(************** 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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In what regions of the EM \ spectrum do these transitions lie?" }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subtitle"], Cell[TextData[{ "Strategy.\n", StyleBox["Assume harmonic oscillator behavior for the vibrations. The IR \ stretching frequency is determined by the force constant and reduced mass (\ \[Mu]):\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\[Nu]\ = \ 1\/\(2 \[Pi]\)\ \@\(k\/\[Mu]\)\), FontWeight->"Plain"], TraditionalForm]]], "\n\n", StyleBox["The lowest energy transition, which occurs between the ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 0 and ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 1 states, requires ", FontWeight->"Plain"], StyleBox["E", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = ", FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\[Nu].\n\nAssume 3D rigid rotor behavior for the rotations. The \ lowest energy transition, which occurs between the ", FontWeight->"Plain"], StyleBox["l", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 0 and ", FontWeight->"Plain"], StyleBox["l", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 1 (or -1) states, requires \[CapitalDelta]", FontWeight->"Plain"], StyleBox["E", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\((hbar\^2\/\(2 I\))\) \((1)\) \((1 + 1)\)\), FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], RowBox[{ StyleBox["(", FontWeight->"Plain"], FractionBox[ SuperscriptBox[ StyleBox["hbar", FontWeight->"Plain"], "2"], "I"], ")"}]}], TraditionalForm]], FontWeight->"Plain"], " = ", StyleBox["h", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\[Nu].\n\n", FontWeight->"Plain"], "Execution.", StyleBox["\n", FontWeight->"Plain"], "Vibration.", StyleBox[" (", FontWeight->"Plain"], "Note: ", StyleBox["masses of atomic isotopes are given in Engel, inside back cover). \ ", FontWeight->"Plain"], StyleBox["The reduced mass of 1H-35Cl is given by", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[{ \(m1H\ = \ \((1.0078\ amu)\) \((1.661\ 10\^\(-27\)\ kg/ amu)\)\), "\[IndentingNewLine]", \(m35Cl\ = \ \((34.9688\ amu)\)\ \((1.661\ 10\^\(-27\)\ kg/ amu)\)\), "\[IndentingNewLine]", \(\[Mu]\ = \ \((m1H\ m35Cl)\)\/\(m1H\ + \ m35Cl\)\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(1.6739558000000004`*^-27\ kg\)], "Output", CellLabel->"Out[1]="], Cell[BoxData[ \(5.808317680000001`*^-26\ kg\)], "Output", CellLabel->"Out[2]="], Cell[BoxData[ \(1.6270638575918797`*^-27\ kg\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell["The vibration frequency is", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(k\ = \ 516\ \((kg\ m\ s\^\(-2\))\) \((m\^\(-1\))\)\), "\[IndentingNewLine]", \(\[Nu]\ = \ \((1\/\(2\ \[Pi]\))\) \@\(k\/\[Mu]\)\)}], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(\(516\ kg\)\/s\^2\)], "Output", CellLabel->"Out[4]="], Cell[BoxData[ \(8.962778906876739`*^13\ \@\(1\/s\^2\)\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ "This frequency lies inside the IR portion of the EM spectrum (approx ", Cell[BoxData[ \(TraditionalForm\`10\^\(13 - 16\)\ Hz\)]], ")\n\n", StyleBox["Rotation.", FontWeight->"Bold"], " The frequency \[Nu] is given by ", Cell[BoxData[ \(TraditionalForm\`\((hbar\^2\/\(h\ I\))\)\ = \ \((h\/\(4\ \ \(\[Pi]\^2\) I\))\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Irot\ = \ 2.644\ 10\^\(-47\)\ kg\ m\^2\), "\[IndentingNewLine]", \(h\ = \ 6.626\ 10\^\(-34\)\ \((kg\ \(m\^2\) s\^\(-2\))\) \((s)\)\), "\[IndentingNewLine]", \(\[Nu]\ = \ \((h\/\(4\ \[Pi]\^2\ Irot\))\)\)}], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(2.644`*^-47\ kg\ m\^2\)], "Output", CellLabel->"Out[6]="], Cell[BoxData[ \(\(6.6260000000000015`*^-34\ kg\ m\^2\)\/s\)], "Output", CellLabel->"Out[7]="], Cell[BoxData[ \(6.347902447183531`*^11\/s\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[{ "This frequency lies inside the microwave portion of the EM spectrum \ (approx ", Cell[BoxData[ \(TraditionalForm\`10\^\(7 - 13\)\ Hz\)]], ")" }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{729, 651}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.5, StyleDefinitions -> "ArticleModern.nb" ] (******************************************************************* Cached data follows. 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