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The second law of thermodynamics, as formulated in the middle of the 19th century by William Thomson (Lord Kelvin) and Rudolf Clausius, states that it is impossible to gain mechanical energy (work) from heat flowing from a cold to a hot body. Clausius postulated that the opposite is the case: it requires input of mechanical (or electric) energy to transport heat from a low- to a high-temperature object. In modern terms: a heat pump, air conditioner, and refrigerator, are devices that move heat from a cold to a warm place, the second law states that they need energy to do this.

Thomson formulated the second law in a slightly different, but equivalent way. He stated that it is impossible in a cyclic process to extract work from a single source of heat. In a cyclic process the heat source ends in a thermodynamic state that is the same as the initial state; the heat source does not lose any net internal energy. In order that a cyclic process is in agreement with the first law of thermodynamics (i.e., conserves energy), it is necessary that the heat generated by the work is returned to the heat source.

Consequences of second law

If the second law would not hold, there would be no fear of energy shortage. For example, it would be possible—as already pointed out by Lord Kelvin—to propel ships by energy extracted from sea water. After all, the oceans contain immense amounts of internal energy. When one could extract just a small portion of it—whereby a slight cooling of the sea water would occur—and use this to propel a ship (a form of work), then ships could move without any net consumption of energy. It would not violate the first law of thermodynamics, because the rotation of the ship's propellers would again heat the water and in total the energy of the supersystem "ship-plus-ocean" would be conserved, in agreement with the first law. Unfortunately, it is not possible, no work can be extracted from the water because it would be a single source of heat. Clausius would explain the violation of the law by observing that ships are warmer than sea water (or at least they are not colder) and hence it needs work to transport heat from the sea to the ship.

PD Image
T_h > T_c. Second law (Kelvin): if W > 0, Q_c ≠ 0.

Without the second law, one could conceive a similar setup on land where energy, extracted from the earth, would charge batteries, and heat, dissipated by electric currents generated by the batteries, would be given back to the earth. Such a device is also out of the question because the second law forbids it.

The second law is summarized in the figure. Two heat reservoirs are shown, one of absolute temperature T_h (the hot reservoir) and the other of temperature T_c (the cold reservoir), T_h >T_c. The reservoirs are coupled by a heat engine (green circle), a construct that converts heat Q_h into work W. The "rest heat" Q_c is delivered to the cold reservoir. This idealized representation of power-generating machines was invented by Sadi Carnot who used it for the study of steam engines. But this schematic representation applies to many machines, for instance also to an automobile, a vehicle with an internal combustion engine. The high-temperature heat bath is formed by the cylinders which are hot because of the combustion of gasoline. The cold heat bath is formed by the environment of the car—the rest heat is delivered to the surroundings through the car's radiator. The cyclically moving pistons, that perform the actual work, form the heat engine.

When net work W is performed by the engine on the surroundings (depicted by W outgoing in the figure), the Kelvin principle states that Q_c ≠ 0, because otherwise there would be a single heat source. The Clausius principle states that for the engine to perform work it is necessary that T_h is larger than T_c. Hence, the second law states that it is not possible to convert all the heat Q_h delivered by the hot reservoir into work, part of it becomes non-zero rest heat Q_c absorbed by the low temperature reservoir. In the case of a car it means that only part of the combustion energy delivered by the gasoline is converted into work, and that a running car by necessity heats up its environment by its rest heat.

It can be shown that the efficiency η ≡ W / Q_h is bounded:

$\eta \leq {\frac {T_{\mathrm {h} }-T_{\mathrm {c} }}{T_{\mathrm {h} }}}$

Thus, when the car cylinders operate at 427 °C = 700 K and the surroundings are 27 °C = 300 K, then η ≤ 400/700 = 57%. It is important to note that this limit to the efficiency is a consequence of the second law of thermodynamics, and can only be raised by higher T_h not by a better streamline of the car or other design improvements.

Mathematical expression of the second law

We consider a single thermodynamic system of absolute temperature T, and let DQ be a small amount of heat entering the system. In the article entropy it is proved from the Clausius/Kelvin principle that a thermodynamic system is characterized, not only by its usual parameters volume, pressure, etc., but also by the state variable S, the entropy of the system. The differential dS is defined by

$dS\;{\stackrel {\mathrm {def} }{=}}\;{\frac {DQ}{T}}>0.$

When DQ leaves the system,

$dS=-{\frac {DQ}{T}}<0.$

Reversible processes

The fact that entropy S is a state function implies that the following holds for a reversible cyclic process,

$\oint {\frac {DQ}{T}}\equiv {\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}{\frac {DQ}{T}}+{\int \limits _{2}\limits ^{1}}_{{\!\!}^{(II)}}{\frac {DQ}{T}}=\oint dS=0.$

This equation is the mathematical expression of the second law of thermodynamics for the special case of reversible processes. The cycle consists of two different paths in state space, denoted by I and II. The path integrals start and end at common points in state space, indicated by 1 and 2.

