Physical Chemistry I
Physical Chemistry I CH 431
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This 3 page Class Notes was uploaded by Sienna Shields on Thursday October 15, 2015. The Class Notes belongs to CH 431 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/224004/ch-431-north-carolina-state-university in Chemistry at North Carolina State University.
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Date Created: 10/15/15
Chemistry 431 Lecture 4 The Partition Function Statistical Thermodynamics NC State University Molecular Partition Functions In general gJ is the degeneracy cl is the energy 9312 We assume that the energy ofthe lowest energy level the ground state is an Recall that 3 1kT Examples ATwo level system Bln nite energy ladder Two Level System Assume that degenemcy O 1lie single state is found at each levell q 1e BE Note that as T 0 q a 1 andasTgtooqgt2 The ratio ofthe population in the two is states e BEwhere c is the energy difference between the two states i l Ensemble Partition Function We distinguish here between the partition function of the ensemble Q and that of an individual molecule q Since Q represents a sum over all states accessible to the system it can written as QW VJ 2 ewww where the indices ijk represent energy levels of different particles The molecular partition function q represents the energy levels of one individual molecule We can rewrite the above sum as Q qqJqk or Q q for N particles Note that q means a sum over states or energy levels accessible to molecule i and q means the same for molecule j 7017 Z 67 The molecular partition function counts the energy levels accessible to molecule ionly 39 Qcounts not only the states of all ofthe molecules but all of the possible combinations of occupations of those states However ifthe particles are not distinguishable then we will have counted N states too many N NN1N2 This factor is exactly how many times we can swap the indices in QNVT and get the same value again provided that the particles are not distinguishable If we consider 3 particles we have ijkjik kij kjijki ikj or 6 3 Thus we write the partition function as Q q t mng xhable pamclex jvi rmt mngnxhable pamclex Translational Partition Function The translational partition function isthe most important one for statistical thermodynamics Pressure is caused bytranslational motion ie momentum exchange with the walls of a container Forthis reason it is important to understand the origin ofthe translational partition function Translational energy levels are so closely spaced asthat they are essentially a continuous distribution The quantum mechanical description ofthe energy levels is obtained fromthe quantum mechanical particle in a box 922008 Particleinabox energy levels I Theenergylevelsare 2 7 rifn2n2 nnwn12m WW BUB y z xv z x I The box is a cube of length a The average quantum numbers will be very large for a typical molecule This is very different than what we find forvibration and electronic levels where the quantum numbers are small ie only one or a few levels are populated Many translational levels are populated thermally The translational partition function is q 5 2 frank quotm l m m m 7 h 2 2 2 m m The three summations are identical and so they can be written as the cube of one summation 3 m 2 2 trans ex I77 7 quot21 p Smasz The fact that the energy levels are essentially continuous and that the average quantum number is very large allows us to rewrite the sum as an integral 3 N 112712 mmv 7 q eXP Smaszdn The translational partition function is proportionalto volume The sum started at 1 and the integral at 0 This difference is not important ifthe average value ofn is ca 109 lfwe have the substitution a h28ma2kTwe can rewrite the integral as 3 3 frans m 7 quot 7 W q e dn E This is a Gaussian integral The solution of Gaussian integrals is discussed the math section of the Website lfwe now plug in fora and recognize that the volume of the box is V a3 we have 32 qm 21 V Probability in the ensemble The ensemble partition function is Q in ergk1 here the ensemble energy is E The population ofa particular state J with energy E is given g erEJkT erEJkT p 7 J E e EJkT Q 10 This knovm as the Boltzmann distribution The normalization constant ofthe above probability is 1Q The sum of all of the probabilities must equal 1 Calculation of average properties The importance ofthe canonical ensemble is evident once we begin to calculate average thermodynamic quantities The basic approach is to sum over the probability ofa state being occupied times the value of the property in a given state In general for an average I re ertg M we can write ltMgt Pij l M could be energy or pressure etc PJ is the Boltzmann probability given by PJ sagal 922008 Ave rage energy lfwe denote the average energy ltEgt then on EJE MJ as 2 E 1210 PJE Jig 843539 Me n e i This can be written compactly as 3an we Consistency check with kinetic theory of gases The average energy per molecule is given by 77 mm 7 2 mm 3B Vw BT V W gm lermsindependenlnfT V which agrees with the kinetic theory of gases The second step follows from the fact that lnabc lna lnb lnc We can rewrite the logarithm as a sum The terms that do not depend on temperature will vanish
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