User talk:Paul Wormer/scratchbook1

Entropy is a function of the state of a thermodynamic system. It is a size-extensive quantity with dimension energy divided by temperature (SI unit: joule/K). Entropy has no clear analogous mechanical meaning—unlike volume, a similar size-extensive state parameter with dimension energy divided by pressure. Moreover entropy cannot directly be measured, there is no such thing as an entropy meter, whereas state parameters as volume and temperature are easily determined. Consequently entropy is one of the least understood concepts in physics.

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The state variable "entropy" was introduced by Rudolf Clausius in 1865 when he gave a mathematical formulation of the second law of thermodynamics. He coined its name from the classical Greek ἐν + τροπή (en = in, at; tropè = change, transformation). On purpose he chose a term similar to "energy", because of the close relationship between the two concepts.

The traditional way of introducing entropy is by means of a Carnot engine, an abstract engine conceived by Sadi Carnot (1824) as an idealization of a steam engine. Carnot's work foreshadowed the second law of thermodynamics. This "engineering" manner of introducing entropy will be discussed below. In this approach, entropy is the amount of heat (per degree kelvin) gained or lost by a thermodynamic system that makes a transition from one state to another.

In the 1870s, Ludwig Boltzmann gave a definition of entropy in the context of the kinetic gas theory, a branch of physics that developed into statistical thermodynamics. Boltzmann's definition of entropy was furthered by John von Neumann to a quantum statistical definition. This point of view will also be discussed in the present article. In the statistical approach the entropy of an isolated (constant energy) system is k_BlnP, where k_B is Boltzmann's constant, P is the number of different wave functions of the system belonging to the system's energy (P is the degree of degeneracy, the probability that the state is described by one of the P wave functions), and the function ln stands for the natural (base e) logarithm.

Not satisfied with the engineering type of argument, the mathematician Constantin Carathéodory gave in 1909 a new axiomatic formulation of entropy and the second law. This was based on Pfaffian differential equations. His axiom replaced the earlier Kelvin-Planck and the equivalent Clausius formulation of the second law and his method did not invoke Carnot engines. Caratheodory's work was taken up by Max Born, and it reached some textbooks. Since it requires more mathematical knowledge than the traditional approach based on Carnot engines, and since this mathematical knowledge is not needed by most students of thermodynamics, the traditional approach is still dominant in the majority of textbooks.

Definition

The state of a thermodynamic system (a point in state space) is characterized by a number of variables, such as pressure p, temperature T, amount of substance n, volume V, etc. Any thermodynamic parameter can be seen as a function of an arbitrary independent set of other thermodynamic variables, hence the terms "property", "parameter", "variable" and "function" are used interchangeably. The number of independent thermodynamic variables of a system is equal to the number of energy contacts of the system with its surroundings.

An example of a reversible (quasi-static) energy contact is offered by the prototype thermodynamical system, a gas-filled cylinder with piston. Such a cylinder can perform work on its surroundings,

$DW=pdV,\quad dV>0,$

where dV stands for a small increment of the volume V of the cylinder, p is the pressure inside the cylinder and DW stands for a small amount of work. Work by expansion is a form of energy contact between the cylinder and its surroundings. This process can be reverted, the volume of the cylinder can be decreased, the gas is compressed and the surroundings perform work DW = pdV on the cylinder.

The small amount of work is indicated by D, and not by d, because DW is not necessarily a differential of a function. However, when we divide DW by p the quantity DW/p becomes obviously equal to the differential dV of the differentiable state function V. State functions depend only on the actual values of the thermodynamic parameters (they are local), and not on the path along which the state was reached (the history of the state). Mathematically this means that integration from point 1 to point 2 along path I in state space is equal to integration along a different path II,

$V_{2}-V_{1}={\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}dV={\int \limits _{1}\limits ^{2}}_{{\!\!}^{(II)}}dV\;\Longrightarrow \;{\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}{\frac {DW}{p}}={\int \limits _{1}\limits ^{2}}_{{\!\!}^{(II)}}{\frac {DW}{p}}$

The amount of work (divided by p) performed along path I is equal to the amount of work (divided by p) along path II. This condition is necessary and sufficient that DW/p is a differentiable state function. So, although DW is not a differential, the quotient DW/p is one.

Reversible absorption of a small amount of heat DQ is another energy contact of a system with its surroundings; DQ is again not a differential of a certain function. In a completely analogous manner to DW/p, the following result can be shown for the heat DQ (divided by T) absorbed by the system along two different paths (along both paths the absorption is reversible):

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

(1)

Hence the quantity dS defined by

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

is the differential of a state variable S, the entropy of the system. In a later subsection equation (1) will be proved from the Clausius/Kelvin principle. Observe that this definition of entropy only fixes entropy differences:

$S_{2}-S_{1}\equiv \int _{1}^{2}dS=\int _{1}^{2}{\frac {DQ}{T}}$

Note further that entropy has the dimension energy per degree temperature (joule per degree kelvin) and recalling the first law of thermodynamics (the differential dU of the internal energy satisfies dU = DQ − DW), it follows that

$dU=TdS-pdV.\,$

(For convenience sake only a single work term was considered here, namely DW = pdV, work done by the system). The internal energy is an extensive quantity, that is, when the system is doubled, U is doubled too. The temperature T is an intensive property, independent of the size of the system. The entropy S, then, is an extensive property. In that sense the entropy resembles the volume of the system.

An important difference between V and S is that the former is a state function with a well-defined mechanical meaning, whereas entropy is introduced by analogy and is not easily visualized. Indeed, as is shown in the next subsection, it requires a fairly elaborate reasoning to prove that S is a state function, i.e., equation (1) to hold.