In order to show conversely that this equation yields the Clausius principle, we consider the heat engine (green circle) in the figure as our system and assume that both heat baths are so large (or the engine so small) that one full cycle of the engine does not change the temperatures of the baths. Then for one cycle of the engine one can write,

$\oint dS=\oint {\frac {DQ_{\mathrm {h} }}{T_{\mathrm {h} }}}-\oint {\frac {DQ_{\mathrm {c} }}{T_{\mathrm {c} }}}={\frac {Q_{\mathrm {h} }}{T_{\mathrm {h} }}}-{\frac {Q_{\mathrm {c} }}{T_{\mathrm {c} }}}=0\quad \qquad \qquad (1)$

where we defined

${\frac {Q_{\mathrm {in} }}{T_{\mathrm {h} }}}\equiv \oint {\frac {DQ_{\mathrm {h} }}{T_{\mathrm {h} }}}={\frac {1}{T_{\mathrm {h} }}}\oint DQ_{\mathrm {h} },$

and the analogous definition holds for Q_c. Note that by definition Q_h and Q_c are positive amounts of heat. The work W, on the other hand, is positive or negative for work performed by or on the system, respectively. It follows from equation (1) that

${\frac {Q_{\mathrm {h} }}{Q_{\mathrm {c} }}}={\frac {T_{\mathrm {h} }}{T_{\mathrm {c} }}}>1,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (2)$

that is, Q_h > Q_c irrespective of the direction of the heat flow.

When the heat flow is as given in the figure (from hot to cold), the first law states that

$W=Q_{\mathrm {h} }-Q_{\mathrm {c} }>0,\,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (3)$

that is, the heat engine performs work on its surroundings.

When the heat flow is in the opposite direction, from cold to hot, the first law of thermodynamics states

$W=Q_{\mathrm {c} }-Q_{\mathrm {h} }<0,\,$

meaning that the surroundings perform work on the heat engine. Hence, in correspondence with the Clausius principle, work on the system is needed to transport heat from the cold to the hot reservoir, i.e, when the figure (with directions of lines reverted) represents a heat pump.

The efficiency η follows from equations (2) and (3)

$\eta \equiv {\frac {W}{Q_{\mathrm {h} }}}={\frac {Q_{\mathrm {h} }-Q_{\mathrm {c} }}{Q_{\mathrm {h} }}}=1-{\frac {Q_{\mathrm {c} }}{Q_{\mathrm {h} }}}=1-{\frac {T_{\mathrm {c} }}{T_{\mathrm {h} }}}={\frac {T_{\mathrm {h} }-T_{\mathrm {c} }}{T_{\mathrm {h} }}},$

and since T_h > T_c > 0, it follows that η < 1.

It must be stressed that this inequality for η holds for reversible processes in which there are no entropy losses. Often the fact that η is less than unity is given in the same discussion as in which it is pointed out that entropy strives toward a maximum. From such an exposition of the second law the incorrect impression may be gathered that the inequality for η is solely due to entropy losses. It is true that in spontaneous (irreversible) processes, in which the entropy increases, the efficiency η is further reduced; the equation for η becomes an upper bound. That is, for irreversible processes the equality, valid for reversible processes, becomes the following inequality,

$\eta <{\frac {T_{\mathrm {h} }-T_{\mathrm {c} }}{T_{\mathrm {h} }}}.$

Spontaneous, irreversible, processes

Many, in fact most, thermodynamic processes are spontaneous and irreversible. A well-known spontaneous process is the flow of heat from a hot to a cold body. The opposite process—the transport of heat from a cold to a hot body—needs work (by the Clausius principle), the process is not spontaneous and accordingly not the reverse of the spontaneous flow of heat from hot to cold bodies. Another example of an irreversible process is Count Rumford's seminal cannon boring experiment where work is converted by friction into heat. It is impossible to revert this process, which is intuitively clear, but also contradicts the Kelvin principle, the impossibility of obtaining work from a single source of heat. The Joule-Thomson effect is yet another example of an irreversible process.

References

C. S. Helrich, Modern Thermodynamics with Statistical Mechanics, Springer (2009). Google books