(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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They deduced that the distribution of energy \ \[Epsilon] as a function of positions q and momenta p in a system of \ molecules could be represented as an exponential function of temperature, f(\ \[Epsilon]) = C", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", StyleBox[\(\(\[Epsilon](p, q)\)\/\(k\ T\)\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]], TraditionalForm]]], ", where T is temperature and k is an universal constant (Boltzmann \ constant). A connection to thermodynamic quantities could be made by \ observing that the average energy of a system of molecules would be the \ normalized (multi-dimensional) integral of the energy of the system weighted \ by the energy distribution function. The normalization factor was named the \ phase integral, sum of states (Zustandessumme) or ", StyleBox["partition function, q = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Integral]\(\[ExponentialE]\^\(-\(\(\[Epsilon](p, q)\)\/\(k\ T\)\)\)\) \[DifferentialD]p \ \[DifferentialD]q\)]], StyleBox[".", FontSlant->"Italic"], StyleBox[" ", FontVariations->{"CompatibilityType"->0}] }], "Text", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell["History", "Subsection"], Cell[TextData[{ StyleBox["Historically, the concept of matter in motion can be traced to \ the ancient Greeks atomists, particularly Democritus in the 5th Century BC \ and revived in the 17th Century by Galileo. Daniel Bernoulli gave a \ quantitative account of gas pressure and an explanation of heat due to \ molecular motion in 1738, referred to as the ", FontVariations->{"CompatibilityType"->0}], StyleBox["kinetic molecular theory of matter", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[". The thread was picked up by Clerk Maxwell in 1859 who, by \ applying statistics to a physical system for the first time, laid the \ foundation for a ", FontVariations->{"CompatibilityType"->0}], StyleBox["statistical mechanics", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" model of matter. From statistics average values of quantities of \ interest are obtained from sums and/or integrals of probabilistic \ distribution functions of the quantities. Maxwell used an exponential \ distribution in kinetic energy ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "m", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["v", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], " to discuss the distribution in velocities of colliding gas molecules. \ Boltzmann generalized the energy to treat a broader class of systems in 1877 \ and provided a kinetic molecular explanation of entropy in terms of \ probabilities of mechanics complementary to Maxwell's explanation of \ mechanical energy in terms of statistics. Boltzmann's connection between \ mechanics and thermodynamics is summarized in his equation S = k log W, which \ is engraved on his tombstone. W is a measure of the number of possible states \ of a system and k is a conversion factor (the universal Boltzmann constant) \ to make the connection to thermodynamical quantities. S is the entropy of the \ system and the log is the natural logarithm. Gibbs completed the classical \ statistical mechanics edifice of the 19th Century by generalizing the choice \ of independent variables for thermodynamic systems. The natural thermodynamic \ variables in Boltzmann's system are energy, volume and number of subsystems. \ While volume and number are convenient to measure and control, energy is not. \ A more practical set of variables is temperature, volume and number which are \ natural to the Helmholtz thermodynamic energy. This is the set used in this \ study and has the fundamental statistical mechanical relation A(T,V,N) = -NkT \ Log[Q(T,V,N)], with Q(T,V,N) the ", StyleBox["canonical partition function", FontSlant->"Italic"], ". Other quantities of interest may be derived from the fundamental \ relation through their thermodynamic definitions: S = -", Cell[BoxData[ \(TraditionalForm\`dA\/dT\)]], ", U = A +TS, Cv = ", Cell[BoxData[ \(TraditionalForm\`dU\/dT\)]], ", etc. Gibbs also introduced the Gibbs \"free\" energy G(T,P,N) as a \ combination of energy and entropy to govern chemical process that are \ constrained conveniently at fixed temperature and pressure (reactions in a \ beaker open to the atmosphere, say). His work established the mathematical \ framework for the thermodynamicist Max Planck to develop quantum mechanics at \ the beginning of the 20th Century.