(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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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Alberty", "Subtitle"], Cell["\<\ Department of Chemistry Massachusetts Institute of Technology Cambridge, MA 02139\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "\tProtein molecules interact with species in their aqueous environment. \ They are always in equilibrium with hydrogen ions, but they may also bind \ other ligands at specific sites. This package is concerned with proteins \ like hemoglobin that bind oxygen molecules at specific sites that are linked \ with acid dissociations in the protein. The simple system considered here is\ \n", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["pK", RowBox[{"1", StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SubscriptBox["pK", RowBox[{"2", StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]], TraditionalForm]]], "\t\t\t\t\t\t\t\t\t\t\t\t\t(1)\n M = HM = ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["H", "Text", FontSlant->"Plain", FontVariations->{"CompatibilityType"->0}], "2"], TraditionalForm]]], "M\nK ||\n ", Cell[BoxData[ \(TraditionalForm\`MO\_2\)]], "= ", Cell[BoxData[ \(TraditionalForm\`HMO\_2\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], "TraditionalForm"], \(MO\_2\)}], TraditionalForm]]], "\n ", Cell[BoxData[ \(TraditionalForm\`pK\_\(1 MO2\)\)]], " pK", Cell[BoxData[ \(TraditionalForm\`\_\(2 MO2\)\)]], "\nwhere K = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[ RowBox[{ RowBox[{"[", StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["]", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], " ", "[", SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"]}], "TraditionalForm"], "]"}], "/", \([\ MO\_2\)}], TraditionalForm]]], "] and the hydrogen ions have been omitted in the acid disociations. This \ is a hypothetical system, but the pKs and their enthalpies of reaction have \ been chosen to be the same as those determined by Antonini and coworkers [1] \ for human hemoglobin.\n\tThere are three ways to make thermodynamic \ calculations on a system like this, depending on the thermodynamic variables \ that are specified. When the temperature and pressure are specified, the \ criterion for spontaneous change and equilibrium is provided by the Gibbs \ energy G. In this approach, which is the one used in chemical \ thermodynamics, the system is described in terms of species, like the eight \ species involved in the system considered here. \n\tIn biochemistry it is \ convenient to specify in addition the pH, and in this case the system is \ described in terms of reactants, which are sums of species. When this is \ done the above system can be described in terms of reactants (pseudoisomer \ groups), and system (1) is described by a single biochemical reaction:\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], "TraditionalForm"], \(MO\_2\)}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], "TraditionalForm"], StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "+ ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], " \t\t\t\t\t\t\t\t\t\t\t\t(2)\nwhere the sub av indicates an average \ number. The equilibrium constant ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", "Text", FontSlant->"Plain", FontVariations->{"CompatibilityType"->0}], "O2"], TraditionalForm]]], "' for this biochemical reaction is referred to as an apparent equilibrium \ constant because it is a function of pH as well as temperature.\n", Cell[BoxData[ \(TraditionalForm\`K\_O2\)]], "'=", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"[", " ", SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"]}], "TraditionalForm"], StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "][", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "] /", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"[", SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"]}], "TraditionalForm"], \(MO\_2\)}], TraditionalForm]]], "] \t\t\t\t\t\t\t\t\t\t\t(3)\nWhen the pH is specified, the Gibbs energy \ no longer provides the criterion for spontaneous change and equilibrium, and \ it is necessary to use a Legendre transform to define a transformed Gibbs \ energy G' that does. \n\tIn discussing a ligand like oxygen molecules, it is \ often convenient to specify the equilibrium concentration of molecular \ oxygen. In this case the system can simply be described in terms of ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], "TraditionalForm"], \(MO\_\(2 av\)\)}], TraditionalForm]]], ". When temperature, pressure, pH, and [", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "] are specified, it is necessary to use a Legendre transform to define a \ further transformed Gibbs energy G'' to provide the criterion for spontaneous \ change and equilibrium. These three types of calculations, in