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'''Entropy''' is a function of the state of a [[thermodynamics|thermodynamic system]]. It is a size-extensive<ref>A size-extensive property of a system becomes ''x'' times larger when the system is enlarged by a factor ''x'', provided all intensive parameters remain the same upon the enlargement. Intensive parameters, like temperature, density, and pressure,  are independent of size.</ref> 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.<ref>It is reported that in a conversation with Claude Shannon, John (Johann) von Neumann  said: "In the second place, and more important, nobody knows what entropy really is [..]”.  M. Tribus, E. C. McIrvine, ''Energy and information'', Scientific American, vol. '''224''' (September 1971), pp. 178–184.</ref>
The '''Jahn-Teller effect''' is the distortion of a highly symmetric—but non-linear—molecule to lower symmetry and lower energy. The effect occurs if the molecule is in a degenerate state of definite energy, that is, more than one wave function is eigenstate with the same energy of the [[molecular Hamiltonian]]. In other words, energy degeneracy of a state implies that two or more orthogonal wave functions describe the  state. Due to Jahn-Teller distortion, the molecule is lowered in symmetry and the energy degeneracy is lifted. One or more wave functions become non-degenerate eigenstates of lower energies, while others wave function rise in energy.
{{Image|Carnot title page.jpg|right|300px}}
The state variable "entropy" was introduced by [[Rudolf Clausius]] in the 1860s<ref>R. J. E. Clausius, ''Abhandlungen über die mechanische Wärmetheorie'' [Treatise on the mechanical theory of heat], two volumes, F. Vieweg, Braunschweig, (1864-1867).</ref>
when he gave a mathematical formulation of  the [[second law of thermodynamics]]. He derived the name  from the classical Greek ἐν + τροπή  (en = in, at; tropè = change, transformation). On purpose Clausius 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  in 1824 by [[Sadi Carnot]]<ref>S. Carnot, ''Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Reflections on the motive power of fire and on machines suited to develop that power)'', Chez Bachelier, Paris (1824).</ref> as an idealization of a steam engine. Carnot's work foreshadowed the [[second law of thermodynamics]]. The  "engineering" manner of introducing entropy through an engine 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.
The effect is named after [[Hermann Jahn|H. A. Jahn]] and [[Edward Teller|E. Teller]] who predicted it in 1937<ref>H. A. Jahn and E. Teller, ''Stability of Polyatomic Molecules in Degenerate Electronic States'', Proc. Royal Soc. vol. '''161''', pp. 220&ndash;235 (1937)</ref>. It took some time before the effect was experimentally observed, because it was masked by other molecular interactions. However, there are now numerous unambiguous observations that agree well with theoretical predictions. These range from the excited states of the simplest non-linear molecule H<sub>3</sub> through moderate sized organic molecules, like ions of substituted [[benzene]], to complex [[crystal]]s and localized impurity centers in solids.


In 1877 [[Ludwig Boltzmann]]<ref> L. Boltzmann,  ''Über die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht'', [On the relation between the second fundamental law of the mechanical theory of heat and the probability calculus with respect to the theorems of heat equilibrium]  Wiener Berichte vol. '''76''', pp. 373-435 (1877)</ref> 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]]<ref>J. von Neumann, ''Mathematische Grundlagen der Quantenmechnik'', [Mathematical foundation of quantum mechanics] Springer, Berlin (1932)</ref> to a quantum statistical definition. The quantum statistical point of view, too, will  be reviewed  in the present article. In the statistical approach the  entropy of an isolated (constant energy) system is ''k''<sub>B</sub> ln''&Omega;'', where ''k''<sub>B</sub>  is [[Boltzmann's constant]], ''&Omega;'' is the number of different wave functions of the system belonging to the system's energy  (''&Omega;'' is the degree of degeneracy, the probability that a state is described by one of the ''&Omega;'' wave functions), and the function ln stands for the natural (base ''e'') [[logarithm]].
<font style = "font-size: 80%; font-style: oblique; font-weight: bolder" >
 
The Jahn-Teller effect is based on a quantum mechanical mechanism and no classical description of it exists.  From here on some knowledge of quantum mechanics is prerequisite to the reading of this article.</font>
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.<ref>C.  Carathéodory,  ''Untersuchungen über die Grundlagen der Thermodynamik''  [Investigation on the foundations of thermodynamics],  Mathematische Annalen, vol. '''67''',  pp. 355-386 (1909).</ref> His theory was based on [[Pfaffian differential equations]]. His axiom replaced the earlier Kelvin-Planck and the equivalent Clausius formulation of the second law and did not need Carnot engines. Caratheodory's work was taken up  by [[Max Born]],<ref>M. Born, Physikalische Zeitschrift, vol. 22, p. 218, 249, 282 (1922)</ref> and it is treated in a few  textbooks.<ref>H. B. Callen, ''Thermodynamics and an Introduction to Thermostatistics.'' John Wiley and Sons, New York, 2nd edition, (1965); E. A. Guggenheim, ''Thermodynamics'', North-Holland, Amsterdam, 5th edition (1967)</ref> 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 introductory works on thermodynamics.
 
