Heat engine are devices that convert thermal energy or heat to mechanical work. The most common example with which you no doubt have some familiarity is the automobile engine. In an engine the heat produced from the burning of the fuel is converted into mechanical energy that powers the vehicle. You are probably also familiar with the consequence of the second law of thermodynamics that a heat engine cannot convert all the input heat to mechanical work but must reject some heat, because you are aware of the waste heat carried by the hot gases that come out the exhaust. A heat engine can never be 100% efficient, because the Kelvin statement of the second law of thermodynamics forbids the construction of an engine that converts heat completely to work without rejecting some heat to a lower temperature reservoir. We paraphrased the statement of the first law of thermodynamics as, "you can't get something for nothing." Our study of the second law of thermodynamics tells us that you cannot break even, because all heat engines must reject some waste heat.

It is important to remember that the Kelvin statement of the second law and the descriptions of heat engines refer to devices that operate in cycles, meaning the devices return to some initial state after going through some processes and then repeat the cycle. This means the internal energy of the system is the same at the end of a cycle as it was at the beginning of the cycle.

Heat engines operating in cycles return to the initial state at the end of a cycle, so the internal energy of the system is unchanged. This means the first law of thermodynamics for a cyclic process is expressed as DU = Q - W = 0 so that Q = W. This tells us that a heat engine that took in a given amount of heat and converted it all to work would be possible without violating the first law of thermodynamics. However, the second law of thermodynamics would not allow such an engine, because the second law requires that a heat engine reject some heat to a low temperature reservoir.

Efficiency of a heat engine is calculated in just exactly the manner in which you would expect it to be calculated, i. e. we divide the output or "what we get" by the input or "what we put in." The output is the work done by the engine, W, and the input is the heat taken in, Q_H, so the efficiency is given by e = W / Q_H. The efficiency is commonly expressed as a percent rather than as a fraction, so that an efficiency calculation that gave a result of 1/4 would be expressed as 25%. Indeed, 25% efficiency is about as high a value for efficiency that an ordinary automobile engine can attain.

The Clausius statement of the second law of thermodynamics is equivalent to the Kelvin statement, although at first glance it appears to be quite different. The Clausius statement tells us that heat will not flow from a colder to a hotter body unless some other effect is also involved. This statement conforms to our common sense, because objects do not get colder when we place them in the refrigerator unless the refrigerator was "plugged in" so that the compressor could run. Work must be done by the refrigerator to remove heat from the warm food placed in it.

The Carnot cycle consists of two isothermal steps in which the temperature is held constant and two adiabatic steps in which there is no heat transfer to or from the engine. Heat is transferred to the engine from a higher temperature reservoir during one isothermal process, and heat flows from the engine to a lower temperature reservoir during the other isothermal process. Analysis of the performance of the Carnot cycle leads to a simple relationship for the efficiency of a Carnot engine that depends only upon the temperature at which the heat is transferred into the engine (the temperature of the high temperature reservoir), T_H, and the temperature at which the heat is rejected by the engine (the temperature of the low temperature reservoir), T_L. In terms of these two temperatures, measured in Kelvin, the efficiency of a Carnot engine is e_c = (T_H - T_L) / T_H . For this calculation we must use the absolute temperature measured in Kelvin. We calculate the Kelvin temperature by adding 273.2 to the Celsius temperature.

A complete understanding of the concept of entropy requires considerable mathematical analysis, but we can understand the significance of entropy by defining entropy as the measure of the disorder of the system. According to the entropy statement of the second law of thermodynamics, the entropy of the universe or of an isolated system can only increase or remain constant; it cannot decrease. This means systems and the universe proceed to higher levels of disorder. In the case of heat engines, the implication of this statement is that heat rejected to the lower temperature reservoir by an engine is no longer available for that engine, i.e. your automobile engine cannot take the heat from its own exhaust as input heat and continue to run.

We learned in Chapter 10 that the first law of thermodynamics is merely a statement of conservation of energy. All too often inventors claim to have developed a machine or an engine that operates in a cycle and produces more work than the heat that is put into it thereby having efficiency greater than 100%. For an engine that operates in cycles, the internal energy does not change in a complete cycle, so the claim that more work is obtained than heat supplied violates the first law of thermodynamics. Such a machine is called a perpetual motion machine of the first kind, and it violates the first law of thermodynamics.

Inventors also claim to have produced engines that do not reject waste heat to a lower temperature reservoir in violation of the second law of thermodynamics. Such machines are called perpetual motion machines of the second kind, and are impossible.