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HW3 - By direct summation calculate the partition function...

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Chemistry 252 Homework 3. Due Wed. February 9 1. By using differentiation product rule prove that if the partition function is given by Q = Q 1 Q 2 the energy of a system is given by E = E 1 + E 2 2. Consider a system that can have only two quantum states with energies E 1 = 0 and E 2 = ε . A) Write the partition function for this system. B) Calculate the average energy of this system as a function of β . C) Calculate the heat capacity of this system as a function of temperature T . 3. The rotational energy levels of a diatomic molecule are given by E J = B J (J+1) . The degeneracy of each level is 2J+1 (i.e. there are 2J+1 states per level). Consider the rotations of the HF molecule B = 20.93 cm -1 . Make a table of Boltzmann factors e β E J for the 4 values of temperature T = 10 K, 30 K, 100 K, 300 K; and J = 1, 2, … 8 (a total of 32 entries).
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Unformatted text preview: By direct summation calculate the partition function and the average rotational energy of the HF molecule at these temperatures. Use your own discretion to which of the terms retain in the calculation in order to be accurate within about 1%. Compare your results to the approximate value given by “classical mechanics” E = k B T . 4. Plot the probabilities of an HF molecule to be in a rotational level J at T = 100 K and T = 300 K. 5. In class we derived the partition function of the harmonic oscillator: Q vib ( T ) = e − h ω /2 k B T 1 − e − h / k B T . Starting from this equation calculate the average vibrational energy of I 2 gas at 300 K. ( h = 214.2 cm-1 )....
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