statmech1 - Chem 444 Instructor Nancy Makri PROBLEM...

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Chem. 444 Instructor: Nancy Makri PROBLEM STATMECH2 A low-frequency “breathing” mode of a protein has frequency 1 69.5 cm - . What is the vibra- tional temperature Θ of this mode? Round off your number to an integer. Assume the spectrum corresponding to this mode is harmonic and choose the zero of energy at the ground vibrational level, such that the energy eigenvalues are given by the expression n E n ϖ = . Express the Boltzmann factor / n B E k T e - in terms of the vibrational temperature. Evaluate the Boltz- mann factor at room temperature for several energy levels (up to quantum numbers for which the Boltzmann factor drops below 0.01). Notice that the Bolzmann factor decreases by / T e each time the quantum number increments by 1. Sum your results to obtain an approximation to the partition function. Then normalize all the Boltzmann factors by the partition function to obtain occupation probabilities. Does the value of the partition function give you a reasonable estimate of the number of energy levels that are populated at this temperature?
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