5.62 Physical Chemistry II Spring 2008

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1 MIT OpenCourseWare Physical Chemistry II Spring 8 For information about citing these materials or our Terms of Use, visit:

2 5.6 Lecture #17: Chemical Equilibrium. II. Examples Readings: Hill, pp Maczek, pp Metiu, pp Dissociation of a Diatomic Molecule AB A + B p )(p K B p ) p = (q A / N)(q B / N) e +ΔD RT [unitless] = (p A (q AB / N) (p AB p ) = ( q trans,b N)(q trans,a N) g,b g,a q rot,b q rot,a q vib,b q vib,a e +ΔD RT (q trans,ab N) g,ab q rot,ab q vib,ab = (πm B )3/ (kt) 5/ (πm A ) 3/ (kt) 5/ h 3 p h 3 p h 3 p (πm AB ) 3/ (kt) 5/ g,b g,a 1 1 σθ rot,ab 1 1( 1 e θ vib,ab T ) e +ΔD RT T g,ab p is in units of bar because the standard state p = 1 bar = 1 5 pascal. But all terms in statistical mechanical expression for are evaluated in S. I. units. Be careful! = (πµ) 3/ (kt) 5/ g,b g,a σθ rot (1 e θ vib T )e +ΔD RT h 3 p g,ab T where µ= m Am B m = A m B reduced mass m A + m B m AB kg/molecule for SI I I p = I [p s in bar] m I =.169 kg mol 1 µ I =.6345 kg mol 1 g, I = 4 g, I = 1 σ I = p I ω e = 14.5 cm 1 θ vib = 38.6K B e =.3737 cm 1 θ rot =.5377K

3 5.6 Spring 8 Lecture #17, Page D,I = 144cm 1 = 17889K (determined by laser spectroscopy!) [Be careful about units here!] ΔD = p(d ) r (D ) = (17889K) = 17889K p r σ θ rot = (π.634 ) 3/ 6 13 (kt) 5/ (1 e 38.6 T ) e T h T 1 bar = 1 5 pascal. 1 pascal = 1N/m. 1N = 1kg m s = ( T 5/ ) (16).174 ( 1 e 38.6/T ) e 17889/T T =.537 T 3/ ( 1 e 38.6/T ) e 17889/T T[K] (calc) (expt) % error ±.1 3% (4.68±.3) 1.6% % % (1.84±.17) % probably more accurate M. J. Perlman and G. K. than expt because is Rollefson, J. Chem. so small at low T, that Phys. 9, 36 (1941) partial pressue of dissociated I atoms is too small to measure accurately revised 1/1/8 1:55 PM

4 5.6 Spring 8 Lecture #17, Page 3 3 Isotope Exchange Reaction H + D HD = = ( q HD N) ( N) ( q trans,hd / N ) N ( q H N) q D ( q trans,h ) q trans,d ( = πm HD ) 3 (kt) 5 h 6 p e+δd RT ( N) g,hd g,h g,d q vib,hd q vib,h q vib,d h 3 p h ( πm H ) 3/ (kt) 5/ πm D 3 p q rot,hd e +ΔD RT q rot,h q rot,d ( ) 3/ (kt) 5/ g,hd g,h g,d D σ D ( 1 e θ vib,h T )( 1 e θ vib,d T ) kt hcb e H σ H hcb e ( 1 e θ vib,hd T ) hcb e HD σ HD kt kt e +ΔD RT m H = amu σ H = g (H ) = 1 m HD = 3 amu σ HD = 1 g (HD) = 1 m D = 4 amu σ D = g (D ) = 1 D (H ) = 36,1 cm 1 D (HD) = 36,394 cm 1 D (D ) = 36,74 cm 1 H ω e (H ) = 441 cm 1 θ vib = 6337 K B e (H ) = 6.8 cm 1 ω e (HD) = 3813 cm 1 HD = 5419 K B e (HD) = 45.7 cm 1 θ vib (D ) = 3116 cm 1 D ω B e (D ) = 3.4 cm 1 e θ vib = 4487 K ΔD = p ( D ) r ( D ) p r ΔD = ( 36, 394) [ 36,1 + 36, 74 ] = 54 cm 1 = 78K revised 1/1/8 1:55 PM