Proof that entropy is a state function

When equation (1) has been proven, the entropy S is shown to be a state function. The standard proof, as given now, is physical, by means of Carnot cycles, and is based on the Clausius/Kelvin formulation of the second law given in the introduction.

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Fig. 1. T > T₀. (I): Carnot engine E moves heat from heat reservoir R to "condensor" C and needs input of work DW_in. (II): E generates work DW_out from the heat flow from C to R.

An alternative, more mathematical proof, postulates the existence of a state variable S with certain properties and derives the existence of thermodynamical temperature and the second law from these properties.

In figure 1 a finite heat bath C ("condensor") of constant volume and variable temperature T is shown. It is connected to an infinite heat reservoir R through a reversible Carnot engine E. Because R is infinite its temperature T₀ is constant, addition or extraction of heat does not change T₀. It is assumed that always T ≥ T₀. One may think of the system E-plus-C as a ship and the heat reservoir R as the sea. The following argument then deals with an attempt of extracting energy from the sea in order to move the ship, i.e., with an attempt to let E perform net outgoing work in a cyclic (i.e., along a closed path in the state space of C) process.

A Carnot engine performs reversible cycles (in the state space of E, not be confused with cycles in the state space of C) and per cycle either generates work DW_out when heat is transported from high temperature to low temperature (II), or needs work DW_in when heat is transported from low to high temperature (I), in accordance with the Clausius/Kelvin formulation of the second law.

The definition of thermodynamical temperature (a positive quantity) is such that for II,

${\frac {DW_{\mathrm {out} }}{DQ}}={\frac {T-T_{0}}{T}},$

while for I

${\frac {DW_{\mathrm {in} }}{DQ_{0}}}={\frac {T-T_{0}}{T_{0}}}.$

The first law of thermodynamics states for I and II, respectively,

$-DW_{\mathrm {in} }-DQ_{0}+DQ=0\quad {\hbox{and}}\quad DW_{\mathrm {out} }+DQ_{0}-DQ=0$

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Fig. 1. Two paths in the state space of the "condensor" C.

For I,

${\begin{aligned}{\frac {DW_{\mathrm {in} }}{DQ_{0}}}&={\frac {DQ-DQ_{0}}{DQ_{0}}}={\frac {DQ}{DQ_{0}}}-1\\&={\frac {T-T_{0}}{T_{0}}}={\frac {T}{T_{0}}}-1\;\Longrightarrow DQ_{0}=T_{0}\left({\frac {DQ}{T}}\right)\end{aligned}}$

For II we find the same result,

${\begin{aligned}{\frac {DW_{\mathrm {out} }}{DQ}}&={\frac {DQ-DQ_{0}}{DQ}}=1-{\frac {DQ_{0}}{DQ}}\\&={\frac {T-T_{0}}{T}}=1-{\frac {T_{0}}{T}}\;\Longrightarrow DQ_{0}=T_{0}\left({\frac {DQ}{T}}\right)\end{aligned}}$

In figure 2 the state diagram of the "condensor" C is shown. Along path I the Carnot engine needs input of work to transport heat from the colder reservoir R to the hotter C and the absorption of heat by C raises its temperature and pressure. Integration of DW_in = DQ − DQ₀ (that is, summation over many cycles of the engine E) along path I gives

$W_{\mathrm {in} }=Q_{\mathrm {in} }-T_{0}{\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}{\frac {DQ}{T}}\quad {\hbox{with}}\quad Q_{\mathrm {in} }\equiv {\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}DQ.$

Along path II the Carnot engine delivers work while transporting heat from C to R. Integration of DW_out = DQ − DQ₀ along path II gives

$W_{\mathrm {out} }=Q_{\mathrm {out} }-T_{0}{\int \limits _{2}\limits ^{1}}_{{\!\!}^{(II)}}{\frac {DQ}{T}}\quad {\hbox{with}}\quad Q_{\mathrm {out} }\equiv {\int \limits _{2}\limits ^{1}}_{{\!\!}^{(II)}}DQ$

Assume now that the amount of heat Q_out extracted (along path II) from C and the heat Q_in delivered (along I) to C are the same in absolute value. In other words, after having gone along a closed path in the state diagram of figure 2, the condensor C has not gained or lost heat. That is,

$Q_{\mathrm {in} }+Q_{\mathrm {out} }=0,\,$

then

$W_{\mathrm {in} }+W_{\mathrm {out} }=-T_{0}{\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}{\frac {DQ}{T}}-T_{0}{\int \limits _{2}\limits ^{1}}_{{\!\!}^{(II)}}{\frac {DQ}{T}}.$

If the total net work W_in + W_out is positive (outgoing), this work is done by heat obtained from R, which is not possible because of the Clausius/Kelvin principle. If the total net work W_in + W_out is negative, then by inverting all reversible processes, i.e., by going down path I and going up along II, the net work changes sign and becomes positive (outgoing). Again the Clausius/Kelvin principle is violated. The conclusion is that the net work is zero and that

$T_{0}{\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}{\frac {DQ}{T}}+T_{0}{\int \limits _{2}\limits ^{1}}_{{\!\!}^{(II)}}{\frac {DQ}{T}}=0\;\Longrightarrow \;{\int \limits _{1}\limits ^{2}}_{{\!\!}^{(I)}}{\frac {DQ}{T}}={\int \limits _{1}\limits ^{2}}_{{\!\!}^{(II)}}{\frac {DQ}{T}}.$

From this independence of path it is concluded that

$dS\equiv {\frac {DQ}{T}}$

is a state (local) variable.

References

• M. W. Zemansky, Kelvin and Caratheodory—A Reconciliation, American Journal of Physics Vol. 34, pp. 914-920 (1966) [1]