Rotational Partition Function, Equipartition

5.62 Lecture #12: Rotational Partition Function. Equipartition Readings: Hill, pp. 153-159; Maczek, pp. 47-53 Metiu, pp. 131-142 DEGREES OF FREEDOM A molecule with n atoms has 3n "degrees of freedom" or 3n coordinates to describe its position and therefore has 3n ways of incorporating energy due to nuclear motion where n is the number of atoms in the molecule. For a diatomic or a linear polyatomic molecule: 3 TRANSLATIONAL degrees of freedom 2 ROTATIONAL degrees of freedom 3n–5 VIBRATIONAL degrees of freedom 3n TOTAL degrees of freedom For a diatomic molecule 3n – 5 = 1 vibrational degree of freedom MOLECULAR ROTATIONAL PARTITION FUNCTION — qrot — DIATOMIC εrot(J) = J(J + 1) hcBe for J = 0,1,2, … gJ = 2J + 1 ∞ qrot = ∑ g(ε) e–ε/kT = ∑ (2J +1) exp[–hcBe J(J+1)/kT] J=0 allowed rotational energies Question: How do you do the summation? Two cases … (Low-T limit case [next Lecture]) Case 1:εrot/kT  1 or hcBeJ(J + 1)/kT  1 [More precisely, we want (E(J + 1) – E(J))  kT at E(J) ≈ kT.) rotational states are closely spaced in energy compared to kT — since energy spacings are so close together, can consider 5.62 Spring 2008 Lecture 12, Page 2 εrot as continuous and use Euler-MacLaurin Summation Formula (draw a picture!) this case is the classical or high-temperature limit. n 1 ∑ f ( J ) = ∫ mn f(J)dJ + [ f(m) + f(n)] + residue… 2 J =m so: ∞ qrot = ∫ 0 (2J + 1) exp[–hcBe J(J+1)/kT] dJ + 1 [1 + 0] + … 2 J=0 J=∞ substitute ω = J(J+1) thus qrot = = dω = (2J+1)dJ ∫ ∞ 0 exp [ −hcBeω / kT ] dω + −kT − hcBe ω kT e hcBe ∞ 0 + 1 +… 2 1 +… 2 ⎛ −kT ⎞ 1 = 0−⎜ + +… ⎝ hcBe ⎟⎠ 2 kT 1 kT 1 + ≈ usually can ignore the hcBe 2 hcBe 2 1 What happens for a ∆ state where Jmin = 2 rather than 0? qrot = Hold on — One correction needed to qrot … SYMMETRY NUMBER ≡ σ ≡ # of equivalent orientations in space which leave appearance of molecule unchanged — # of indistinguishable orientations in which molecules can be found as a result of rotation. We divide by σ because otherwise we would be overcounting by counting indistinguishable orientations. A homonuclear molecule, O2, has σ = 2 because an end over end (half) rotation by π does not alter the appearance of O2. The symmetry number is rigorously based on the nuclear spins. We’ll see the details later. So q rot = kT σhcBe for hcBe  kT or εrot  kT revised 3/3/08 9:19 AM 5.62 Spring 2008 Lecture 12, Page 3 where σ ≡ symmetry # = 1 for heteronuclear diatomics = 2 for homonuclear diatomics Really this is qrot-nuc, but we’ll refer to it as qrot. hcBe k Define θrot = “rotational temperature”, θrot (has units of K) q rot = So kT T = σhcBe σθrot for θrot  T Molecular Rotational Partition Function for Diatomics also written as … 8π 2 IkcT qrot = σh 2 because Be = h 8π 2 Ic I = µRe2 Let's go back and check whether dropping extra terms in Euler-MacLaurin series was a good approximation … q rot = kT 1 + +… σhcBe 2 MOLECULE Be (cm ) σ θrot (K) at T = 300K qrot = T/σθrot + 1/2 HCl 10.59 1 15.24 19.688 + 0.5 2.5 CO 1.93 1 2.77 108.30 + 0.5 0.4 I2 0.037 2 0.1065 1408.5 + 0.5 0.04 –1 % error (neglect of 1/2) As Be becomes smaller or equivalently as θrot becomes smaller compared to T, dropping extra terms becomes better approximation; also, discrete to continuous approximation becomes better. revised 3/3/08 9:19 AM 5.62 Spring 2008 Lecture 12, Page 4 Contributions of Rotation to Thermodynamic Functions for εrot  kT q rot ⎛ kT ⎞ kT N = → Q rot = ( q rot ) = ⎜ σhcBe ⎝ σhcBe ⎟⎠ N ⎛ kT ⎞ A rot = −kTln Q rot = −NkTln q rot = −NkTln ⎜ ⎝ σhcBe ⎟⎠ ⎛ ∂A ⎞ ⎛ kT ⎞⎞ ∂ ⎛ p rot = − ⎜ ⎟ = ⎜ NkTln ⎜ ⎟⎟ = 0 ⎝ ∂V ⎠N,T ∂V ⎝ ⎝ σhcBe ⎠⎠N,T because εrot does not depend on V ⎛ ∂ ln Q rot ⎞ ∂ ln q rot E rot = kT2 ⎜ ⎟ = NkT2 ⎝ ∂T ⎠N,V ∂T = NkT2 ⎛1⎞ ∂ ln(k / σhcBe ) ∂ ln T + nkT2 = NkT2 ⎜ ⎟ + 0 ⎝ T⎠ ∂T ∂T average rotational energy of a diatomic molecule Erot = NkT for θrot  T or εrot  kT 1 (not 2 NkT; why?) a "quantum" result (but based on the approximation of replacing a sum by an integral) CLASSICAL EQUIPARTITION RESULT FOR ROTATIONAL ENERGY ⎛1 ⎞ 2 ⎜ NKT⎟ = NkT ⎝2 ⎠ Each degree of translational and rotational energy contributes (1/2)kT to total energy. For a diatomic molecule, there are 2 rotational degrees of freedom [Why 2?]. – Therefore, 2(1/2 NkT) = NkT. This is why CV for monatomic gases is ~(3/2)R and for most diatomic gases at moderate T is ~(5/2)R! Quantum and classical approach lead to same result for rotation at 300 K. Why? Because rotational energy levels are very closely spaced compared to kT. We calculated revised 3/3/08 9:19 AM 5.62 Spring 2008 Lecture 12, Page 5 qrot by approximating a sum over energy levels as an integral over energy levels. Rotational energy levels are so closely spaced that they "look" continuous compared to kT at room temperature for most molecules. Erot does not depend on the properties of the molecule in the classical limit! “Quantum” result for Cv ⎛ ∂E ⎞ ⎛ ∂T ⎞ rot CVrot = ⎜ ⎟ = Nk ⎜ ⎟ = Nk = R if N = Na (or CV = nR) N,V ⎝ ∂T ⎠ ⎝ ∂T ⎠ for εrot  kT or θrot  T (εrot needs to be better defined, see below) 1. More about high temperature limit, which is the requirement that permits the sum, ∞ qrot = ∑ g (ε ( J )) e−hcBJ ( J +1) kT , J min to be replaced by an integral, qrot = ∫ ∞ J min dJ ( 2J + 1) e− hcBJ ( J +1) kT + 1 ⎡( 2J + 1) + 0⎤⎦ . 2 ⎣ min It is necessary that ∆εrot  kT at εrot ≈ kT. The rotational energy level spacing must be small relative to kT. ∆εrot (J ) = hcB [( J +1)( J + 2 ) − J ( J +1)] = hcB2 ( J +1) . This spacing must be small relative to kT when εrot = kT εrot = hcBJ ( J +1) = kT Thus hcBJ(J+1)  hcB2(J + 1), which requires that J  2. This means that we want kT  εrot (J = 2) = 6hcB. kT  6hcB is the requirement that specifies when it is OK to replace sum by integral. revised 3/3/08 9:19 AM 5.62 Spring 2008 2. Lecture 12, Page 6 Some useful stuff concerning fractional populations in rotational levels. A. fraction of population in J-th level fJ = θrot = B. ( 2J + 1) e−θ rot J ( J +1) T T σθrot hcB k Most populated J 2 2e−θrot J ( J +1) T − (2J +1) (θrot T ) e−θrot J ( J +1) T dfJ =0= dJ qrot Thus 2 θ 2 = (2J +1) rot T 1/2 ⎛ 2T ⎞ ⎜ ⎟ −1 ⎝ θrot ⎠ J max = . 2 For T / θrot = 100 Jmax = 6.5 C. Fractional population in most populated J level ⎡ 2T ⎤ ⎢θ ⎥ e ⎣ rot ⎦ 1/2 fJmax = For T θrot = 100 D. −θrot 4T ⎛⎡ ⎤1/2 ⎞⎛⎡ ⎤1/2 ⎞ ⎜⎢ 2T ⎥ −1⎟⎜⎢ 2T ⎥ ⎟ ⎜ ⎟⎜ θ ⎟ ⎝⎣ θrot ⎦ ⎠⎝⎣ rot ⎦ ⎠ ⎛ 2θ ⎞ ⎛θ⎞ ≈ ⎜ rot ⎟ e−1/2 = 0.85 ⎜ ⎟ . ⎝ T ⎠ ⎝T ⎠ 1/2 T θrot 1/2 fJmax = 0.085. T . This is a simple-minded way of asking θrot about the population of the “last” thermally accessible level. Fractional population in J = revised 3/3/08 9:19 AM 5.62 Spring 2008 Lecture 12, Page 7 ⎛ T ⎞⎛ T fJ =T θrot ⎞ θrot T ⎛ T ⎞ − ⎜⎝ θrot ⎟⎠ ⎜⎝ θrot +1⎟⎠ 2 + 1 ⎜⎝ θ ⎟⎠ e rot = T θrot = 2e ⎛ T ⎞ −⎜ +1 ⎝ θrot ⎠⎟ . For T / θrot ≈ 100 fJ =T θrot = 2e−101 = 3 × 10 −44. This is a very small fractional population. It would be more appropriate to ask for the fractional population of the J′ value for which J′ ∑ (2J +1) = T θrot , J min because this sets the total number of significantly populated J, MJ levels equal to qrot. Using J′ ∑ (2J +1) = ( J′ +1) 2 J =0 ( ) and T/θrot = 100, we get J′ = 9 and fJ ′=9 19e−9 10 /100 = = 0.077 . 100 – Low-T limit results for E and CV next time. But what do you know without any – equations about E and CV in the limit T → 0K? revised 3/3/08 9:19 AM