppose these two devices are combined as shown in (Figure). From the first law, these quantities are related by. It extracts heat from the hot reservoir, does work W, and discards heat Q to the cold reservoir. The refrigerator removes heat Q from a cold reservoir at a temperature and transfers all of it to a hot reservoir at a temperature Now consider a real heat engine working in the same temperature range.
Let us first assume that the Clausius statement is false, so that the perfect refrigerator of (Figure)(b) does exist. To prove the equivalence of the Kelvin and Clausius statements, we show that if one statement is false, it necessarily follows that the other statement is also false. A “ perfect refrigerator,” shown in (Figure)(b), which works without such external aid, is impossible to construct. Heat transfer in the direction of increasing temperature always requires some energy input. The Clausius statement is related to the everyday observation that heat never flows spontaneously from a cold object to a hot object. Thus, the Clausius statement becomes: It is impossible to construct a refrigerator that transfers heat from a cold reservoir to a hot reservoir without aid from an external source. We can show that the Kelvin statement is equivalent to the Clausius statement if we view the two objects in the Clausius statement as a cold reservoir and a hot reservoir. Neither of these devices is achievable in reality. (b) A “perfect refrigerator” transports heat from a cold reservoir to a hot reservoir without work input. (a) A “perfect heat engine” converts all input heat into work. The first law does not exclude the possibility of constructing a perfect engine, but the second law forbids it. Despite advancing technology, we are not able to build a heat engine that is efficient. The Kelvin statement is a manifestation of a well-known engineering problem. However, if the gas were returned to its initial state (that is, made to complete a cycle), it would have to be compressed and heat would have to be extracted from it.
Another example is a chamber of gas that can absorb heat from a heat reservoir and do work isothermally against a piston as it expands. Without completing a cycle, the substance in the engine is not in its original state and therefore an “other effect” has occurred. For example, an engine can absorb heat and turn it all into work, but not if it completes a cycle. Note that “without any other effect” is a very strong restriction. This statement describes an unattainable “ perfect engine,” as represented schematically in (Figure)(a). This is known as the Kelvin statement of the second law of thermodynamics. It is impossible to convert the heat from a single source into work without any other effect. Second Law of Thermodynamics (Kelvin statement)