\nModern statistical mechanics is based on \ the extension of classical mechanical distribution functions to quantized \ systems." }], "Text", FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["Heat Capacity", "Subsection"], Cell[TextData[{ "Euler's theorem for homogeneous functions for which f(\[Lambda]", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", \[Lambda]", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...) = ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\^n\)]], "f(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",..) states that nf = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+i\( x\_i\) df\/dx\_i\)]], ". Application of Euler's theorem to the average energy of a molecular \ system gives immediately a striking and powerful result called the ", StyleBox["equipartition theorem", FontSlant->"Italic"], ", namely each quadratic term in the classical energy expression \ (Hamiltonian) of a molecular system contributes ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "kT to its average molecular energy, or ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "RT to the average molar energy. The beauty of the equipartition theorem \ lies in the fact that simple molecular motion is governed by simple energy \ expression power laws in the variables of the motion. Thus the classical \ Hamiltonian for a collection of N non-interacting diatomic molecules modeled \ as two-body systems of total mass M, reduced mass \[Mu] with intraparticle \ separation r translating and rotating freely in three dimensions and \ vibrating as a harmonic oscillator with force constant k is a sum of N \ identical terms \[Epsilon] = ", Cell[BoxData[ \(TraditionalForm\`p\_x\^2\/\(2 M\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`p\_y\^2\/\(2 M\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`p\_z\^2\/\(2 M\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`p\_\[Theta]\^2\/\(2 \[Mu]\ r\^2\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`p\_\[Phi]\^2\/\(2 \[Mu]\ \(r\^2\) \(sin\^2\) \ \[Theta]\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`p\_r\^2\/\(2 \[Mu]\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\(k\ r\^2\)\/2\)]], ". The p terms are various linear and angular momentum variables. Although \ energy cannot be measured easily, its derivative, known as the heat capacity \ Cv = ", Cell[BoxData[ \(TraditionalForm\`\((dE\/dT)\)\_V\)]], " can be and it provides a particularly sensitive test of theoretical \ models. From the equipartition theorem we see that Cv = ", Cell[BoxData[ \(TraditionalForm\`nN\/2\)]], "k = ", Cell[BoxData[ \(TraditionalForm\`n\/2\)]], "R for N = 1 mol (Avogadro's constant), where R is the molar gas constant \ and n is the number of quadratic terms in the Hamiltonian. In the case of the \ diatomic molecule there are 7 terms in quadratic variables so the classical \ heat capacity would be Cv = ", Cell[BoxData[ \(TraditionalForm\`7\/2\)]], "R. However this value is much larger than that observed for dihydrogen at \ room temperature. Furthermore, Cv is not observed to be constant with \ changing temperature. Further progress required a new description of \ molecules, namely the ", StyleBox["quantum mechanics", FontSlant->"Italic"], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Quantum Statistical Mechanics", "Section"], Cell[TextData[{ "In the 20th Century quantum mechanics was developed to describe the \ behavior of molecular systems. In the simplest approximation, the various \ types of molecular motion may be considered as independent, resulting in a \ sum of energies for the various types. Analogous to a classical mechanical \ system, the types of motion are categorized as translation, rotation, \ vibration, to which may be added electronic and possibly nuclear excitation \ (arranged in order of increasing quantized energy level differences): ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_molecule\ = \ \[Epsilon]\_translation\ \ + \ \[Epsilon]\_rotation\ + \ \[Epsilon]\_vibration\ + \ \ \[Epsilon]\_electronic + \ \[Epsilon]\_nuclear\)]], ".\nFrom the quantum mechanical eigensolutions for free translation, free \ rotation, harmonic oscillation and electronic and nuclear structure, \ statistical mechanical partition functions may be constructed in terms of \ universal (Boltzmann, Avogadro) constants and the mechanical and bonding \ parameters characteristic of the molecular system (mass, bonding distance and \ vibrational frequencies). The corresponding partition function is then a \ product of partition functions for the various types of molecular motion:" }], "Text"], Cell[BoxData[ \(q\_molecule\ = \ \(q\_translation\) \(q\_rotation\) \(q\_vibration\) \ \(q\_electronic\) q\_nuclear\)], "Text"], Cell[CellGroupData[{ Cell["Temperature effects", "Subsection"], Cell[TextData[{ "At