==Classical definition==
The state (a point in state space) of a thermodynamic system  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,
:<math>
DW = pdV, \quad dV > 0,
</math> 
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'' < 0 ''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 in state space), 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,
:<math>
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}
</math>
The amount of work (divided by ''p'') performed reversibly 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 the differential of a 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 reversibly by the system along two different paths (along both paths the absorption is reversible):
 
<div style="text-align: right;" >
<div style="float: left; margin-left: 35px;" >
<math>{\int\limits_1\limits^2}_{{\!\!}^{(I)}}\frac{DQ}{T} = {\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DQ}{T} .
</math>
</div>
<span id="(1)" style="margin-right: 200px; vertical-align: -40px; ">(1)</span>
</div>
<br><br>
Hence the quantity ''dS'' defined  by
:<math>
dS \;\stackrel{\mathrm{def}}{=}\; \frac{DQ}{T}
</math>
is the differential of a state variable ''S'', the ''entropy'' of the system. In the next subsection  equation (1) will be proved from the Clausius/Kelvin principle.  Observe that this definition of entropy  only fixes entropy differences:
:<math>
S_2-S_1 \equiv \int_1^2 dS = \int_1^2 \frac{DQ}{T}
</math>
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'' &minus; ''DW''), it follows that
:<math>
dU = TdS - pdV.\,
</math>
(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. The temperature ''T'' is an intensive property, independent of the size of the system. It follows that the entropy ''S'' is an extensive property. In that sense the entropy resembles the volume of the system. We reiterate that volume 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., that equation [[#(1)|(1)]] holds.
 
===Proof that entropy is a state function===
Equation [[#(1)|(1)]] gives the sufficient condition that the entropy ''S''  is  a state function. The standard proof of equation (1), as given now, is physical,  by means of an engine making [[Carnot cycle]]s, and is based on the Kelvin formulation of the [[second law of thermodynamics]].
{{Image|Entropy.png|right|250px}}
 
Consider the figure. A system, consisting of an arbitrary closed system ''C'' (only heat goes in and out) and a reversible heat engine ''E'', is coupled to a large heat reservoir ''R'' of constant temperature ''T''<sub>0</sub>. The system ''C'' undergoes a cyclic state change 1-2-1. Since no work is performed on or by ''C'', it follows that
:<math>
Q_\mathrm{I} =Q_\mathrm{II} \quad\hbox{with}\quad Q_\mathrm{I} \equiv \int_1^2 DQ_\mathrm{I},\quad  Q_\mathrm{II} \equiv \int_1^2 DQ_\mathrm{II}.
</math>
For the heat engine ''E'' it holds (by the definition of [[thermodynamic temperature]]) that
:<math>
\frac{DQ_\mathrm{I}}{DQ^0_\mathrm{I}} = \frac{T_\mathrm{I}}{T_0}\quad\hbox{and}\quad
\frac{DQ_\mathrm{\mathrm{II}}}{DQ^0_\mathrm{II}} = \frac{T_\mathrm{II}}{T_0} .
</math>
Hence
:<math>
\frac{Q^0_\mathrm{I}}{T_0} \equiv \frac{1}{T_0} \int_1^2 DQ^0_\mathrm{I} = \int_1^2 \frac{DQ_\mathrm{I}}{T_\mathrm{I}}
\quad\hbox{and}\quad
\frac{Q^0_\mathrm{II}}{T_0} \equiv \frac{1}{T_0} \int_1^2 DQ^0_\mathrm{II} = \int_1^2 \frac{DQ_\mathrm{II}}{T_\mathrm{II}} .
</math>
From Kelvin's principle it follows that ''W'' is necessarily less or equal zero, because there is only the single heat source ''R''.  Invoking the first law of thermodynamics we get,
:<math>
W = Q^0_\mathrm{I} - Q^0_\mathrm{II} \le 0\; \Longrightarrow\; \frac{Q^0_\mathrm{I}}{T_0} \le \frac{Q^0_\mathrm{II}}{T_0},
</math>
so that
:<math>
\int_1^2 \frac{DQ_\mathrm{I}}{T_\mathrm{I}} \le \int_1^2 \frac{DQ_\mathrm{II}}{T_\mathrm{II}}
</math>
Because the processes inside ''C'' and  ''E'' are assumed reversible, all arrows can be reverted and in the very same way it is shown that
:<math>
\int_1^2 \frac{DQ_\mathrm{II}}{T_\mathrm{II}} \le \int_1^2 \frac{DQ_\mathrm{I}}{T_\mathrm{I}},
</math>
so that  equation (1) holds (with a slight change of notation, subscripts are transferred to the respective integral signs):
:<math>{\int\limits_1\limits^2}_{{\!\!}^{(I)}}\frac{DQ}{T} = {\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DQ}{T} .
</math>
 
