Ro-vibrational coupling in high temperature thermochemistry of the BBr molecule.

Transcription

1 Ro-vibrational coupling in high temperature thermochemistry of the BBr molecule. Marcin Buchowiecki Institute of Physics, University of Szczecin, Poland AMOC2018

2 Table of contents 1 2

3 Summation of ro-vibrational levels. Future prospects - study of molecular states near dissdociation (Perspective: Accurate ro-vibrational calculations on small molecules, J. Tennyson JCP 145, , 2016). For high temperatures, typically T 1000K, experience shows that it is necessary to incude, at least approximately, all states in the summation PH 3 (phosphine) million rotation-vibration energy levels; NH 3 (ammonia) million rotation-vibration energy levels. reduction of computational cost - not all rotational levels used above certain total angular momentum number; Tennyson et al CH 4 - up to 3000K (+ significant differences between methods; the differences in PESs); Nikitin et al

4 True bound, resonance and scattering states or what is a molecule.

5 True bound, resonance and scattering states or what is a molecule. Only true bound states at high T: BBr, 6000K - entropy; ro-vibrational separation with exact classical expression for vibrational partition function on Manning Rosen PEC (J.-F. Wang et al. Chem. Phys. Lett. 686 (2017) 131) Partition functions and equilibrium constants of 291 diatomic molecules up to K; summation of ro-vibrational levels (P.S. Barklem, R. Collet, Astron. Astrophys. 588 (2016) A96) Partition functions of 51 molecules up to 6000K (in some cases; not only diatomic) (R.R. Gamache et al. J. Quant. Spectrosc. Radiat. Transfer 203 (2017) 70)

6 True bound, resonance and scattering states or what is a molecule. Bound and resonance/metastable states: N + 2, NO, O 2, CN, C 2, CO, CO + - summation up to 50000K (Y. Babou et al. Int. J. Thermophys. 30 (2009) 416) V J (r) = V (r) + J(J + 1)/(2µr 2 ) Often idea of excluded volume used: K = B b 0 Water dimer 400K-1000K (C. Leforestier J. Chem. Phys. 140 (2014) )

7 True bound, resonance and scattering states or what is a molecule.

8 True bound, resonance and scattering states or what is a molecule. [9] M. Capitelli et al. ESA Scientific Technical Review 246 (2005) [6] B.J. McBride NASA Tech. Paper (2002) Y. Babou et al. Int. J. Thermophys. 30 (2009) 416

9 True bound, resonance and scattering states or what is a molecule. Water dimer and trimer - bound and metastable states in absorption spectrum. Spectroscopic model - metastable dimer is considered as two monomers almost freely rotating near each other - absoprtion spectrum modeled as double absorption of the monomer broadened due to a short lifetime of the metastable molecule. T.A. Odnitsova, M. Yu. Tretyakov, J. Quant. Spect. Radiat. Transf. 120 (2013) 134

10 True bound, resonance and scattering states or what is a molecule. Bound, resonance, and scattering states: Partition function, K, Na 2, Li 2 ; ground and 1 excited state. F.H. Mies, P.S. Julienne J. Chem. Phys. 77 (1982) 6162

11 True bound, resonance and scattering states or what is a molecule. Resonance and scattering states from semiclassical approximation for the scattering phase shift F.H. Mies, P.S. Julienne J. Chem. Phys. 77 (1982) 6162

12 F.H. Mies, P.S. Julienne J. Chem. Phys. 77 (1982) 6162 True bound, resonance and scattering states or what is a molecule.

13 True bound, resonance and scattering states or what is a molecule. MgH, partition function, heat capacity, up to 6000K, summation + semiclassical approximation (T. Szidarovszky, A.G. Csaszar J. Chem. Phys. 142 (2015) ) N 2, heat capacity, entalpy up to 25000K, classical approach via virial coefficient (R. Phair, L. Biolsi, P.M. Holland Int. J. Thermophys. 11 (1990) 201) O 2, heat capacity, entalpy up to 25000K, classical approach via virial coefficient; good comparison with JANAF (up to 6000K) but O 2 is easy - plot form Babou (L. Biolsi, P.M. Holland Int. J. Thermophys. 17 (1996) 191) Na 2, heat capacity, viscosity, thermal conductivity, classical approach via virial coefficient; also repulsive PEC used; at 6000K the heat capacity is twice the NIST result (L. Biolsi, P.M. Holland Int J. Thermophys. 31 (2010) 831)

14 True bound, resonance and scattering states or what is a molecule. continued: For Na 2 at 2500K: 31.70J/(molK) (NIST) vs J/(molK) (Biolsi,Holland) but without repulsive part 25.43J/(molK) Comparison not reliable - NIST values do not take into account unbound states - discrepancy must have different source M.W. Chase J. Phys. Chem. Ref. Data, Monograph 9, 1998 Repulsive part of potential cannot be ignored for the heat capacity of Ag 2 (M.L. Biolsi, L. Biolsi, P.M. Holland Paper #368 ACS 228th National Meeting, 2004)

15 Classical approach to thermochemistry/thermophysics. The ro-vibrational partition function of diatomic molecule can be calculated classically from Hamiltonian in spherical coordinates following the procedure of Hill with the condition for bound states as Qrovib B = 1 π (2µ β )3/2 exp( βv )γ(3/2, βv )r 2 dr, (1) σ σ is given by the V (σ) = D e condition, γ is the incomplete gamma function, and V = V lim r V with lim r V = D e for each PEC considered here.

