Entropy

:For other senses of the term entropy, see entropy (disambiguation).

Thermodynamic definition of entropy

In this section, we discuss the original definition of entropy, as introduced by Clausius in the context of classical thermodynamics. Clausius defined the change in entropy of a thermodynamic system, during a reversible process in which an amount of heat dQ is introduced at constant absolute temperature T, as

Related Topics:
Reversible process - Heat - Absolute temperature

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: dS = rac{dQ}{T} ,!

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This definition makes sense when absolute temperature has been defined.

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

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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 infinitesimally 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 waves are created, so the gas is not in equilibrium, and the transformation is irreversible.

Related Topics:
Temperature - Volume - Reversible - Quasistatic - Infinitesimal - Thermodynamic equilibrium - Irreversible - Piston - Pressure wave

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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.

Related Topics:
Heat engine - Cyclic process - Mechanical work - Positive - Conservation of energy

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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.

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Definition of temperature

In thermodynamics, absolute temperature is defined in the following way. Suppose we have two heat reservoirs, which are systems sufficiently large that their temperatures do not change when energy flows into or out of them. A reversible cycle exchanges heat with the two heat reservoirs. If the cycle absorbs an amount of heat Q from the first reservoir and delivers an amount of heat Q′ to the second, then the respective reservoir temperatures T and T′ obey

Related Topics:
Absolute temperature - Heat reservoir

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:rac{Q}{T} = rac{Q'}{T'} ,!

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Proof: Introduce an additional heat reservoir at an arbitrary temperature T0, as well as N cycles with the following property: the j-th such cycle operates between the T0 reservoir and the Tj reservoir, transferring energy dQj to the latter. From the above definition of temperature, the energy extracted from the T0 reservoir by the j-th cycle is

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:dQ_{0,j} = T_0 rac{dQ_j}{T_j} ,!

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Now consider one cycle of the heat engine, accompanied by one cycle of each of the smaller cycles. At the end of this process, each of the N reservoirs have zero net energy loss (since the energy extracted by the engine is replaced by the smaller cycles), and the heat engine has done an amount of work equal to the energy extracted from the T0 reservoir,

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:W = sum_{j=1}^N dQ_{0,j} = T_0 sum_{j=1}^N rac{dQ_j}{T_j} ,!

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If this quantity is positive, this process would be a perpetual motion machine of the second kind, which is impossible. Thus,

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:sum_{i=1}^N rac{dQ_i}{T_i} le 0 ,!

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Now repeat the above argument for the reverse cycle. The result is

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:sum_{i=1}^N rac{dQ_i}{T_i} = 0 ,! (reversible cycles)

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Now consider a reversible cycle in which the engine exchanges heats dQ1, dQ2, ..., dQN with a sequence of N heat reservoirs with temperatures T1, ..., TN. A negative dQ means that energy flows from the reservoir to the engine, and a positive dQ means that energy flows from the engine to the reservoir. We can show (see the box on the right) that

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:sum_{i=1}^N rac{dQ_i}{T_i} = 0 ,!.

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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,

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:oint rac{dQ}{T} = 0 ,! (reversible cycles)

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where the integral is taken over the entire cycle, and T is the temperature of the system at each point in the cycle.

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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.

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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

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:S_X = S_R + int_R^X rac{dQ}{T} ,!

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

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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

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:Delta S ge int rac{dQ}{T} ,!

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where the equality holds if the transformation is reversible.

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Notice that if dQ = 0, then ΔS ≥ 0. This is the Second Law of Thermodynamics, which we have discussed earlier.

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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".

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Measuring entropy

In real experiments, 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.

Related Topics:
Experiment - Measure - Calorimetry

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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

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

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

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

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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,

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

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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.

Related Topics:
First order phase transition - Latent heat

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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.

Related Topics:
Spectroscopically - Ice

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