the order in \ which we will discuss them they are:" }], "Text"], Cell[TextData[{ "I. At specified T, P, and pH, the transformed Gibbs energy G' provides \ the criterion for equilibrium\nII. At specified T, P, pH, and [", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "], the further transformed Gibbs energy G'' provides the criterion for \ equilibrium\nIII. At specified T and P, the Gibbs energy G provides the \ criterion for equilibrium\n\nThe third type of calculation is the most basic, \ but in the absence of experimental methods for detecting species directly, \ the approaches in I and II are used. However, approaches I and II can yield \ the standard thermodynamic properties of species, as described in this \ package. \n\tThe Gibbs energies G', G'', and G are the most important \ thermodynamic properties of these systems because if they can be determined \ as functions of their natural variables (the variables indicated in I, II, \ and III), all the other thermodynamic properties can be obtained by taking \ partial derivatives. \n\tThis Introduction cannot provide more information \ on the ideas and equations behind the calculations presented here, but all \ the background is provided in ref. 2. More specifically, the thermodynamics \ of protein-ligand binding is discussed in references 3-5. This last article \ emphasizes the importance of Maxwell relations in providing relationships \ between thermodynamic properties. Maxwell relations (second cross partial \ derivatives) apply to exact differentials. " }], "Text"], Cell[TextData[{ "\tMaxwell relations are important because they show the relationships \ between different properties. Since properties like G, G', and G'' cannot be \ measured directly, it is important that some Maxwell relations yield \ experimental properties that are related to properties like G''. A prime \ example of this is the average number of oxygen molecules bound ", Cell[BoxData[ \(N\&_\_O2\)]], ", which can be determined spectrophotometrically. Another example is ", Cell[BoxData[ \(\(\[CapitalDelta]\_r\) N\_H\)]], ", which can be obtained as the difference between the acid titration \ curves of the protein in the absence of oxygen and saturated with oxygen. If \ ", Cell[BoxData[ \(N\&_\_O2\)]], " can be determined at a series of pHs and at least two temperatures, all \ the other thermodynamic properties can be calculated, including the standard \ properties of the species involved. " }], "Text"], Cell[TextData[{ "\tStandard further transformed thermodynamic properties like ", Cell[BoxData[ \(\(\[CapitalDelta]\_f\) G'' \[Degree]\)]], " are very complicated functions of T, pH, and ", Cell[BoxData[ RowBox[{"[", RowBox[{ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], FormBox[\(\_2\), "TraditionalForm"]}], "]"}]]], ". Taking partial derivatives yields even more complex functions. These \ could never be done by hand, but ", StyleBox["Mathematica ", FontSlant->"Italic"], "makes it possible to make them.\n\tData and programs for calculations on \ the thermodynamics of metabolism are freely available in MathSource [6] if \ you have ", StyleBox["Mathematica", FontSlant->"Italic"], " in your personal computer. More ", StyleBox["Mathematica", FontSlant->"Italic"], " programs for the thermodynamics of biochemical reactions are available in \ the ", StyleBox["Mathematica", FontSlant->"Italic"], " Journal by Akers and Goldberg [7].. \n\nReferences\n\n(1) E. Antonini, \ J. Wymna, M. Brunori, C. Fronticelli, E. Bucci, and A. Rossi-Fanelli, J. \ Biol. Chem. 240, 1090-1103 (1965).\n(2) R. A. Alberty, Thermodynamics of \ Biochemical Reactions, Wiley, Hoboken, NJ (2003).\n(3) R. A. Alberty, \ Biophys. Chem. 62, 141-159 (1996).\n(4) R. A. Alberty, Biophys. Chem. 63, \ 119-132 (1997).\n(5) R. A. Alberty, Biophys. Chem., in press.\n(6) R. A. \ Alberty, BasicBiochemData2, (2002) \ http://www.mathsource.com/cgi/msitem?0211-622. ", StyleBox["\n", FontSlant->"Italic"], "(7) D. L. Akers and R. N. Goldberg, Math. J. 8, 86-113 (2001). \ http://www.mathematica-journal.com/issue/v8i1/" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[BoxData[ \(\(BeginPackage["\"];\)\)], "Input"], Cell[BoxData[ \(\(Off[General::spell, General::spelll];\)\)], "Input"], Cell[BoxData[{ \(\(calcappeqconst::usage = "\";\)\), "\n\ ", \(\(reactionprops293::usage = \*"\"\\"";\)\), "\n", \(\(reactionprops313::usage = \*"\"\\"";\)\), "\[IndentingNewLine]", \(\(calcstdfrtrGe::usage = \*"\"\\"";\)\), "\n", \(\(calcstdfrtrGe293::usage = "\";\)\), "\n", \(\(calcstdfrtrGe313::usage = "\";\)\), "\n", \(\(nO2::usage = "\";\)\), "\n", \ \(\(taboxycols293::usage = "\";\)\), "\n", \(\ \(taboxycols313::usage = "\";\)\), "\n", \(\(frtrGbyint::usage = "\";\)\), "\n", \(\(calcnegRTlnPM::usage = \*"\"\< calcnegRTlnPM derives the function of \ temperature, pH, and concentration of molecular oxygen that yields -\!\(RTlnP\ \_M\). 