Entropy (classical thermodynamics)

Definition

According to the Clausius equality, for a closed homogeneous system, in which only reversible processes take place,

${\displaystyle \oint \frac{\delta Q}{T}=0.}$

With ${\displaystyle T}$ being the uniform temperature of the closed system and ${\displaystyle \delta Q}$ the incremental reversible transfer of heat energy into that system.

That means the line integral ${\int }_L\frac{\delta Q}{T}$ is path-independent.

A state function ${\displaystyle S}$, called entropy, may be defined which satisfies

${\displaystyle \mathrm{d}S=\frac{\delta Q}{T}.}$

Entropy measurement

The thermodynamic state of a uniform closed system is determined by its temperature T and pressure P. A change in entropy can be written as

${\displaystyle \mathrm{d}S={\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }T}\right)}_P\mathrm{d}T+{\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }P}\right)}_T\mathrm{d}P.}$

The first contribution depends on the heat capacity at constant pressure C_P through

${\displaystyle {\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }T}\right)}_P=\frac{C_P}{T}.}$

This is the result of the definition of the heat capacity by δQ = C_P dT and T dS = δQ. The second term may be rewritten with one of the Maxwell relations

${\displaystyle {\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }P}\right)}_T=-{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_P}$

and the definition of the volumetric thermal-expansion coefficient

${\displaystyle {\alpha }_V=\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_P}$

so that

${\displaystyle \mathrm{d}S=\frac{C_P}{T}\mathrm{d}T-{\alpha }_{V}V\mathrm{d}P.}$

With this expression the entropy S at arbitrary P and T can be related to the entropy S₀ at some reference state at P₀ and T₀ according to

${\displaystyle S(P,T)=S(P_0,T_0)+{\int }_{T_0}^T\frac{C_P(P_0,T^{\mathrm{\prime }})}{T^{\mathrm{\prime }}}\mathrm{d}{T}^{\mathrm{\prime }}-{\int }_{P_0}^P{\alpha }_V(P^{\mathrm{\prime }},T)V(P^{\mathrm{\prime }},T)\mathrm{d}{P}^{\mathrm{\prime }}.}$

In classical thermodynamics, the entropy of the reference state can be put equal to zero at any convenient temperature and pressure. For example, for pure substances, one can take the entropy of the solid at the melting point at 1 bar equal to zero. From a more fundamental point of view, the third law of thermodynamics suggests that there is a preference to take S = 0 at T = 0 (absolute zero) for perfectly ordered materials such as crystals.

S(P, T) is determined by followed a specific path in the P-T diagram: integration over T at constant pressure P₀, so that dP = 0, and in the second integral one integrates over P at constant temperature T, so that dT = 0. As the entropy is a function of state the result is independent of the path.

The above relation shows that the determination of the entropy requires knowledge of the heat capacity and the equation of state (which is the relation between P,V, and T of the substance involved). Normally these are complicated functions and numerical integration is needed. In simple cases it is possible to get analytical expressions for the entropy. In the case of an ideal gas, the heat capacity is constant and the ideal gas law PV = nRT gives that α_VV = V/T = nR/p, with n the number of moles and R the molar ideal-gas constant. So, the molar entropy of an ideal gas is given by

${\displaystyle S_m(P,T)=S_m(P_0,T_0)+C_P\ln \frac{T}{T_0}-R\ln \frac{P}{P_0}.}$

In this expression C_P now is the molar heat capacity.

The entropy of inhomogeneous systems is the sum of the entropies of the various subsystems. The laws of thermodynamics hold rigorously for inhomogeneous systems even though they may be far from internal equilibrium. The only condition is that the thermodynamic parameters of the composing subsystems are (reasonably) well-defined.

Temperature-entropy diagrams

Entropy values of important substances may be obtained from reference works or with commercial software in tabular form or as diagrams. One of the most common diagrams is the temperature-entropy diagram (TS-diagram). For example, Fig.2 shows the TS-diagram of nitrogen, depicting the melting curve and saturated liquid and vapor values with isobars and isenthalps.

