solns21b-ps3-Jan11 - Chemistry 21b Problem set # 3 Out: 19...

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Chemistry 21b Problem set # 3 Out: 19 January 2011 Due: 26 January 2011 Problems are worth: 1a=10, 1b=10, 1c=10, 1d=10; 2a=15, 2b=15; 3a=10, 3b=10, 3c=10. 1. At low temperatures, the rotational part of the partition function Q ( T ) = s J =0 (2 J + 1)exp( - E J /kT ) dominates the non-translational degrees of freedom, where E J = BJ ( J + 1) cm 1 for a simple diatomic or linear molecule in a 1 Σ (that is, closed shell) electronic state. Roughly, this function tells you over how many states the population is distributed, and is central to the Boltzmann distribution ( N J /g J ) = ( N/Q ) e E J /kT , where N J =population in state J , g J =degeneracy of state J , E J =energy of state J , N =total population, and Q =the partition function) and statistical thermodynamics, as you’ll see later in Ch 21c (or may already know). (a.) Derive an alternative, approximate form of this partition function by replacing the summation with an integral and by taking J to be a continuous variable. This formula should depend only on the temperature T and the rotation constant B . (b.) Evaluate the values of Q ( T ) at T =20 and T =200 K, both by the direct summation (truncated at some appropriate values of J , use a mathematical package for convenience!) and through use of your approximate formula for the HCCCN molecule. The rotational constant for this molecule is B =4549.06 MHz. (c.) Recall that the fractional population in level J is given by p ( J ) = (2 J + 1)exp( - E J /kT ) /Q ( T ) . Derive a formula for the value of J = J max at which the maximum population occurs for any speciFed temperature, based upon your integral form of the partition function. (d.) ±ind J max for T =20 and T =200 K for HCCCN. 2. Prove that the following relationships, asserted in class, are in fact true for a complete, orthonormal basis set: (a) s f | f >< f | = 1 (the closure relation) (b) < m | ∂x | k > = - m ¯ h 2 ( E m - E k ) < m | x | k > .
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This note was uploaded on 01/03/2012 for the course CH 21b taught by Professor List during the Fall '10 term at Caltech.

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solns21b-ps3-Jan11 - Chemistry 21b Problem set # 3 Out: 19...

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