Given that the {html_definer} of the vibrational stretch of {HTML} is {nu_html_outpu}, calculate the force constant, k, in N m-1;
", "advice": "a) If necessary, convert wavenumber to Hz
\n\\[{\\rm wavenumber}\\times {c}={\\nu}\\]
\n\\[\\nu=\\var{wavenumber}~\\rm cm^{-1}\\times 3\\times 10^{10}~cm~s^{-1}=\\var{nu_value2_mantissa}\\times 10^{\\var{nu_value2_log}}~Hz\\]
\n\nCalculate $\\mu$
\n\\[\\mu=\\frac{m_1\\times m_2}{m_1+m_2}=\\var{reduced_mass}~\\rm g~mol^{-1}\\]
\nConvert from g mol-1 to kg mol-1....
\n\\[\\frac{\\var{reduced_mass}{~\\rm g~mol^{-1}}}{1000}=\\var{reduced_mass}\\times 10^{-3}~\\rm kg~mol^{-1}\\]
\n...then divide by Avogadro's number to obtain a value in kg molecule-1;
\n\\[\\frac{\\var{reduced_mass}\\times 10^{-3}~\\rm kg~mol^{-1}}{6.022\\times 10^{23}~{\\rm molecules~mol^{-1}}}=\\var{reduced_mass_mantissa}\\times10^{\\var{reduced_mass_log}}~\\rm kg~molecule^{-1}\\]
\nRearrange for $k$
\n\\[\\nu=\\frac{1}{2\\pi}\\sqrt{\\frac{k}{\\mu}}\\]
\n\\[(2\\pi v)^2\\mu=k\\]
\nthen substitute values in for each of the variables;
\n\\[k =(2\\pi \\times \\var{nu_value2_mantissa}\\times 10^{\\var{nu_value2_log}})^2\\times\\var{reduced_mass_mantissa}\\times10^{\\var{reduced_mass_log}}{\\rm ~kg~molecule^{-1}} =\\var{force_constant}~\\rm N~m^{-1}\\]
\n\n(b) Note that;
\n\\[{\\rm Zero~point~energy}=\\frac{1}{2}h\\nu\\]
\nso;
\n\\[{\\rm Zero~point~energy}=\\frac{1}{2}\\times 6.626 \\times 10^{-34}~{\\rm J~s} \\times \\var{nu_value2_mantissa}\\times 10^{\\var{nu_value2_log}}~{\\rm Hz}\\]
\n\\[=\\var{ZPE_mantissa}\\times 10^{\\var{ZPE_log}}~{\\rm J}\\]
\n", "rulesets": {"": []}, "extensions": [], "variables": {"reduced_mass_kg": {"name": "reduced_mass_kg", "group": "Ungrouped variables", "definition": "reduced_mass/(6.022)", "description": "", "templateType": "anything"}, "html_definer": {"name": "html_definer", "group": "Ungrouped variables", "definition": "[(\"wavenumber\"),\"frequency\"][randomiser_units]", "description": "", "templateType": "anything"}, "randomiser_units": {"name": "randomiser_units", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "randomi
", "templateType": "anything"}, "reduced_mass_log": {"name": "reduced_mass_log", "group": "Ungrouped variables", "definition": "floor(log(reduced_mass_kg))-26\n", "description": "", "templateType": "anything"}, "force_constant": {"name": "force_constant", "group": "Ungrouped variables", "definition": "siground(((2*3.14*nu_list[0]*3*10^(10))^(2))*(get((rot_constants[randomiser]),\"reduced mass\",0)/(6.022*10^(23)*1000)),4)", "description": "", "templateType": "anything"}, "randomiser": {"name": "randomiser", "group": "Ungrouped variables", "definition": "8", "description": "random(0..9)
", "templateType": "anything"}, "ZPE": {"name": "ZPE", "group": "Ungrouped variables", "definition": "0.5*nu_list[1]*6.626*10^(-34)", "description": "", "templateType": "anything"}, "nu_list": {"name": "nu_list", "group": "Ungrouped variables", "definition": "[\n ( \n (siground((get((rot_constants[randomiser]),\"nu\",0)),4)),\n (siground((get((rot_constants[randomiser]),\"nu\",0)*3*10^(10)),4))\n )\n]", "description": "", "templateType": "anything"}, "nu_value2_log": {"name": "nu_value2_log", "group": "Ungrouped variables", "definition": "floor(log(nu_list[1]))", "description": "", "templateType": "anything"}, "reduced_mass": {"name": "reduced_mass", "group": "Ungrouped variables", "definition": "siground(get((rot_constants[randomiser]),\"reduced mass\",0),4)", "description": "", "templateType": "anything"}, "HTML": {"name": "HTML", "group": "Ungrouped variables", "definition": "Molecule_identifiers[randomiser]", "description": "0 = CO
1 = H35Cl
2= H81Br
3= HI
4= Na35Cl
5= Na79Br
6= LiF
7= Li35Cl
8= 63CuF
9= 63Cu35Cl
Rotational constants data from;
\n<https://catalog.data.gov/dataset/nist-computational-chemistry-comparison-and-benchmark-database-srd-101>
\naugmented with reduced masses calculated from IUPAC Green Book.
