5 The Second Law and Entropy
Understand how the second law constrains heat-to-work conversion, how entropy describes process direction, and how reservoir temperatures set ideal engine and appliance performance limits.
What the restricts
The expresses energy conservation: energy transferred as heat or work changes a system’s energy but is not created or destroyed. The adds a restriction that the first law alone cannot provide: it determines which energy transformations can occur naturally and limits how much heat can become useful work.
Heat flows spontaneously from a warmer object to a cooler one. The reverse transfer is possible only when work is supplied, as in a . Similarly, a cyclic engine cannot take heat from a single reservoir and convert it all into work. These are equivalent forms of the .
Heat engines and efficiency
A operates in a cycle, absorbing heat from a hot reservoir, producing work, and rejecting some heat to a colder reservoir. Let be the heat absorbed from the hot reservoir, the magnitude of the heat rejected to the cold reservoir, and the net work produced.
Because the engine returns to its initial state each cycle, its change in internal energy over a cycle is zero. The first law therefore gives
The engine’s is the fraction of input heat converted to work:
The requires for any cyclic engine operating between ordinary hot and cold reservoirs. Thus, no such engine can convert all its absorbed heat into work.
Refrigerators and heat pumps
A uses work to move heat from a colder region to a warmer one. A uses the same kind of cycle but focuses on delivering heat to the warmer region. For both devices, is removed from the cold region, is delivered to the hot region, and the required input work is
Their performance is measured by a (COP), not by engine efficiency:
A COP can exceed : it compares heat moved with work supplied, and the device moves heat rather than converting work entirely into heat. For the same cycle, .
and process direction
, represented by , is a thermodynamic state property that helps express the quantitatively. For a reversible path between two states,
where is absolute temperature in kelvins. is a state function, so this reversible-path calculation gives the change between the states even when the actual process is irreversible.
For an isolated system—or, more generally, the system and its surroundings taken together—the states
stays constant in an ideal reversible process and increases in an irreversible one. A system’s own can decrease, provided the increase elsewhere is at least as large. For instance, a lowers the of its cold interior, but its work input and heat rejection cause a net increase in the surroundings and the overall process.
Reversible limits and real devices
A reversible engine operating between hot and cold reservoirs at temperatures and has the greatest possible efficiency for those temperatures. Its efficiency is the :
The temperatures must be in kelvins. Any real engine operating between the same reservoirs has efficiency no greater than this limit; friction, finite-temperature differences during heat transfer, and other irreversible effects make it lower.
The corresponding maximum COPs for ideal reversible refrigerators and heat pumps are
For example, between reservoirs at and , the maximum engine efficiency is , or . The ideal COP is , while the ideal heat-pump COP is . Real devices perform below these limits.
The does not say that energy disappears: the first law still accounts for all of it. Rather, energy transferred as heat becomes less fully available for conversion into useful work as irreversible processes generate . The temperature difference between reservoirs matters because it determines the theoretical opportunity to produce work.