sufficiently high thermal temperatures characterized by the mechanical \ parameters of the system the statistical mechanical sums (partition \ functions) may be approximated as classical limiting integrals. For lower \ temperatures sums must be used. The temperatures of thermal excitation depend \ on the type of motion and the values of parameters characteristic of each \ molecule. For molecular dihydrogen, these characteristic excitation \ temperatures are:\n\n", Cell[BoxData[ \(\(\[Theta]\_\(\(translation\)\(\ \)\(=\)\(\ \)\)\) \(\(\ \)\(h\^2\)\)\ \/\(2*\[Pi]Mk\) = 1.5119402601410126`*^-18 K, \ \(\[Theta]\_\(\(rotation\)\(\ \)\(=\)\(\ \)\)\) h\^2\/\(8*\(\[Pi]\^2\) Ik\) = \ 87.4409 K, \[Theta]\_\(\(vibation\)\(\ \)\) = \ \(h\[Nu]\/k\ = \ 6338.31 K\), \[Theta]\_\(\(electronic\)\(\ \)\) = \ \ \(\[Epsilon]\_electronic\/k\ = \ 51966 \( \(K\)\(.\)\(\ \)\)\)\)], "Text"], "\nNuclear excitation energies are of the order of ", Cell[BoxData[ \(TraditionalForm\`10\^6\)]], "K. The diatomic partition functions for translation, rotation, vibration \ and electronic excitation have rather simple forms upon integration or \ summation (", Cell[BoxData[ \(g\_e\)]], " is the degeneracy of the electronic state):" }], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ \(q\_t[ T]\ = \ \ \((T\/\[CapitalTheta][t])\)\^\(3\/2\)\)], "DisplayFormula",\ FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(q\_r[ T]\ = \ \(\[Sum]\+\(J = 0\)\%\(n\_J\)\((2 J + 1)\) \ \[ExponentialE]\^\(-\(\(J*\((J + 1)\)*\[CapitalTheta][r]\)\/T\)\)\)\/\[Sigma]\ \ \[Rule] \ \(T\/\[Sigma]\[CapitalTheta][ r]\) \((high\ temperature)\)\)], "DisplayFormula"], Cell[BoxData[ \(q\_v[ T]\ = \ \ \(\[Sum]\+\(v = 0\)\%\[Infinity] \[ExponentialE]\^\(-\(\ \[CapitalTheta][v]\/T\)\)\ = \ \(\(\ \)\(1\)\)\/\(1\ - \[ExponentialE]\^\(-\ \(\[CapitalTheta][v]\/T\)\)\)\)\)], "DisplayFormula"], Cell[BoxData[ \(q\_e[ T]\ = \ \[Sum]\+\(i = 1\)\%\(n\_e\)g\_e[\([i]\)] \[ExponentialE]\^\(\ \(\[CapitalTheta][e]\)[\([i]\)]\/T\)\)], "DisplayFormula"], Cell["\<\ Under the assumption that the individual contributions to the total energy of \ the molecule are not interdependent, the basic partition function for a \ molecule is the product of the the contributing partition functions:\ \>", "Text"], Cell[BoxData[ \(Q[T]\ = \ q\_t[T] q\_r[T] q\_v[T] q\_e[T]\)], "Text"], Cell[TextData[{ "The heat capacity of ", Cell[BoxData[ \(TraditionalForm\`7\/2\)]], "R for a diatomic molecule would be achieved only at sufficiently high \ temperatures to populate the higher translational, rotational and vibrational \ energy states of the molecules. In the case of all gas molecules we could be \ confident that translation would be fully activated at room temperature \ (300K) and that rotational excited states of the molecule would be somewhat \ occupied but that the molecules would be largely in the ground vibrational \ and electronic states. Thus the expected heat capacity of gaseous dihydrogen \ at room temperature would be somewhat less than ", Cell[BoxData[ \(TraditionalForm\`5\/2\)]], "R. This is close to the value observed in the laboratory." }], "Text"], Cell[TextData[{ "Evidence of this discussion of heat capacity in terms of activation of \ quantized energy levels was actually observed a half century before the \ invention of quantum mechanics by Clerk Maxwell. Maxwell noted that as \ dihydrogen gas was cooled below room temperature, its heat capacity \ diminished, approaching the value of a monatomic gas (", Cell[BoxData[ \(TraditionalForm\`3\/2\)]], "R) at cryogenic temperatures. However, since Maxwell did not have the \ quantum-mechanical statistical mechanical theory at his disposal, he did not \ deduce the cause of the change in heat capacity. The increase in heat \ capacity of dihydrogen as temperature approaches ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_rotation\)]], " (87K) is shown in the theoretical statistical mechanical heat capacity \ vs. temperature graph below. An integral approximation has been employed for \ the translational partition function and the quantized sum of terms given \ above has been used for the rotational partition function." }], "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -3.20924e-017 0.00333333 0 0.103006 [ [.16667 -0.0125 -6 -9 ] [.16667 -0.0125 6 0 ] [.33333 -0.0125 -9 -9 ] [.33333 -0.0125 9 0 ] [.5 -0.0125 -9 -9 ] [.5 -0.0125 9 0 ] [.66667 -0.0125 -9 -9 ] [.66667 -0.0125 9 0 ] [.83333 -0.0125 -9 -9 ] [.83333 -0.0125 9 0 ] [1 -0.0125 -9 -9 ] [1 -0.0125 9 0 ] [1.025 0 0 -6.28125 ] [1.025 0 10 6.28125 ] [-0.0125 .10301 -6 -4.5 ] [-0.0125 .10301 0 4.5 ] [-0.0125 .20601 -6 -4.5 ] [-0.0125 .20601 0 4.5 ] [-0.0125 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