", "templateType": "json"}, "reduced_mass_mantissa": {"name": "reduced_mass_mantissa", "group": "Ungrouped variables", "definition": "siground((reduced_mass_kg)/(10^(reduced_mass_log+26)),3)", "description": "", "templateType": "anything"}, "Molecule_identifiers": {"name": "Molecule_identifiers", "group": "Ungrouped variables", "definition": "[\n (\n html(\"\"+\"C\"+\"O\"+\"\"),\n html(\"\"+\"H\"+\"\"+\"35\"+\"\"+\"Cl\"+\"\"),\n html(\"\"+\"H\"+\"\"+\"81\"+\"\"+\"Br\"+\"\"),\n html(\"\"+\"H\"+\"I\"+\"\"),\n html(\"\"+\"Na\"+\"\"+\"35\"+\"\"+\"Cl\"+\"\"),\n html(\"\"+\"Na\"+\"\"+\"79\"+\"\"+\"Br\"+\"\"),\n html(\"\"+\"Li\"+\"F\"+\"\"),\n html(\"\"+\"Li\"+\"\"+\"35\"+\"\"+\"Cl\"+\"\"),\n html(\"\"+\"\"+\"63\"+\"\"+\"Cu\"+\"F\"+\"\"),\n html(\"\"+\"\"+\"63\"+\"\"+\"Cu\"+\"\"+\"35\"+\"\"+\"Cl\"+\"\")\n )\n]", "description": "", "templateType": "anything"}, "nu_html": {"name": "nu_html", "group": "Ungrouped variables", "definition": "[\n (\n html(\"\"+{nu_value}+\" \"+\"cm\"+\"\"+\"-1\"+\"\"+\"\"),\n html(\"\"+{nuval2man}+\" \"+\"× \"+\"10\"+\"\"+{nu_value2_log}+\"\"+\" \"+\"Hz\"+\"\")\n )\n]", "description": "", "templateType": "anything"}, "nu_value": {"name": "nu_value", "group": "Ungrouped variables", "definition": "nu_list[randomiser_units]", "description": "", "templateType": "anything"}, "ZPE_log": {"name": "ZPE_log", "group": "Ungrouped variables", "definition": "floor(log(ZPE))", "description": "", "templateType": "anything"}, "nuval2man": {"name": "nuval2man", "group": "Ungrouped variables", "definition": "nu_list[1]/(10^(nu_value2_log))", "description": "", "templateType": "anything"}, "ZPE_mantissa": {"name": "ZPE_mantissa", "group": "Ungrouped variables", "definition": "siground(ZPE/(10^(ZPE_log)),3)", "description": "", "templateType": "anything"}, "wavenumber": {"name": "wavenumber", "group": "Ungrouped variables", "definition": "nu_list[0]", "description": "", "templateType": "anything"}, "nu_html_outpu": {"name": "nu_html_outpu", "group": "Ungrouped variables", "definition": "nu_html[randomiser_units]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "200"}, "ungrouped_variables": ["randomiser", "Molecule_identifiers", "HTML", "randomiser_units", "nu_list", "nu_value", "nuval2man", "nu_value2_log", "nu_html", "nu_html_outpu", "html_definer", "force_constant", "wavenumber", "reduced_mass", "reduced_mass_log", "reduced_mass_mantissa", "reduced_mass_kg", "ZPE", "ZPE_log", "ZPE_mantissa"], "variable_groups": [{"name": "Chemical element masses", "variables": ["rot_constants"]}], "functions": {"": {"parameters": [], "type": "number", "language": "jme", "definition": ""}, "molecule_name": {"parameters": [["atoms", "list"]], "type": "number", "language": "jme", "definition": "html(join(map(isotope_name(atom),atom,atoms),\"\"))"}, "isotope_name": {"parameters": [["atom", "dict"]], "type": "string", "language": "jme", "definition": "(\"\"+(if(atom[\"isotope\"]<>\"\",\"\"+string(atom[\"isotope\"])+\"\",\"\")+atom[\"symbol\"])+\"\")"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{force_constant}-{force_constant}/50", "maxValue": "{force_constant}+{force_constant}/50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "student_significand (The significand as the student entered it):\n parsenumber(studentanswer[0],\"en\")\n\nsignificand_size (If student's significand is written a*10^n, 1<=a<10, this is n):\n floor(log(abs(student_significand)))\n\nstudent_exponent (The exponent as the student wrote it):\n parsenumber(studentanswer[1],\"en\")\n\nadjusted_exponent (The exponent of the student's number, taking into account the size of their significand): \n student_exponent + significand_size\n\nadjusted_significand (The student's significand, scaled into the range 1..10):\n student_significand/(10^significand_size)\n\nsignificand_feedback (Feedback on the adjusted significand: mark gap 0):\n feedback(\"Significand:\");\n let(result,apply_marking_script(\"numberentry\",string(adjusted_significand), gaps[0][\"settings\"],gaps[0][\"marks\"]),\n concat_feedback(result[\"mark\"][\"feedback\"],0.5)\n )\n\nexponent_feedback (Feedback on the adjusted exponent: mark gap 1):\n feedback(\"Exponent:\");\n let(result,apply_marking_script(\"numberentry\",string(adjusted_exponent), gaps[1][\"settings\"],gaps[1][\"marks\"]),\n concat_feedback(result[\"mark\"][\"feedback\"],0.5)\n )\n\nmark:\n apply(significand_feedback);\n apply(exponent_feedback)\n\ninterpreted_answer: [adjusted_significand, adjusted_exponent]", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate the zero-point vibrational energy in units of Joules;
\n[[0]] $\\times$ 10[[1]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ZPE_mantissa}-{ZPE_mantissa}/50", "maxValue": "{ZPE_mantissa}+{ZPE_mantissa}/50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ZPE_log}+{ZPE_log}/50", "maxValue": "{ZPE_log}-{ZPE_log}/50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}]}]}], "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}]}