HW8 soln - hw8 solution PDF

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HW 8 "! = "# !

%$1.

and

!

For N2: (! =

"& = ! "! − ' %& + ' %&(! ! (

$

%&(! 14.324&'(−1 −3 = = 6.073 × 10 & % 2358.6&'(−1

"! = !

2358.6&'(−1

4 /6.073 × 10−3 0

= 97094&'(−1

1 1 "& = 97094&'(−1 − 32358.6&'(−14 + 314.324&'(−14 = 95918&'(−1 ! 2 4

For O2:

(! =

%&(! 12.073&'(−1 = 7.639 × 10−3 = 1580.4&'(−1 % &

"! = !

1580.4&'(−1

4 /7.639 × 10

−3

0

= 51721&'(−1

1 1 "& = 51721&'(−1 − 31580.4&'(−14 + 312.073&'(−14 = 50934&'(−1 ! 2 4

b. The molecular orbital energy level diagrams of N2 and O2 are shown below:

c. The bond dissociation energy for N2 is higher than that for O2 , which is consistent with N2 has a triple bond, while O2 has a double bond. d. The maximum value of n (%)*%) occurs when ∆1 6 ++1/2 = 0 = %2 − 2(%012 + 1)% 2(3 '

'

So: %)*% = (% − 1 = ( 45.&78 !

×10−39

− 1 = 81.4, and %)*% = 81

2. i) For the P-branch (J decreasing) of a rovibrational spectra, the lines are given by 32& (5) = 32− 29: 5

Therefore, the spacing between the two transitions are:

∆32& (5) = ;32 − 29: (6)< − ;32 − 29: ( 7 )< = 29:

So we have

∆32& (2863.8232 − 2842.6358)'('( = 10.5937'('( = 2 2 We can therefore calculate the vibrational frequency for the transition to the J=6 state: : 5 =(2863.8232 + 12 × 10.5937)&'('( = 2990.9476&'('( 2 = 32& ( 3 5) + 29 9: =

The vibrational frequency is related to the spring constant: 32= The reduced mass of 1H35Cl is :=

)" :)#$

)" ;)#$

=

@ 1 ? ⟹ @ = (2>'3 2)) A 2>' A

(':*)=) ×(8@:*)= ):'.55&@$:×'&%&'AB = *)= ':*)=:;:8@:*)=

1.6144; × 10C(7 =>,

So:

@ = C2>&(2.998 × 10(* '( ∙ E'()(2990.9476&'('()F (1.6144& × 10')+ @G) = 512.45&@G ∙ E') )

ii)

1 (6.626 × 10',- 5 ∙ E) ℏ 2> H= = = 2.6423 × 10'-+ @G ∙ () = AJ ) 4>'9: 4>( 2.998 × 10(* '( ∙ E'( )(10.5937&'('() H 2.6423 × 10'-+ @G ∙ () = 1.279; × 10−10 ? = 1.279;Å J=? =? A 1.6144; × 10−27 =>

3. The Lewis structure of HCN is

H

C

N

i) HCN is a linear molecule and has 3N-5 = 3(3)-5 = 4 normal modes. ii) The 4 normal modes are:

4. The Lewis structure of nitrogen dioxide, NO2 is:

N O

O

i) NO2 is a nonlinear molecule (with bent structure similar to water) and has 3N-6 = 3(3)-6 = 3 normal modes. ii) The vibrational motions are similar to those of water:

5. In an electronic excited state an electron has typically been promoted from a bonding orbital to an orbital with less bonding character. In the ground state the interactions between the electrons and nuclei and electron- electron are optimum therefore affording the largest nuclear-nuclear repulsion (shortest bond). Therefore, the bond lengths in the excited state will typically be longer. 6. We have:

< A./ >&= M N. AON/ PQ 0

'0

(AO .1.2 + AO782 )&N.1.2(R; J)T3456 (J)PU.1.2 PU3456 = M N.1.2 # (R; J)T3456 # (J) 0

'0 0

0

'0 0

'0

= M T3456 # (J) VM N.1.2 #(R; J) AO .1.2 &N.1.2( & R; J)PU.1.2W T3456 (J)PU3456

+ M T3456 # (J )AO 782 VM N.1.2 #(R; J)N.1.2 &(R; J)PU.1.2W T3456 (J)PU3456 0

Since the electronic portions of the wave functions, N , are orthogonal such that '0

'0

.1.2 0 X./ = M N. N/ PQ = M N.1.2 # &(R; J)N.1.2 &(R; J)PU= .1.2 0

0

'0

'0

The average value of the transition dipole moment is given by:

< A./ >&= M T3456 #(J)VM N.1.2 #( R; J)AO .1.2 &N.1.2 ( & R; J)PU.1.2WT3456 (J)PU3456 0

0

'0

'0

We can make the Franck-Condon approximation and: = M N.1.2 # (R; J* )A O .1.2 &N.1.2 &(R; J)PU.1.2 O.1.2 &N.1.2 (R; J* )&PU.1.2 M N.1.2 # (R; J)A 0

0

'0

'0

So we get < A./ >&= M T3456 #(J)T3456 (J)PU3456M N.1.2 # (R; J* )A O &N.1.2 (R; J* )&PU.1.2 .1.2 0

0

'0

'0

The first integral is the vibrational overlap integral....