 
 
 
 
 
<!--
{{Image|Entropy.png|right|350px|Fig. 1.  ''T'' > ''T''<sub>0</sub>. (I): Carnot engine ''E'' moves  heat from  heat reservoir ''R'' to closed system  ''C'' and needs input of work ''DW''<sub>in</sub>. (II): ''E'' generates work ''DW''<sub>out</sub> from the heat flow from ''C'' to ''R''. }}.
In figure 1  a closed system ''C'' (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 ''T''<sub>0</sub> is constant, addition or extraction of heat does not change ''T''<sub>0</sub>.  Without loss of generality it can be assumed that ''T'' > ''T''<sub>0</sub>, then path I in figure 1 gives transport of heat from low to high temperature and path II form high to low. If it is assumed that
 
A Carnot engine performs reversible cycles (in the state space of ''E'', not to be confused with cycles in the state space of ''C'') and per cycle either generates work ''DW''<sub>out</sub> when heat is transported from ''C'' to ''R'' (II), or needs work  ''DW''<sub>in</sub> 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,
:<math>
\frac{DW_\mathrm{out}}{DQ} = \frac{T-T_0}{T},
</math>
while for  I
:<math>
\frac{DW_\mathrm{in}}{DQ_0} = \frac{T-T_0}{T_0}.
</math>
 
The first law of thermodynamics states for  I and II, respectively,
:<math>
-DW_\mathrm{in}  -DQ_0 + DQ=0\quad\hbox{and}\quad DW_\mathrm{out} + DQ_0-DQ=0
</math>
{{Image|Cycle entropy.png|right|150px|Fig. 1. Two paths in the state space of the "condensor" C.}}
For  I,
:<math>
\begin{align}
\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{align}
</math>
 
For II we find the same result,
:<math>
\begin{align}
\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{align}
</math>
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''<sub>in</sub> = ''DQ'' &minus; ''DQ''<sub>0</sub> (that is, summation  over many cycles of the engine E) along path I gives
:<math>
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.
</math>
Along path II the Carnot engine delivers work while transporting heat from C to R. Integration of ''DW''<sub>out</sub> = ''DQ'' &minus; ''DQ''<sub>0</sub> along path II gives
:<math>
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
</math>
 
Assume now that the  amount of heat ''Q''<sub>out</sub> extracted (along path II) from C and the heat ''Q''<sub>in</sub> 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,
:<math>
Q_\mathrm{in} + Q_\mathrm{out} = 0, \,
</math>
then
:<math>
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}.
</math>
If the total net work ''W''<sub>in</sub> + ''W''<sub>out</sub>  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''<sub>in</sub> + ''W''<sub>out</sub> 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
:<math>
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}.
</math>
From this independence of path it is concluded that
:<math>
dS \equiv \frac{DQ}{T}
</math>
is a state (local) variable.
-->
 
==Footnotes==
<references />
==References ==
* M. W. Zemansky, ''Kelvin and Caratheodory—A Reconciliation'', American Journal of Physics  Vol. '''34''',  pp. 914-920 (1966)  [http://dx.doi.org/10.1119/1.1972279]

Revision as of 07:08, 4 February 2010

The Jahn-Teller effect is the distortion of a highly symmetric—but non-linear—molecule to lower symmetry and lower energy. The effect occurs if the molecule is in a degenerate state of definite energy, that is, more than one wave function is eigenstate with the same energy of the molecular Hamiltonian. In other words, energy degeneracy of a state implies that two or more orthogonal wave functions describe the state. Due to Jahn-Teller distortion, the molecule is lowered in symmetry and the energy degeneracy is lifted. One or more wave functions become non-degenerate eigenstates of lower energies, while others wave function rise in energy.

The effect is named after H. A. Jahn and E. Teller who predicted it in 1937[1]. It took some time before the effect was experimentally observed, because it was masked by other molecular interactions. However, there are now numerous unambiguous observations that agree well with theoretical predictions. These range from the excited states of the simplest non-linear molecule H3 through moderate sized organic molecules, like ions of substituted benzene, to complex crystals and localized impurity centers in solids.

The Jahn-Teller effect is based on a quantum mechanical mechanism and no classical description of it exists. From here on some knowledge of quantum mechanics is prerequisite to the reading of this article.

  1. ↑ H. A. Jahn and E. Teller, Stability of Polyatomic Molecules in Degenerate Electronic States, Proc. Royal Soc. vol. 161, pp. 220–235 (1937)
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