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At specified T, P, and pH, the transformed Gibbs energy G' \ provides the criterion for equilibrium\ \>", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Ia. Calculate the apparent equilibrium constant for the reaction ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], "TraditionalForm"], \(MO\_2\)}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], "TraditionalForm"], StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "+ ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], " " }], "Subsection"], Cell[TextData[{ "The apparent equilibrium constant for reaction (2) is given by\n", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "O2"], TraditionalForm]]], "' = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox["KP", StyleBox["M", FontSlant->"Plain"]], "TraditionalForm"], "/", \(P\_MO2\)}], TraditionalForm]]], "\t\t\t\t\t\t\t\t\t\t\t\t\t(4)\nwhere K = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[ RowBox[{ RowBox[{"[", StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "]"}], " ", "[", SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"]}], "TraditionalForm"], "]"}], "/", \([MO\_2\)}], TraditionalForm]]], "]. ", Cell[BoxData[ FormBox[ StyleBox[\(P\_M\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], "is the binding polynomial for ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], FontVariations->{"CompatibilityType"->0}], StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], " that is given by\n", Cell[BoxData[ FormBox[ StyleBox[\(P\_M\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], "= 1 + ", Cell[BoxData[ \(TraditionalForm\`10\^\(\(-pH\) + pK1M\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`10\^\(\(-2\) pH + pK1M + pK2M\)\)]], "\t\t\t\t\t\t\t\t\t(5)\nThe binding polynomial for ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], FontVariations->{"CompatibilityType"->0}], StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], "TraditionalForm"], SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"]}], TraditionalForm]]], " is given by \n", Cell[BoxData[ FormBox[ StyleBox[\(P\_MO2\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], "= 1 + ", Cell[BoxData[ \(TraditionalForm\`10\^\(\(-pH\) + pK1MO2\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`10\^\(\(-2\) pH + pK1MO2 + pK2MO2\)\)]], "\t\t\t\t\t\t\t\t(6)\nThe following program derives the function of T and \ pH that gives ", Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["K", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "O2"], "'"}]]], ". The values of the five properties in system (1) are given as functions \ of temperature in the program calcappeqconst." }], "Text"], Cell[BoxData[ \(calcappeqconst := Module[{pK1M, pK2M, pK1MO2, pK2MO2, k, pM, pMO2}, (*This\ program\ derives\ the\ function\ of\ temperature\ \ and\ pH\ that\ yields\ the\ apparent\ equilibrium\ constant\ for\ the\ \ dissociation\ of\ molecular\ oxygen\ from\ HavMO2 . \ \ Energies\ are\ in\ \ joules\ per\ \(\(mole\)\(.\)\)*) \[IndentingNewLine]pK1M = 7.85 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pK2M = 5.46 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); pK1MO2 = 6.67 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); pK2MO2 = 6.04 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]k = 10^\(-5\)* Exp[\((\((60.7*10^3)\)/8.3145)\)*\((1/293.15 - 1/t)\)]; \[IndentingNewLine]pM = \((1 + 10^\((\(-pH\) + pK1M)\) + 10^\((\(-2\)*pH + pK1M + pK2M)\))\); \[IndentingNewLine]pMO2 = \((1 + 10^\((\(-pH\) + pK1MO2)\) + 10^\((\(-2\)*pH + pK1MO2 + pK2MO2)\))\); \[IndentingNewLine]k* pM/pMO2]\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(calcappeqconst\)], "Input"], Cell[BoxData[ \(\(\((1 + 10\^\(\(\(7.85`\)\(\[InvisibleSpace]\)\) - pH - \ 1969.1986250228626`\ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - \ 1\/t)\)\) + 10\^\(\(\(13.309999999999999`\)\(\[InvisibleSpace]\)\) - 2\ pH - \ 1640.1282977644003`\ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - \ 1\/t)\)\))\)\ \[ExponentialE]\^\(7300.499128029345`\ \ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - 1\/t)\)\)\)\/\(100000\ \ \((1 + 10\^\(\(\(6.67`\)\(\[InvisibleSpace]\)\) - pH - 1969.1986250228626`\ \ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - 1\/t)\)\) + \ 10\^\(\(\(12.71`\)\(\[InvisibleSpace]\)\) - 2\ pH - 1640.1282977644003`\ \ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - 1\/t)\)\))\)\)\)], \ "Output"] }, 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At specified T, P, pH, and [", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "], the further transformed Gibbs energy G'' provides the criterion for \ equilibrium" }], "Section"], Cell[CellGroupData[{ Cell[TextData[{ "IIa", StyleBox[". Calculate the standard further transformed Gibbs energy of \ formation of ", "Subsection"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]] }], "Subsection"], Cell[TextData[{ "\tWhen these independent variables are specified, the system can be \ represented by ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]], ", where av indicates an average number. The standard further transformed \ Gibbs energy of formation of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]], " is given by\n", Cell[BoxData[ \(\(\[CapitalDelta]\_f\) G'' \[Degree]\)]], " = -RTln", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["P", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "M"], TraditionalForm]]], "- RTln(1 + ", Cell[BoxData[ RowBox[{"[", RowBox[{ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, 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pK2MO2, k, pM, pMO2, kprimeO2}, (*This\ program\ derives\ the\ function\ of\ \ temperature, \ pH, \ and\ concentration\ of\ molecular\ oxygen\ that\ yields\ the\ \ standard\ further\ transformed\ Gibbs\ energy\ of\ formation\ of\ HavMO2av . \ \ \ Energies\ are\ in\ joules\ per\ \ \(\(mole\)\(.\)\)*) \[IndentingNewLine]pK1M = 7.85 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pK2M = 5.46 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); pK1MO2 = 6.67 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); pK2MO2 = 6.04 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]k = 10^\(-5\)* Exp[\((60.7*10^3/8.3145)\)*\((1/293.15 - 1/t)\)]; \[IndentingNewLine]pM = \((1 + 10^\((\(-pH\) + pK1M)\) + 10^\((\(-2\)*pH + pK1M + pK2M)\))\); \[IndentingNewLine]pMO2 = \((1 + 10^\((\(-pH\) + pK1MO2)\) + 10^\((\(-2\)*pH + pK1MO2 + pK2MO2)\))\); \[IndentingNewLine]kprimeO2 = k*pM/pMO2; 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" }], "Text"], Cell[BoxData[ \(\(taboxycols293 = \(\(nO2 /. t \[Rule] 293.15\) /. co2 \[Rule] { .0000001, .000001, .00001, .0001, .001, .01}\) \ /. pH \[Rule] {5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9};\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[Transpose[taboxycols293], TableHeadings \[Rule] {{5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9}, {"\<.0000001\>", "\<.000001\>", "\<.00001\>", "\<.0001\>", \ "\<.001\>", "\<.01\>"}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\".0000001\"\>", "\<\".000001\"\>", \ "\<\".00001\"\>", "\<\".0001\"\>", "\<\".001\"\>", "\<\".01\"\>"}, {"5", "0.00203401994074513`", "0.019974541859407738`", "0.1693086535132365`", "0.6708540190732498`", "0.953230937996039`", "0.995117582313652`"}, {"5.5`", "0.001560963893593734`", "0.015393382310060735`", "0.13520277153591775`", "0.6098937668156129`", "0.9398823248670273`", "0.9936443545194699`"}, {"6", "0.0011704899480094106`", "0.011582880671538846`", "0.1048940312873646`", "0.539565504591662`", "0.9213750930276652`", "0.9915387723926142`"}, {"6.5`", "0.0011752170800911107`", "0.011629169612112232`", "0.10527350510262266`", "0.5405678244224105`", "0.9216669179788224`", "0.991572559122118`"}, {"7", "0.0018304059382646472`", "0.018007411525741045`", "0.15496022274724`", "0.6471122716525858`", "0.9482873405311785`", "0.9945763078630293`"}, {"7.5`", "0.0035253933782026263`", "0.03416977664321247`", "0.26133112950590714`", "0.7796320878968585`", "0.9725113550187244`", "0.9971814040569713`"}, {"8", "0.006087244925740583`", "0.057710753817385906`", "0.3798267471115439`", "0.8596397933960584`", "0.9839345207517817`", "0.9983698823155293`"}, {"8.5`", "0.00822250892148163`", "0.0765594894496052`", "0.45327344365229505`", "0.8923652416831098`", "0.9880820135347842`", "0.998795279292905`"}, {"9", "0.009295198154488888`", "0.08577621897412313`", "0.48406827361612725`", "0.9036832443343827`", "0.9894541572061404`", "0.9989353104771145`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9}, { ".0000001", ".000001", ".00001", ".0001", ".001", ".01"}}]]]], "Output"] }, Open ]], Cell[TextData[{ "If the average number of oxygen molecules bound by the protein is measured \ at a specified pH and a series of concentrations of molecular oxygen, the \ value of the apparent equilibrium constant at that pH and temperature can be \ calculated by plotting 1/", Cell[BoxData[ \(TraditionalForm\`N\&_\_O2\)]], " versus 1/[", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "]. For example, at 293.15 K and pH 5 the the average numbers of oxygen \ molecules bound at the six oxygen concentrations are given by " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Transpose[taboxycols293]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({0.00203401994074513`, 0.019974541859407738`, 0.1693086535132365`, 0.6708540190732498`, 0.953230937996039`, 0.995117582313652`}\)], "Output"] }, Open ]], Cell["The reciprocal binding numbers are given by", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(recipnos = 1/\(Transpose[taboxycols293]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({491.63726469351445`, 50.063726469351465`, 5.906372646935145`, 1.4906372646935147`, 1.0490637264693514`, 1.0049063726469352`}\)], "Output"] }, Open ]], Cell["The reciprocal concentrations are given by", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(recipconc = 1/{ .0000001, .000001, .00001, .0001, .001, .01}\)], "Input"], Cell[BoxData[ \({1.