16 Classical approach to thermochemistry/thermophysics. At high temperatures the influence of unbound states is significant, especially heat capacity is a quantity sensitive to unbound states. The inclusion of unbound states is easy in the classical theory because it involves change of exp( βv ) in Q rovib = 1 2 π for exp( βv ) exp( βd e ) ( ) 2µ 3/2 exp( βv )r 2 dr. (2) β 0 Q rovib = 1 2 π ( ) 2µ 3/2 [exp( βv ) exp( βd e )]r 2 dr. (3) β 0

17 Classical approach to thermochemistry/thermophysics. Let H=T+V be the Hamiltonian of internal motion of diatomic molecule given as a sum of kinetic energy T and potential energy V, by using equality e βv = (e βv 1) + 1 the internal partition function is Q = 1 (2π) 3 = 1 (2π) 3 e βt dp e β(t +V ) dpdr = 1 (2π) 3 (e βv 1)dr + 1 (2π) 3 e βt dp e βv dr e βt dpdr, the second term is a partition function of non-interacting atoms (with V = 0) which means that the rest contains all interactions (truly bound, metastable, and scattering states). (4)

18 Classical approach to thermochemistry/thermophysics. Other approaches - use second virial coefficient: B(T ) (e βv 1)dr K = B b B m D.E. Stogryn, J.O. Hirschfelder, J. Chem. Phys. 77 (1959) Q = Q tr B, K = K(B) + K(M) + K(C) (repulsive part of PEC - negative effect) F.H. Mies, P.S. Julienne, J. Chem. Phys. 77 (1982) K = B R. Phair, L. Biolsi, P.M. Holland, P.M. Holland, Int. J. Thermophys 11 (1990) 201. idea of excluded volume: K = (B b 0 )/RT C. Leforestier, J. Chem. Phys. 140 (2014)

19 BBr high temperature thermochemistry. The potential energy curve (PEC) for the ground electronic state 1 Σ was modeled with the improved Manning-Rosen oscillator ( ) V g (r) = D e 1 eαre 1 2 e αr, (5) 1 where D e = a.u., r e = a.u., and α = a.u. The low lying excited states are described with the Morse PECs, the 3 Π state with V e1 (r) = ( 1 e (r 3.517)) , and the 1 Π state with V e2 (r) = ( 1 e (r 3.545)) M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

20 BBr high temperature thermochemistry. The other thermochemical quantities that is internal thermal energy E int, internal heat capacity C p,int, and internal entropy S int are calculated as E int = d ln Q dβ, (6) C p,int = β 2 d 2 ln Q dβ 2, (7) S int = ln Q + βe int, (8) with P = 1atm. C p = C p,int + (5/2)R, (9) S = S int + (5/2)R + ln [ ( ) ] m 3/2 1 R, (10) 2πβ βp

21 BBr high temperature thermochemistry. Table: The heat capacities of the BBr molecule T(K) Cp,int B (g) C p,int B (g2e) C p,int(g2e) C p (g2e) C p,nist 300K K K K K K K K K K M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

22 BBr high temperature thermochemistry. V g (r) K K 3000K 2000K 500K r Figure: The ground state potential energy curve and the related thermal energies (dots) related to energy at inflection point (dashed line). M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

23 BBr high temperature thermochemistry. % Q B int (g) E B int (g) C B p,int (g) S B int (g) T(K) Figure: Influence of ro-vibrational coupling on the partition function Qint B B (g), the thermal energy Eint (g), the heat capacity at constant pressure Cp,int B B (g), and the entropy Sint (g) given as the percent difference between the exact value with respect to separated rotations value. M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

24 BBr high temperature thermochemistry Q B int (g) Q B int (g+2e) Q int (g+2e) T(K) Figure: The partition function of the bound molecule on the ground electronic state Qint B (g) compared with the partition function based on the ground and excited electronic states Qint B (g + 2e), and with the partition function based on both all electronic states and unbound states Q int (g + 2e). M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

25 BBr high temperature thermochemistry E B int (g) E B int (g+2e) E int (g+2e) T(K) Figure: The energy of the bound molecule on the ground electronic state Eint B (g) compared with the energy based on the ground and excited electronic states Eint B (g + 2e), and with the energy based on both all electronic states and unbound states E int (g + 2e). M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

26 BBr high temperature thermochemistry. 4 3 C B p,int (g) C B p,int (g+2e) C p,int (g+2e) T(K) Figure: The heat capacity of the bound molecule on the ground electronic state Cp,int B (g) compared with the heat capacity based on the ground and excited electronic states Cp,int B (g + 2e), and with the heat capacity based on both all electronic states and unbound states C p,int (g + 2e). M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236

27 BBr high temperature thermochemistry S B int (g) S B int (g+2e) S int (g+2e) T(K) Figure: The entropy of the bound molecule on the ground electronic state Sint B (g) compared with the entropy based on the ground and excited electronic states Sint B (g + 2e), and with the entropy based on both all electronic states and unbound states S int (g + 2e). M. Buchowiecki, Chem. Phys. Lett. 692 (2018) 236