PPT Partition function.pptx

Partition function indicates the mode of distribution of particles in various energy states. It plays a role similar to the wave function of the quantum mechanics,which contains all the dynamical information about the system.

1. Presented by: Dr. Sharayu M. Thorat Associate Professor Shri Shivaji College of Arts, Commerce & Science, Akola PARTITION FUNCTIONS

2. According to Boltzmann distribution law, the fraction of molecules which is in the most probable state at temperature T, having energy ε i is given by Where n i is number of molecules having energy ε i at temperature T, N is total number of molecules g i is the degeneracy (or statistical weight) of the energy level ε i The denominator of above equation which gives the sum of the terms for all energy levels is called partition function, represented by q. Thus partition function q =

3. Molecular Partition Functions for an Ideal gas: Molecules are associated with energy of different types. All these forms of energy must be taken into account while mentioning partition function. The molecular energy levels needed for the evaluation of the molecular partition functions are obtained from the solution of the Schrodinger equation. According to Born-Oppenheimer approximation, the total energy of a molecule is composed of contributions from the translational, rotational, vibrational and electronic modes of motion. ε = ε tran + ε rot + ε vib + ε electr The equation holds good if there is no coupling between the different modes of motion. Electronic energy ε electr can be obtained for the simple atoms. q= The molecular partition function q is given by ………..(1) The total partition function of a molecule is the product of the translational, rotational, vibrational and electronic contribution .

4. i.e. q = q trans x q rot x q vib xq elec …………(2) The Translational Partition Function For a particle of mass m, moving in an infinite 3-dimensional box of sides a, b and c, assuming that the potential is zero within the box, the energy levels obtained by the solution of the Schrodinger equation are given by the expression ε nx ny nz = ε tran = ………….(3) where each of the quantum numbers n x , n y ,n z vary from one to infinity. Using equations (1) & (3) the translational partition function ,neglecting degeneracy is given by q tran = …………(4) Where the triple summation is taken over all integral values of n x ,n y & n z from one to infinity. The motion of the particles in the three x, y & z directions being independent, we can replace the triple summation as a product of three summations.

5. Thus, q tran = x x ………..(5) The spacing between the energy levels of a particle in a three dimensional box is very small compared with the thermal energy ,kt. Hence we can replace the summation by integration. Accordingly, q tran = dn x dn y x x dn z ………..(6) dx = From calculus, the standard integral ……….(7) Using this result, the three integrals in equation 6 which are identical except for the constants a,b and c can be calculated giving q tran = a/h(2 𝝿mkt) 1/2 x b/h(2𝝿mkt) 1/2 x c/h(2𝝿mkt) 1/2 = 3/2 x abc = x V ………..(8)

6. Equation 8 ⟹ Translational partition function depends upon volume & temperature V= volume of the box in which the molecule moves Equation 8 can be written as q trans = q trans x V ……….(9) q trans = partition function per unit volume If M is the mass per mole then m = M/N A , N A is Avogadro’s number, k = R/N A Hence x q trans = N A 3 ……….(10) In the case of a perfect monoatomic gas ,the molar partition function is given by Z = ……………(11)

7. Partition function for diatomic molecules: Rotational partition function The partition function for rotational energy of a diatomic molecule is given by ………..(1) From quantum mechanical principles, the rotational energy 𝟄 r for a diatomic molecule at the J th quantum level is given by ……….(2) Where I= moment of inertia = 𝞵r 2 As axis of rotation is defined by two co-ordinates ,which means that there are two rotational degrees of freedom. Each quantum level of rotation will bring in two possible modes of distribution of rotational energy. Thus statistical weight factor J is given by 2J+1 Hence equation 1 becomes, f r = Σ ( 2J+1) e – {J(J+1)h 2 }/ 8 𝝿 2 IkT ……….(3) Since the levels are closely spaced, the summation can be replaced by integration

8. f’ r = ∫ ( 2J+1)e – {J(J+1)h 2 }/ 8 𝝿 2 IkT dJ ∞ ∴ ∫ ( 2J+1)e – J(J+1) 𝛽 dJ = ∞ ……….( 4) where 𝛽= h 2 /8𝝿 2 IkT Suppose G= J(J+1), On differentiating it we get, dG =(2J+1) dJ Hence equation 2 becomes, fr = ∫e -G𝛽 dG = 1/𝛽 = 8𝝿 2 IkT/h 2 ∞ ..…….(5) The value of fr is valid for hetero- nuclear molecules like NO, HCl etc whereas in the case of homonuclear molecules like O 2 , N 2 etc where the molecule when reversed, becomes indistinguishable from initial state, the partition function is to be divided by the number of symmetry viz 2. ∴ fr = In general , when the symmetry number is δ ,partition function is fr = ………..(6)

9. Equation 6 holds good for diatomic molecules, other than hydrogen & deuterium. fr = For homonuclear diatomic molecules like H 2 , N 2 ,O 2 etc σ =2 For heteronuclear diatomic molecules like CO, NO ,HCl etc σ =2 fr = Where B is rotational constant =

10. Vibrational Partition function The partition function for vibrational energy of a diatomic molecule is given by fv = Σ g v e - 𝜀v/ kT As the statistical weight of each vibrational level is unity, we have, Σ e - 𝜀v/ kT fv = At the nth quantum level, the vibrational energy of a diatomic molecule is given by, 𝜀 v = 〔n+1/2〕 h ν Where ν = fundamental frequency of vibration n = an integer 0,1,2,3……etc ………(1) ……..(2) ∴ from equation 2 we have, fv = = = ……..(3) For a diatomic molecule vibrating as a simple harmonic oscillator, the vibrational energy levels are obtained by the solution of the Schrödinger's wave equation .

11. = ∵ The quantity h ν / kT is very small & as a first approximation, f v = ………(4) The value of ν is equal to cw, where c is velocity of light & wcm -1 is the vibration frequency in wave numbers of the given oscillator. Hence f v = ………(5) Equation 5 may be used for the vibrational partition function of a diatomic molecule at all temperatures ; the only approximation involved is that the oscillations are supposed to be harmonic in nature.