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Given that the rotational constant, B, of {HTML} is {B_html_rotconst_output};  

", "advice": "

a) If presented with a wavenumber, you will need to start by converting this into a frequency while noting that; 

\n

\\[c=\\lambda\\nu\\]

\n

and;

\n

\\[{\\rm wavenumber}=\\frac{1}{\\lambda}\\]

\n

therefore

\n

\\[\\frac{\\nu}{c}=\\frac{1}{\\lambda}={\\rm wavenumber}\\] and \\[{\\rm wavenumber}~\\times~c~=\\nu\\] 

\n

so

\n

\\[{\\var{wavenumber}~{\\rm cm^{-1}}}~\\times~3~\\times~10^{10}{~\\rm cm~s^{-1}}={\\var{B_value2_mantissa}~\\times~10^{\\var{B_value2_log}}~{\\rm Hz}}=B\\]

\n

Revise how the interval between transitions in an observed spectrum (see figure), which is sometimes referred as the \"line spacing\", connects with the transitions between energy levels in a molecule. Examine how the photon energy (the value on the x axis) associates with transitions between rotational levels.  

\n

\n

\n

There is a simple relationship between photon energy and the J involved with transitions (where the J used in the equation is that of the lower state involved in the transition); 

\n

\\[\\nu=2B(J+1)\\]

\n

The energy of a photon is proportional to the frequency of light (remember, E=h$\\nu$). The above relationship therefore holds whether you know B in units of Hz (frequency), or units of J (energy). However, you must be consistent throughout the calculation. For example, if you specify B in units of Hz, you will get a result for $\\nu$ in Hz. If you specify B in Joules, the result of your calculation will be an energy in Joules instead. The question specificies B in units of Hz so we will proceed as follows; 

\n

\\[\\nu=2B(J+1)\\] so

\n

\\[\\nu=2B(J+1)=2~\\times~\\var{B_value2_mantissa}\\times 10^{\\var{B_value2_log}}\\times(\\var{random_trans}+1)~{\\rm Hz}={\\var{freq_trans_mantissa}}\\times10^{{\\var{freq_trans_log}}}~{\\rm Hz}\\]

\n

\n

(b) To calculate the moment of inertia, I; 

\n

\\[I=\\frac{h}{8\\pi^{2}B}\\]

\n

\\[I=\\frac{h}{8\\pi^{2}B}=\\frac{6.626~\\times~10^{-34}{\\rm~J~s}}{8\\pi^{2}~\\times\\var{B_value2_mantissa}\\times 10^{\\var{B_value2_log}}~{\\rm Hz}}=\\var{mantissa_inertia_x}\\times ~10^{\\var {log_inertia_x}}~{\\rm kg~m^{2}}\\]

\n

(c) To calculate the reduced mass in g mol-1, where m1 is the mass of the first atom and m2 is the mass of the second atom, each expressed in units of g mol-1

\n

\\[\\frac{m_{1}~\\times~m_{2}}{m{_1}+m_{2}}=\\var{reduced_mass}~\\rm g~mol^{-1}\\]

\n

(d) To calculate the reduced mass in kg molecule-1, divide the reduced mass in g mol-1 by (the Avogadro number multiplied by 1000)

\n

\\[\\frac{\\var{reduced_mass}~\\rm {g~mol^{-1}}}{1000~{\\rm g~kg^{-1}}~\\times~6.022~\\times~10^{23}~{\\rm molecules~mol^{-1}}}=\\var{mantissa_reduced_mass_kg}\\times 10^{\\var{log_reduced_mass_kg}}\\rm ~kg~ molecule^{-1}\\]

\n

(e) To determine the bond length from the moment of inertia and reduced mass;

\n

\\[I=\\mu~r^{2}~~~~~~~~~~{\\rm so}~~~~~~~~~ r=\\sqrt{\\frac{I}{\\mu}}\\]  

\n

\\[r=\\sqrt{\\frac{I}{\\mu}}=\\sqrt{\\frac{\\var{mantissa_inertia_x}\\times ~10^{\\var {log_inertia_x}}}{\\var{mantissa_reduced_mass_kg}\\times 10^{\\var{log_reduced_mass_kg}}}}=\\var{bond_length_angstroms}\\times10^{-10}~{\\rm m}=\\var{bond_length_angstroms}~{\\unicode{x212B}}\\]  

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r

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fre1q

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randomi

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Rotational constants data from;

\n

<https://catalog.data.gov/dataset/nist-computational-chemistry-comparison-and-benchmark-database-srd-101>

\n

augmented with reduced masses calculated from IUPAC Green Book. 

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what is the frequency, $\\nu$, in units of Hz, of the {trans} transition in the microwave spectrum of this molecule?

\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": "{freq_trans_mantissa}-{freq_trans_mantissa}/50", "maxValue": "{freq_trans_mantissa}+{freq_trans_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": "{freq_trans_log}-{freq_trans_log}/50", "maxValue": "{freq_trans_log}+{freq_trans_log}/50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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": false, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The moment of inertia, I, is [[0]] $\\times$ 10[[1]] kg m2.

\n

", "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": "{mantissa_inertia_x}-{mantissa_inertia_x}/50", "maxValue": "{mantissa_inertia_x}+{mantissa_inertia_x}/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": "{log_inertia_x}+{log_inertia_x}/50", "maxValue": "{log_inertia_x}-{log_inertia_x}/50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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, "prompt": "

What is the reduced mass, $\\mu$, in units of g mol-1?

", "minValue": "{reduced_mass}-{reduced_mass}/50", "maxValue": "{reduced_mass}+{reduced_mass}/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": "

The reduced mass, $\\mu$, is [[0]] $\\times$ 10[[1]] kg molecule-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": "{mantissa_reduced_mass_kg}-{mantissa_reduced_mass_kg}/50", "maxValue": "{mantissa_reduced_mass_kg}+{mantissa_reduced_mass_kg}/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": "{log_reduced_mass_kg}+{log_reduced_mass_kg}/50", "maxValue": "{log_reduced_mass_kg}-{log_reduced_mass_kg}/50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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": false, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The bond length, r, is [[0]] $\\times$ 10[[1]] m.

", "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": "{bond_length_angstroms}-{bond_length_angstroms}/50", "maxValue": "{bond_length_angstroms}+{bond_length_angstroms}/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": "{log_angstroms}+{log_angstroms}/50", "maxValue": "{log_angstroms}-{log_angstroms}/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/"}]}