`*^7, 1.`*^6, 99999.99999999999`, 10000.`, 1000.`, 100.`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Fit[Transpose[{recipconc, recipnos}], {1, x}, x]\)], "Input"], Cell[BoxData[ \(\(\(0.9999999999999729`\)\(\[InvisibleSpace]\)\) + 0.00004906372646935143`\ x\)], "Output"] }, Open ]], Cell["\<\ The slope of the double reciprocal plot is equal to the apparent \ equilibrium constant under these conditions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(calcappeqconst /. t \[Rule] 293.15\) /. pH \[Rule] 5\)], "Input"], Cell[BoxData[ \(0.00004906372646935146`\)], "Output"] }, Open ]], Cell[TextData[{ "Table 6 Average Number of Oxygen Molecules Bound in ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]], "at 313.15 K as a Function of pH and Concentration of Molecular Oxygen. \ The concentrations of molecular oxygen are molar. 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2\ pH\) + \ 10\^\(\(\(7.85`\)\(\[InvisibleSpace]\)\) - pH\)\)]\)], "Output"] }, Open ]], Cell[TextData[{ "Since integration yields the second term in equation 8, and so the first \ term in ", Cell[BoxData[ FormBox[ SubscriptBox["P", StyleBox["M", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], TraditionalForm]]], " has to be added. " }], "Text"], Cell[BoxData[ \(\(\(calcnegRTlnPM := Module[{pK1M, pK2M, pM}, (*This\ program\ derives\ the\ function\ of\ temperature, \ pH, \ and\ concentration\ of\ molecular\ oxygen\ that\ yields\ - RTlnPM . \ Energies\ are\ in\ joules\ per\ \(\(mole\)\(.\)\)\ *) \[IndentingNewLine]pK1M = 7.85 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pK2M = 5.46 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pM = \((1 + 10^\((\(-pH\) + pK1M)\) + 10^\((\(-2\)*pH + pK1M + pK2M)\))\); \[IndentingNewLine]\(-8.3145\)*t* Log[pM]]\)\(\[IndentingNewLine]\) \)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(calcnegRTlnPM\)], "Input"], Cell[BoxData[ \(\(-8.3145`\)\ t\ Log[ 1 + 10\^\(\(\(7.85`\)\(\[InvisibleSpace]\)\) - pH - \ 1969.198625022863`\ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - 1\ \/t)\)\) + 10\^\(\(\(13.309999999999999`\)\(\[InvisibleSpace]\)\) - 2\ pH - \ 1640.1282977644005`\ \((\(\(0.003411222923418046`\)\(\[InvisibleSpace]\)\) - \ 1\/t)\)\)]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\((calcnegRTlnPM + frtrGbyint)\) /. t \[Rule] {293.15, 313.15}\) /. pH \[Rule] {12, 13, 14}\) /. co2 \[Rule] 0\)], "Input"], Cell[BoxData[ \({{\(-0.17254834139133912`\), \(-0.017255379348808126`\), \ \(-0.0017255433872025688`\)}, {\(-0.0686377923539441`\), \ \(-0.006863858558452048`\), \(-0.0006863866492304645`\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(calcstdfrtrGe /. t \[Rule] {293.15, 313.15}\) /. pH \[Rule] {12, 13, 14}\) /. co2 \[Rule] 0\)], "Input"], Cell[BoxData[ \({{\(-0.17254834139133912`\), \(-0.017255379348808126`\), \ \(-0.0017255433872025688`\)}, {\(-0.0686377923539441`\), \ \(-0.006863858558452048`\), \(-0.0006863866492304645`\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "This shows that experimental data on ", Cell[BoxData[ \(TraditionalForm\`N\&_\_O2\)]], " can be integrated to obtain the oxygen-dependent term in the standard \ further transformed Gibbs energy. Measurements of ", Cell[BoxData[ \(TraditionalForm\`N\&_\_O2\)]], " as a function of pH and [", Cell[BoxData[ FormBox[ StyleBox[\(O\_2\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], "] at at least two temperatures can be used to obtain all the thermodynamic \ properties of the system." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["IIc. Calculate the a", "Subsection"], "verage number of hydrogen ions bound in ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]], " " }], "Subsection"], Cell[TextData[{ "\tAccording to a Maxwell equation, the average number of hydrogen ions \ bound in ", Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}]]], " is given by (1/RTln(10)) times the partial derivative of G'' with respect \ to pH." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(nH = 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Calculate the standard further transformed entropy of formation of", " ", Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}]]] }], "Subsection"], Cell[TextData[{ "The standard further transformed entropy of formation of ", Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}]]], " is calculated by taking the derivative of ", Cell[BoxData[ \(\(\[CapitalDelta]\_f\) G'' \[Degree]\)]], " with respect to T and multiplying by minus 1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(frtrentropy = 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.83761 .33023 .85674 .35905 .90312 .34308 .88449 .31357 Metetra .758 .684 .78 r .25585 .57802 .28077 .61539 .32241 .57119 .29731 .54279 Metetra .757 .731 .829 r .29731 .54279 .32241 .57119 .36334 .53872 .33849 .51387 Metetra .746 .756 .861 r .33849 .51387 .36334 .53872 .40417 .51239 .37971 .48881 Metetra .733 .767 .88 r .37971 .48881 .40417 .51239 .44519 .48963 .42119 .46625 Metetra .721 .77 .891 r .42119 .46625 .44519 .48963 .48657 .46907 .46304 .44537 Metetra .711 .768 .897 r .46304 .44537 .48657 .46907 .52839 .44994 .50533 .42564 Metetra .703 .765 .9 r .50533 .42564 .52839 .44994 .57073 .43175 .54814 .40671 Metetra .697 .761 .901 r .54814 .40671 .57073 .43175 .61362 .41419 .59151 .38835 Metetra .691 .756 .901 r .59151 .38835 .61362 .41419 .65711 .39706 .63547 .37036 Metetra .687 .751 .9 r .63547 .37036 .65711 .39706 .70123 .38019 .68006 .35263 Metetra .684 .747 .899 r .68006 .35263 .70123 .38019 .746 .36348 .72531 .33506 Metetra .681 .742 .897 r .72531 .33506 .746 .36348 .79146 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.31546 Metetra .696 .759 .9 r .59026 .31546 .6132 .34316 .65828 .32443 .63585 .29566 Metetra .693 .753 .898 r .63585 .29566 .65828 .32443 .70401 .30585 .68208 .27601 Metetra .689 .748 .896 r .68208 .27601 .70401 .30585 .75042 .28735 .729 .25642 Metetra .687 .742 .893 r .729 .25642 .75042 .28735 .79755 .26887 .77663 .23687 Metetra .684 .737 .891 r .77663 .23687 .79755 .26887 .8454 .25037 .82499 .21729 Metetra .766 .789 .872 r .17414 .49938 .20262 .52116 .24443 .49294 .21645 .47118 Metetra .756 .793 .883 r .21645 .47118 .24443 .49294 .28633 .467 .25889 .44485 Metetra .746 .793 .891 r .25889 .44485 .28633 .467 .32846 .44276 .30157 .41996 Metetra .737 .79 .896 r .30157 .41996 .32846 .44276 .37092 .41979 .34458 .39618 Metetra .728 .786 .899 r .34458 .39618 .37092 .41979 .41378 .39778 .38798 .37327 Metetra .721 .782 .9 r .38798 .37327 .41378 .39778 .4571 .37651 .43184 .35102 Metetra .715 .776 .901 r .43184 .35102 .4571 .37651 .50092 .35578 .47621 .32929 Metetra .709 .771 .9 r .47621 .32929 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0.00266574, 0.00266574}}] }, Open ]], Cell[TextData[{ "Note that these standard further transformed entropies are with reapect to \ ", Cell[BoxData[ \(\(\[CapitalDelta]\_f\) S\[Degree]\)]], "(M) = 0 at each temperature." }], "Text"], Cell[TextData[{ "Table 11 Standard Further Transformed Entropy of Formation of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]], "at 293.15 K as a Function of pH and Concentration of Molecular Oxygen. \ The entropies are in J ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["K", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], \(-1\)], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`mol\^\(-1\)\)]], ". " }], "Text"], Cell[BoxData[ \(\(frtrentropy293 = \(\(frtrentropy /. t \[Rule] 293.15\) /. co2 \[Rule] { .0000001, .000001, .00001, .0001, .001, .01}\) \ /. pH \[Rule] {5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9};\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[Transpose[frtrentropy293], TableHeadings \[Rule] {{5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9}, {"\<.0000001\>", "\<.000001\>", "\<.00001\>", "\<.0001\>", \ "\<.001\>", "\<.01\>"}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\".0000001\"\>", "\<\".000001\"\>", \ "\<\".00001\"\>", "\<\".0001\"\>", "\<\".001\"\>", "\<\".01\"\>"}, { "5", \(-47.1533109407505`\), \(-50.64747429399138`\), \ \(-79.61327918087498`\), \(-173.81554418165163`\), \(-214.96243085872203`\), \ \(-204.6852925926462`\)}, { "5.5`", \(-67.97553766798683`\), \(-70.61550370752187`\), \ \(-93.40745567425607`\), \(-181.36556627744972`\), \(-231.56317066462043`\), \ \(-223.59232540693858`\)}, { "6", \(-85.04844443087514`\), \(-86.95126255597725`\), \ \(-103.95979418928208`\), \(-181.50413419376386`\), \(-239.77736793371315`\), \ \(-234.65195795934574`\)}, { "6.5`", \(-95.10212493468411`\), \(-96.76231113814875`\), \ \(-111.58991993521026`\), \(-178.8195330294276`\), \(-227.8222285881949`\), \ \(-220.97189893500692`\)}, { "7", \(-95.20611064233775`\), \(-97.29829107163576`\), \ \(-114.91175747042192`\), \(-175.43461734509418`\), \(-200.94744740539747`\), \ \(-188.57417650944672`\)}, { "7.5`", \(-79.632621004521`\), \(-83.51068602701699`\), \ \(-111.9538264059255`\), \(-171.88086125384638`\), \(-180.61763827269095`\), \ \(-165.01216649464345`\)}, { "8", \(-49.82985701822925`\), \(-57.619659069794864`\), \ \(-105.514875805384`\), \(-169.68519977404807`\), \(-171.48657675364657`\), \ \(-154.76508053973404`\)}, { "8.5`", \(-23.312920150309704`\), \(-35.38919775522078`\), \ \(-100.87492326021868`\), \(-168.7710177743858`\), \(-168.21950056221647`\), \ \(-151.15059124811302`\)}, { "9", \(-9.709024789030522`\), \(-24.27269272562708`\), \ \(-98.83826886157891`\), \(-168.45267532012636`\), \(-167.14354911977847`\), \ \(-149.96622723723632`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9}, { ".0000001", ".000001", ".00001", ".0001", ".001", ".01"}}]]]], "Output"] }, Open ]], Cell[TextData[{ "Table 12 Standard Further Transformed Entropy of Formation of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "av"], \(MO\_\(2 av\)\)}], TraditionalForm]]], "at 313.15 K as a Function of pH and Concentration of Molecular Oxygen. \ The entropies are in J ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["K", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], \(-1\)], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`mol\^\(-1\)\)]], ". " }], "Text"], Cell[BoxData[ \(\(frtrentropy313 = \(\(frtrentropy /. t \[Rule] 313.15\) /. co2 \[Rule] { .0000001, .000001, .00001, .0001, .001, .01}\) \ /. pH \[Rule] {5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9};\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[Transpose[frtrentropy313], TableHeadings \[Rule] {{5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9}, {"\<.0000001\>", "\<.000001\>", "\<.00001\>", "\<.0001\>", \ "\<.001\>", "\<.01\>"}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\".0000001\"\>", "\<\".000001\"\>", \ "\<\".00001\"\>", "\<\".0001\"\>", "\<\".001\"\>", "\<\".01\"\>"}, { "5", \(-46.151133914248966`\), \(-46.848512512270425`\), \ \(-53.532767370427784`\), \(-100.14098454787434`\), \(-186.54522757440253`\), \ \(-200.6872130287268`\)}, { "5.5`", \(-66.4950482298958`\), \(-67.03408392297827`\), \ \(-72.24487237277023`\), \(-110.97816657647455`\), \(-197.17912635181807`\), \ \(-216.7235582087489`\)}, { "6", \(-82.28388839734973`\), \(-82.687657513018`\), \ \(-86.61360012700868`\), \(-117.15990018356563`\), \(-195.63729644793074`\), \ \(-217.35166845862986`\)}, { "6.5`", \(-87.57098767600522`\), \(-87.97912654001884`\), \ \(-91.91804773599404`\), \(-120.81334002467212`\), \(-181.77885912368916`\), \ \(-190.90452215764884`\)}, { "7", \(-76.73438408103975`\), \(-77.42058584345021`\), \ \(-83.85065900033925`\), \(-122.18340346027095`\), \(-168.85525320741635`\), \ \(-165.52654285812`\)}, { "7.5`", \(-50.03367862829301`\), \(-51.45739260603521`\), \ \(-64.1751710390897`\), \(-121.79745439013533`\), \(-162.66064147595537`\), \ \(-154.3449461069767`\)}, { "8", \(-23.484058795664907`\), \(-25.8383535973154`\), \ \(-46.01615464013578`\), \(-121.2242592705282`\), \(-160.40506586214474`\), \ \(-150.47877387938115`\)}, { "8.5`", \(-8.939010549881623`\), \(-11.88790694684809`\), \ \(-36.62864068712619`\), \(-120.94666691194077`\), \(-159.65862570969446`\), \ \(-149.22463014133163`\)}, { "9", \(-3.239218069402532`\), \(-6.437370514823539`\), \ \(-33.051459165421235`\), \(-120.84598278208647`\), \(-159.41914557451992`\), \ \(-148.82495800085542`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9}, { ".0000001", ".000001", ".00001", ".0001", ".001", ".01"}}]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\((frtrenthalpy - t*frtrentropy)\) /. t \[Rule] 293.15\) /. co2 \[Rule] {0, .0001}\) /. pH \[Rule] {5, 7, 9}\)], "Input"], Cell[BoxData[ \({{\(-19303.210459209913`\), \(-5153.340525089148`\), \ \(-166.76670197107933`\)}, {\(-22011.775946929527`\), \ \(-7692.144833560553`\), \(-5870.547960536656`\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(calcstdfrtrGe /. t \[Rule] 293.15\) /. co2 \[Rule] {0, .0001}\) /. pH \[Rule] {5, 7, 9}\)], "Input"], Cell[BoxData[ \({{\(-19303.210459209913`\), \(-5153.340525089144`\), \ \(-166.76670197107907`\)}, {\(-22011.775946929516`\), \ \(-7692.144833560547`\), \(-5870.547960536648`\)}}\)], "Output"] }, Open ]], Cell["These agree as expected.", "Text"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ III. At specified T and P, the Gibbs energy G provides the \ criterion for equilibrium\ \>", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "IIIa Calculate the standard reaction properties for M", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], " = M + O", Cell[BoxData[ \(TraditionalForm\`\_2\)]] }], "Subsection"], Cell["\<\ \tThe following program can be used to calculate the thermodynamic \ properties of the five reactions involved in system (1) at any desired \ temperature in the range approximately 273.25 K to 313.15 K. Since these \ reactions are written in terms of species, the program is used here to \ calculate properties at 298.15 K since that is the usual temperature for \ chemical thermodynamic tables. \ \>", "Text"], Cell[BoxData[ \(calcchemrxprops := Module[{k, pK1M, pK2M, pK1MO2, pK2MO2, line1rx, line2rx, line3rx, line4rx, line5rx}, (*This\ program\ calculates\ the\ functions\ for\ \ calculating\ K\ and\ four\ pKs\ and\ the\ reaction\ Gibbs\ energy, \ stndard\ reaction\ enthalpy, \ and\ standard\ reaction\ entropy\ at\ a\ specified\ temperature . \ \ \ The\ energies\ are\ in\ joules\ per\ mole, \ and\ the\ entropies\ are\ in\ joules\ per\ K\ per\ \ \(\(mole\)\(.