To provide a quantitative measure for the direction of spontaneous change, Clausius introduced the concept of entropy as a precise way of expressing the second law of thermodynamics. The Clausius form of the second law states that spontaneous change for an irreversible process in an isolated system (that is, one that does not exchange heat or work with its surroundings) always proceeds in the direction of increasing entropy. For example, the block of ice and the stove constitute two parts of an isolated system for which total entropy increases as the ice melts.

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By the Clausius definition, if an amount of heat *Q* flows into a large heat reservoir at temperature *T* above absolute zero, then the entropy increase is Δ*S* = *Q*/*T*. This equation effectively gives an alternate definition of temperature that agrees with the usual definition. Assume that there are two heat reservoirs *R*1 and *R*2 at temperatures *T*1 and *T*2 (such as the stove and the block of ice). If an amount of heat *Q* flows from *R*1 to *R*2, then the net entropy change for the two reservoirs is

*T*1 >

*T*2. Thus, the observation that heat never flows spontaneously from cold to hot is equivalent to requiring the net entropy change to be positive for a spontaneous flow of heat. If

*T*1 =

*T*2, then the reservoirs are in equilibrium, no heat flows, and Δ

*S*= 0.

The condition Δ*S* ≥ 0 determines the maximum possible efficiency of heat engines—that is, systems such as gasoline or steam engines that can do work in a cyclic fashion. Suppose a heat engine absorbs heat *Q*1 from *R*1 and exhausts heat *Q*2 to *R*2 for each complete cycle. By conservation of energy, the work done per cycle is *W* = *Q*1 – *Q*2, and the net entropy change is

*W*as large as possible,

*Q*2 should be as small as possible relative to

*Q*1. However,

*Q*2 cannot be zero, because this would make Δ

*S*negative and so violate the second law. The smallest possible value of

*Q*2 corresponds to the condition Δ

*S*= 0, yielding as the fundamental equation limiting the efficiency of all heat engines. A process for which Δ

*S*= 0 is reversible because an infinitesimal change would be sufficient to make the heat engine run backward as a refrigerator.

The same reasoning can also determine the entropy change for the working substance in the heat engine, such as a gas in a cylinder with a movable piston. If the gas absorbs an incremental amount of heat *d**Q* from a heat reservoir at temperature *T* and expands reversibly against the maximum possible restraining pressure *P*, then it does the maximum work *d**W* = *P* *d**V*, where *d**V* is the change in volume. The internal energy of the gas might also change by an amount *d**U* as it expands. Then by conservation of energy, *d**Q* = *d**U* + *P* *d**V*. Because the net entropy change for the system plus reservoir is zero when maximum work is done and the entropy of the reservoir decreases by an amount *d**S*reservoir = −*d**Q*/*T*, this must be counterbalanced by an entropy increase of

*d*

*S*

*system*+

*d*

*S*

*reservoir*= 0. For any real process, less than the maximum work would be done (because of friction, for example), and so the actual amount of heat

*d*

*Q*′ absorbed from the heat reservoir would be less than the maximum amount

*d*

*Q*. For example, the gas could be allowed to expand freely into a vacuum and do no work at all. Therefore, it can be stated that with

*d*

*Q*′ =

*d*

*Q*in the case of maximum work corresponding to a reversible process.

This equation defines *S**system* as a thermodynamic state variable, meaning that its value is completely determined by the current state of the system and not by how the system reached that state. Entropy is an extensive property in that its magnitude depends on the amount of material in the system.

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In one statistical interpretation of entropy, it is found that for a very large system in thermodynamic equilibrium, entropy *S* is proportional to the natural logarithm of a quantity Ω representing the maximum number of microscopic ways in which the macroscopic state corresponding to *S* can be realized; that is, *S* = *k* ln Ω, in which *k* is the Boltzmann constant that is related to molecular energy.

All spontaneous processes are irreversible; hence, it has been said that the entropy of the universe is increasing: that is, more and more energy becomes unavailable for conversion into work. Because of this, the universe is said to be “running down.”