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:dS = rac{\delta Q}{T} \!
where
: δ''Q'' is an amount of Heat introduced to the system,
: ''T'' is a constant Absolute Temperature , and
: δ denotes the Inexact Differential .

This expression comes from the Clausius Theorem . This definition makes sense when absolute temperature has been defined. Note that the small amount \delta Q of energy transferred by heating is denoted by
\delta Q rather than dQ, because ''Q'' is not a State Function while the entropy is.

Clausius gave the quantity ''S'' the name "entropy", from the Greek word ''τρoπή'', "transformation". Since this definition involves only differences in entropy, the entropy itself is only defined Up To an arbitrary additive constant.

When a process is irreversible, the above definition must be replaced by the statement that the entropy change is equal to the amount of energy required to return the system to its original state by a reversible transformation at a constant temperature, divided by that temperature. This is explained in more detail below.


HEAT ENGINES

Clausius' identification of ''S'' as a significant quantity was motivated by the study of reversible and irreversible thermodynamic transformations. A thermodynamic transformation is a change in a system's thermodynamic properties, such as Temperature and Volume . A transformation is reversible if it is Quasistatic which means that it is Infinitesimal ly close to Thermodynamic Equilibrium at all times. Otherwise, the transformation is Irreversible . To illustrate this, consider a gas enclosed in a Piston chamber, whose volume may be changed by moving the piston. If we move the piston slowly enough, the density of the gas is always homogeneous, so the transformation is reversible. If we move the piston quickly, Pressure Wave s are created, so the gas is not in equilibrium, and the transformation is irreversible.

A Heat Engine is a thermodynamic system that can undergo a sequence of transformations which ultimately return it to its original state. Such a sequence is called a Cyclic Process , or simply a ''cycle''. During some transformations, the engine may exchange energy with the environment. The net result of a cycle is (i) Mechanical Work done by the system (which can be Positive or negative, the latter meaning that work is done ''on'' the engine), and (ii) heat energy transferred from one part of the environment to another. By the Conservation Of Energy , the net energy lost by the environment is equal to the work done by the engine.

If every transformation in the cycle is reversible, the cycle is reversible, and it can be run in reverse, so that the energy transfers occur in the opposite direction and the amount of work done switches sign.


DEFINITION OF TEMPERATURE

In thermodynamics, Absolute Temperature is ''defined'' in the following way. Suppose we have two Heat Reservoir s, which are systems sufficiently large that their Temperature s do not change when energy flows into or out of them. A reversible cycle exchanges heat with the two heat reservoirs:

Now consider a reversible cycle in which the engine exchanges heats \delta Q_1,\delta Q_2,\cdots,\delta Q_N with a sequence of ''N'' heat reservoirs with temperatures ''T1'', ..., ''TN''. We can show (see the box on the right) that:

:\sum_{i=1}^N rac{\delta Q_i}{T_i} = 0 \!.
where
δQ

T

i


Since the cycle is reversible, the engine is always infinitesimally close to equilibrium, so its temperature is equal to any reservoir with which it is contact. In the limiting case of a reversible cycle consisting of a ''continuous'' sequence of transformations,

:\oint rac{\delta Q}{T} = 0 \,\! (reversible cycles)

where the integral is taken over the entire cycle, and ''T'' is the temperature of the system at each point in the cycle. This is a particular case of the Clausius Theorem .


Entropy as a state function

We can now deduce an important fact about the entropy change during ''any'' thermodynamic transformation, not just a cycle. First, consider a reversible transformation that brings a system from an equilibrium state ''A'' to another equilibrium state ''B''. If we follow this with ''any'' reversible transformation which returns that system to state ''A'', our above result says that the net entropy change is zero. This implies that the entropy change in the first transformation depends ''only on the initial and final states''.

This allows us to define the entropy of any ''equilibrium'' state of a system. Choose a reference state ''R'' and call its entropy ''SR''. The entropy of any equilibrium state ''X'' is

:S_X = S_R + \int_R^X rac{\delta Q}{T} \,\!

Since the integral is independent of the particular transformation taken, this equation is well-defined.


ENTROPY CHANGE IN IRREVERSIBLE TRANSFORMATIONS

We now consider irreversible transformations. It can be shown that the entropy change during any transformation between two ''equilibrium'' states is

:\Delta S \ge \int rac{\delta Q}{T} \,\!

where the equality holds if the transformation is reversible.

Notice that if \delta Q=0, then Δ''S'' ≥ 0. This is the Second Law of Thermodynamics, which we have discussed earlier.

Suppose a system is thermally and mechanically isolated from the environment. For example, consider an insulating rigid box divided by a movable partition into two volumes, each filled with gas. If the pressure of one gas is higher, it will expand by moving the partition, thus performing work on the other gas. Also, if the gases are at different temperatures, heat can flow from one gas to the other provided the partition is an imperfect insulator. Our above result indicates that the entropy of the system ''as a whole'' will increase during these process (it could in principle remain constant, but this is unlikely.) Typically, there exists a maximum amount of entropy the system may possess under the circumstances. This entropy corresponds to a state of ''stable equilibrium'', since a transformation to any other equilibrium state would cause the entropy to decrease, which is forbidden. Once the system reaches this maximum-entropy state, no part of the system can perform work on any other part. It is in this sense that entropy is a measure of the energy in a system that "cannot be used to do work".


MEASURING ENTROPY

In real Experiment s, it is quite difficult to Measure the entropy of a system. The techniques for doing so are based on the thermodynamic definition of the entropy, and require extremely careful Calorimetry .

For simplicity, we will examine a mechanical system, whose thermodynamic state may be specified by its volume ''V'' and pressure ''P''. In order to measure the entropy of a specific state, we must first measure the Heat Capacity at constant volume and at constant pressure (denoted ''CV'' and ''CP'' respectively), for a successive set of states intermediate between a reference state and the desired state. The heat capacities are related to the entropy ''S'' and the temperature ''T'' by

:C_X = T \left( rac{\partial S}{\partial T} ight)_X \,\!

where the ''X'' subscript refers to either constant volume or constant pressure. This may be Integrated Numerically to obtain a change in entropy:

:\Delta S = \int rac{C_X}{T} dT \,\!

We can thus obtain the entropy of any state (''P'',''V'') with respect to a reference state (''P0'',''V0''). The exact formula depends on our choice of intermediate states. For example, if the reference state has the same pressure as the final state,

: S(P,V) = S(P, V_0) + \int^{T(P,V)}_{T(P,V_0)} rac{C_P(P,V(T,P))}{T} dT \,\!

In addition, if the path between the reference and final states lies across any First Order Phase Transition , the Latent Heat associated with the transition must be taken into account.

The entropy of the reference state must be determined independently. Ideally, one chooses a reference state at an extremely high temperature, at which the system exists as a gas. The entropy in such a state would be that of a classical ideal gas plus contributions from molecular rotations and vibrations, which may be determined Spectroscopically . Choosing a ''low'' temperature reference state is sometimes problematic since the entropy at low temperatures may behave in unexpected ways. For instance, a calculation of the entropy of Ice by the latter method, assuming no entropy at zero temperature, falls short of the value obtained with a high-temperature reference state by 3.41 J/(mol·K). This is due to the "zero-point" entropy of ice mentioned earlier.


SEE ALSO



FURTHER READING

  • Goldstein, Martin, and Inge F., 1993. ''The Regrigerator and the Universe''. Harvard Univ. Press. A gentle introduction at a lower level than this entry.