\)\)*) \[IndentingNewLine]k = 10^\(-5\)* Exp[\((\((60.7*10^3)\)/8.3145)\)*\((1/293.15 - 1/t)\)]; \[IndentingNewLine]pK1M = 7.85 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pK2M = 5.46 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pK1MO2 = 6.67 - \((37.7*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]pK2MO2 = 6.04 - \((\(-6.3\)*10^3/\((8.3145*Log[10])\))\)*\((1/293.15 - 1/t)\); \[IndentingNewLine]line1rx = {k, \(-8.3145\)* \ .29815*Log[k], \(-t^2\)* D[\(-8.3145\)*Log[k], t]/1000, \(-D[\(-8.3145\)*t*Log[k], t]\)}; \[IndentingNewLine]line2rx = {pK1M, 8.3145* .29815*Log[10]*pK1M, \(-t^2\)* D[8.3145*Log[10]*pK1M, t]/1000, \(-D[8.3145*t*Log[10]*pK1M, t]\)}; \[IndentingNewLine]line3rx = {pK2M, 8.3145* .29815*Log[10]*pK2M, \(-t^2\)* D[8.3145*Log[10]*pK2M, t]/1000, \(-D[8.3145*t*Log[10]*pK2M, t]\)}; \[IndentingNewLine]line4rx = {pK1MO2, 8.3145* .29815*Log[10]*pK1MO2, \(-t^2\)* D[8.3145*Log[10]*pK1MO2, t]/1000, \(-D[8.3145*t*Log[10]*pK1MO2, t]\)}; \[IndentingNewLine]line5rx = {pK2MO2, 8.3145* .29815*Log[10]*pK2MO2, \(-t^2\)* D[8.3145*Log[10]*pK2MO2, t]/1000, \(-D[8.3145*t*Log[10]*pK2MO2, t]\)}; \[IndentingNewLine]{line1rx, line2rx, line3rx, line4rx, line5rx}]\)], "Input", InitializationCell->True], Cell["\<\ Table 12 Thermodynamic Properties of Chemical Reactions at 298.15 K\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[calcchemrxprops /. t \[Rule] 298.15, TableHeadings \[Rule] {{"\", "\", \ "\", "\", "\"}, \ {"\", \*"\"\<\!\(\[CapitalDelta]\_r\)G\[Degree]\>\"", \*"\"\<\!\(\ \[CapitalDelta]\_r\)H\[Degree]\>\"", \ \*"\"\<\!\(\[CapitalDelta]\_r\)S\[Degree]\>\""}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"K or pK\"\>", "\<\"\\!\\(\[CapitalDelta]\\_r\\)G\ \[Degree]\"\>", "\<\"\\!\\(\[CapitalDelta]\\_r\\)H\[Degree]\"\>", "\<\"\\!\\(\ \[CapitalDelta]\\_r\\)S\[Degree]\"\>"}, {"\<\"K(MO2=M+O2)\"\>", "0.000015183677681604054`", "27.504869671550903`", "60.69999999999999`", "111.33701267298036`"}, {"\<\"pK1M(HM=H+M)\"\>", "7.737349061035698`", "44.16506053016469`", "37.7`", \(-21.683919269376773`\)}, {"\<\"pK2M(H2M=H+HM)\"\>", "5.478824957970161`", "31.2733255271463`", \(-6.3`\), \(-126.02155132365019`\)}, {"\<\"pK1MO2(HMO2=H+MO2)\"\>", "6.557349061035699`", "37.42957903456593`", "37.7`", "0.906996362348039`"}, {"\<\"pK2MO2(H2MO2=H+HMO2)\"\>", "6.058824957970161`", "34.58398592328806`", \(-6.3`\), \(-137.12556070195558`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{"K(MO2=M+O2)", "pK1M(HM=H+M)", "pK2M(H2M=H+HM)", "pK1MO2(HMO2=H+MO2)", "pK2MO2(H2MO2=H+HMO2)"}, { "K or pK", "\!\(\[CapitalDelta]\_r\)G\[Degree]", "\!\(\[CapitalDelta]\_r\)H\[Degree]", "\!\(\[CapitalDelta]\_r\)S\[Degree]"}}]]]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["IIIb Calculate standard thermodynamic properties of species", \ "Subsection"], Cell[TextData[{ "\tEight species are involved in these five chemical reactions, and the \ standard thermodynamic propertie of H", Cell[BoxData[ \(TraditionalForm\`\(\^+\)\)]], " and O", Cell[BoxData[ \(TraditionalForm\`\_2\)]], ",aq are available in the NBS Tables. Since the standard thermodynamic \ properties of the remaining six species cannot be calculated from the \ properties of the five reactions, the properties of M are assigned to be zero \ at each temperature. This is the same as the convention of the usual \ chemical thermodynamic properties for ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["H", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "+"], TraditionalForm]]], ". The entropy of formation of ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["O", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "(aq) at 298.15 K, which is not given in the NBS Tables is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(-11.7\) - 16.4)\)/ .29815\)], "Input"], Cell[BoxData[ \(\(-94.24786181452286`\)\)], "Output"] }, Open ]], Cell[TextData[{ "in J ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SuperscriptBox[ StyleBox["K", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], \(-1\)], "TraditionalForm"], \(mol\^\(-1\)\)}], TraditionalForm]]], "." }], "Text"], Cell["\<\ The other properties can be calculated from Table 12 by using \ LinearSolve. When the species are arranged in the order HM, \ H2M,MO2,HMO2,H2MO2, the stoichiometric number matrix is \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(stoichmat = {{0, 0, \(-1\), 0, 0}, {\(-1\), 0, 0, 0, 0}, {1, \(-1\), 0, 0, 0}, {0, 0, 1, \(-1\), 0}, {0, 0, 0, 1, \(-1\)}}\)], "Input"], Cell[BoxData[ \({{0, 0, \(-1\), 0, 0}, {\(-1\), 0, 0, 0, 0}, {1, \(-1\), 0, 0, 0}, {0, 0, 1, \(-1\), 0}, {0, 0, 0, 1, \(-1\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[stoichmat]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"0", "0", \(-1\), "0", "0"}, {\(-1\), "0", "0", "0", "0"}, {"1", \(-1\), "0", "0", "0"}, {"0", "0", "1", \(-1\), "0"}, {"0", "0", "0", "1", \(-1\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ When the properties are arranged in the order G, H, S, the \ thermodynamic properties of the reactions are given by.\ \>", 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