Partition functions of ideal gases

We showed that if the # of available quantum states is >> N The condition is valid when Examples gases at low densities (independent molecules) 2

Monoatomic ideal gas general MIG 3

Translational partition function MIG If cubic (a=b=c) Particle in a parallelepiped a,b,c nx, ny, nz =1,2,3… 4

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this sum cannot be expressed as a sum of a series but… 6

example: average translational energy MIG 7

Electronic partition function MIG 8 the ground state energy is taken as the zero of energy the other terms are negligible since they are typically in the order of and smaller exceptions: some of the halogens may have contributions from the first terms

Summary MIG 9

Diatomic ideal gas 10

Zeros of energy rotational: J=0 (rotational energy =0) vibrational: a) the bottom of the well or b) the ground vibrational state; in (a) the ground vibrational state is h /2 electronic: the energy is zero when the two atoms are completely separated 11

Dissociation energy (D o )and ground state electronic energy (D e ) 12 Anharmonic oscillator: HCl

vibrational partition function DIG 13 harmonic oscillator approximation vibrational temperature:

average vibrational energy 14 vibrational contribution to C v

fraction of molecules in the jth vibrational state 15 see problems 3.35; 336 (we solved them last class) most molecules are in the ground vibrational state at room T

rotational partition function DIG 16 (sum is over levels) rotational temperature

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rotational partition function DIG 18 because the ratio rotational temperature/T is small for most molecules at ordinary Ts integrating much better at high T; is the high T limit

average rotational energy 19 total rotational contribution to Cv is R; R/2 per rotational degree of freedom for a diatomic fraction of molecules in the Jth rotational state see problem 3.37 that we solved last class

physical meaning of the rotational temperature 20 It gives us an estimate of the temperature at which the thermal energy (kT) equals the separation between rotational levels. At this T, the population of excited rotational states is significant. 88 K for H 2, 15.2 K for HCl and K for CO 2

21 most molecules are in the excited rotational levels at ordinary Ts

Symmetry effects 22 is for heteronuclear DIG For homonuclear DIG the factor of 2 comes from the symmetry of the homonuclear molecule; 2 indistinguishable orientations

23 molecular partition function DIG restrictions: only the ground state electronic state is populated zero (electronic) taken at the separated atoms zero (vibrational) taken at the bottom of the potential well

average energy DIG 24

average Cv DIG 25

Vibrational partition function of a polyatomic molecule 26 n vib is the number of vibrational degrees of freedom 3n-5 for a linear molecule 3n-6 for a nonlinear molecule normal modes are independent

since the normal modes of a polyatomic molecule are independent 27

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Rotational partition function of a linear polyatomic molecule 30 linear J = 0, 1, 2, … degeneracy m j is the distance from nucleus j to the center of mass of the molecule  is 1 for nonsymmetrical molecules (N 2 O, COS) and 2 for symmetrical such as CO 2

Importance of rotational motion 31

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symmetry number 33  is 1 for nonsymmetrical molecules (N 2 O, COS)  is 2 for symmetrical such as CO 2 how about NH 3 ? symmetry number is the number of different ways in which a molecule can be rotated into a configuration indistinguishable from the original For water,  =2, successive 180 o rotations about an axis through the O atom bisecting the two H atoms result in two identical configurations for CH 4, for any axis through one of the four CH bonds there are 3 successive 120 o rotations that result in identical configurations, therefore  = 4x3 =12

Linear polyatomic moment of inertia 34 Example HCN

Non linear rigid polyatomic 35

rigid non-linear polyatomic 36 3 moments of inertia if the three are equal, spherical top only two equal, symmetric top three different, asymmetric top

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38 examples of rotational symmetry

examples 39 spherical top symmetrical top J=A, B, C 3 rotational temperatures

Spherical top molecules 40 Allowed energies: J = 0, 1, 2, … Degeneracy:

Spherical top molecules 41 Can be solved analytically

Rotational partition functions 42 Spherical top Asymmetric top Symmetric top

Average rotational energy (nonlinear polyatomic) 43

Partition function ideal gas of linear polyatomic molecule 44

45 Partition function ideal gas of nonlinear polyatomic molecule Obtain U and Cv

Comparison to experiments 46

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Summary Considering the molecules that constitute a macroscopic material, we construct q, and from q we construct Q, and from Q any thermodynamic property. For example, U and C v are not just numbers in tables. We have some new insights about why different materials have different thermodynamic properties Next, we will discuss the laws that govern the macroscopic thermodynamic properties. 48