5 5.6 Spring 8 Lecture #17, Page 4 What is at T = 98K? = mh 3 m HD m 3 D g,hd H T D T 3 ( 1 e θ vib )( 1 e θ vib ) B H e B e D σ H σ D e +ΔD RT HD T ) HD g,h g,d ( 1 e θ vib (B e ) σ HD = (.3 / ) 3 ( ) 3/ ( ) 3/ 1 ( 1 e 4487/98 ) 1 1 ( 1 e 5419/98 ) ( 1 e 6337/98 ) (6.8)(3.4) e 78/98 (45.7) 1 = 3.7 T[K] [CALC] [EXP] DEPENDENCE OF ON T I I H + D HD Qualitative difference in behaviors: Ι Ι Η + D HD q trans I e +ΔD RT q q trans,hd rot,hd q trans I q rot q vib q vib,hd e +ΔD /RT qtrans,h q trans,d q rot,h q vib,h q rot,d q vib,d revised 1/1/8 1:55 PM

6 5.6 Spring 8 Lecture #17, Page 5 q trans I q transi (ignore factor of in mass) q trans,hd q trans,h q trans,d 3 3/ 1 ignoring 4 mass q rot,h q rot,d q rot,hd functions µ ) HD µ H µ ( / 3 = D (1 / ) ( 1) 1 q vib,h q vib,d q vib,hd q trans q rot q vib e + D RT q trans 1 3, q rot 1 3, q vib 1, D 18,K 1 7 e 18,/T σ e + D RT D 78K 4e 78/T large T dependence and large Small values of : because of 1 7 factor gain in translational entropy due to no gain in entropy except for change in number of moles symmetry # (factor of 4) results in shift of equilibrium 4 at modest T because of small toward separated atoms at high T difference in zero point energy. actually q trans T 5/, 1 1, the q rot T pre-exponential factor is T- dependent as T increases, both pre- exponential and exponential factors increase and shift equilibrium toward dissociation. Recall from 5.6: G (T) = H (T) T S (T) = RTln K(T) K(T) = e S (T)/R H (T)/RT e pre-exponential factor This gives us an intuitive understanding of the T-dependence of equilibrium constants. Mostly, S (T) is determined by change in number of moles (strong T-dependence), secondarily in changes in floppiness (approximately T-independent). Mostly H (T) is determined by bond energies (or differences in dissociation energies), but if you want to compute K(T) from microscopic quantities, use K(T) = e G RT and use statistical mechanics to calculate G (T) directly, not both H (T) and S (T) separately. revised 1/1/8 1:55 PM

7 5.6 Spring 8 Lecture #17, Page 6 In using statistical mechanics to compute equilibrium constants, it is computationally most compact and intuitively most instructive to assemble the relevant factors in ( q C N) c ( q D N) d ( q A N) a ( q B N) b by assembling all of the relevant information factored according to degee of freedom Translation (translation)(electronic)(vibration)(rotation) Key factors are does the number of moles change the only species-specific quantity is mass Electronic Vibration Rotation Key factor is degeneracy of ground state For CO X 1 + C( 3 P) + O( 3 P) g: The electronic factor is usually negligibly T-dependent, unless there are low-lying states. For C, the 3 P state is regular and J = is lowest. For O, the 3 P state is inverted and J = is lowest. So at low-t the degeneracies are 1 and 5, not 9 and 9, but at T where atoms have appreciable population, kt spin-orbit splittings for atoms from the first three rows of the periodic table. Atoms have q vib = 1 (no vibrational d/f). For polyatomic molecules, the lowest frequency vibrations result in small but dominant T-dependence. It is easy to guess whether a molecule has low-frequency vibrations. Atoms have q rot = 1 (no rotational d/f). Generally, all rotations are in the high-t limit. Thus q rot T 3/ (non-linear polyatomic) or T 1 (linear molecule). For isotope effects in a diatomic molecule: q vib, q rot, and zero-point energy ω e [µ] 1/ B e [µ] 1 In a polyatomic molecule, the relationships between atomic masses and ω ei (1 i 3N 6) and A, B, C are more complicated. revised 1/1/8 1:55 PM