Entropy change in irreversible transformations

We now consider inhomogeneous systems in which internal transformations (processes) can take place. If we calculate the entropy S₁ before and S₂ after such an internal process the Second Law of Thermodynamics demands that S₂ ≥ S₁ where the equality sign holds if the process is reversible. The difference S_i = S₂ − S₁ is the entropy production due to the irreversible process. The Second law demands that the entropy of an isolated system cannot decrease.

Suppose a system is thermally and mechanically isolated from the environment (isolated system). 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 allows heat conduction. Our above result indicates that the entropy of the system as a whole will increase during these processes. 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.

An irreversible process degrades the performance of a thermodynamic system, designed to do work or produce cooling, and results in entropy production. The entropy generation during a reversible process is zero. Thus entropy production is a measure of the irreversibility and may be used to compare engineering processes and machines.

Thermal machines

Clausius' identification of S as a significant quantity was motivated by the study of reversible and irreversible thermodynamic transformations. 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 its environment. The net result of a cycle is

1. mechanical work done by the system (which can be positive or negative, the latter meaning that work is done on the engine),
2. heat transferred from one part of the environment to another. In the steady state, 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 heat transfers occur in the opposite directions and the amount of work done switches sign.

Heat engines

Consider a heat engine working between two temperatures T_H and T_a. With T_a we have ambient temperature in mind, but, in principle it may also be some other low temperature. The heat engine is in thermal contact with two heat reservoirs which are supposed to have a very large heat capacity so that their temperatures do not change significantly if heat Q_H is removed from the hot reservoir and Q_a is added to the lower reservoir. Under normal operation T_H > T_a and Q_H, Q_a, and W are all positive.

As our thermodynamical system we take a big system which includes the engine and the two reservoirs. It is indicated in Fig.3 by the dotted rectangle. It is inhomogeneous, closed (no exchange of matter with its surroundings), and adiabatic (no exchange of heat with its surroundings). It is not isolated since per cycle a certain amount of work W is produced by the system given by the first law of thermodynamics

${\displaystyle W=Q_H-Q_a.}$

We used the fact that the engine itself is periodic, so its internal energy has not changed after one cycle. The same is true for its entropy, so the entropy increase S₂ − S₁ of our system after one cycle is given by the reduction of entropy of the hot source and the increase of the cold sink. The entropy increase of the total system S₂ - S₁ is equal to the entropy production S_i due to irreversible processes in the engine so

${\displaystyle S_i=-\frac{Q_H}{T_H}+\frac{Q_a}{T_a}.}$

The Second law demands that S_i ≥ 0. Eliminating Q_a from the two relations gives

${\displaystyle W=\left(1-\frac{T_a}{T_H}\right)Q_H-T_a{S}_i.}$

The first term is the maximum possible work for a heat engine, given by a reversible engine, as one operating along a Carnot cycle. Finally

${\displaystyle W=W_{\text{max}}-T_a{S}_i.}$

This equation tells us that the production of work is reduced by the generation of entropy. The term T_aS_i gives the lost work, or dissipated energy, by the machine.

Correspondingly, the amount of heat, discarded to the cold sink, is increased by the entropy generation

${\displaystyle Q_a=\frac{T_a}{T_H}{Q}_H+T_a{S}_i=Q_{a,\text{min}}+T_a{S}_i.}$

These important relations can also be obtained without the inclusion of the heat reservoirs. See the article on entropy production.

Refrigerators

The same principle can be applied to a refrigerator working between a low temperature T_L and ambient temperature. The schematic drawing is exactly the same as Fig.3 with T_H replaced by T_L, Q_H by Q_L, and the sign of W reversed. In this case the entropy production is

${\displaystyle S_i=\frac{Q_a}{T_a}-\frac{Q_L}{T_L}}$

and the work needed to extract heat Q_L from the cold source is

${\displaystyle W=Q_L\left(\frac{T_a}{T_L}-1\right)+T_a{S}_i.}$

The first term is the minimum required work, which corresponds to a reversible refrigerator, so we have

${\displaystyle W=W_{\text{min}}+T_a{S}_i}$

i.e., the refrigerator compressor has to perform extra work to compensate for the dissipated energy due to irreversible processes which lead to entropy production.