8/3/2019 Chem 373- Assigned question for the angular momentum

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Assigned question for the angular momentum. This is the kind of question that

One might find on the final exam from time to time

1. The rotation of an 1H117I molecule can be pictured as the orbital motion of an H

Atom at the distance 160 pm from a stationary I atom.[This is quite a good picture, to be precise, both atoms rotate around their common

center of mass, which is very close to the I nucleus.]

Suppose that the molecule rotates only in a plane. Calculate the energy needed to excite

the molecule

Em

I

E

I

x Js

x x x kg x m

x J

l

o

=

= =

= = =

2 2

2 34 2

27 12 2

22

2

0 0

2

1 055 10

2 1 008 1 6605 10 160 10

1 30 10

h

h

[m the groundstate

E

l

1

]

( . )

( ) ( . ) ( . )( )

.

2. Calculate the energies of the first four rotational levels of 1H117I free to rotate in three

dimensions, using for its moment of inertia

I Rm m

m m= =

+

2 1 2

1 2

, with and R = 160 pm

El l

I

l l

RR=

+=

+( ) ( ),

1

2

1

2

2 2

22h h

[I = in place of m]

El l

I

l l

RJs

x x kg x x mx

m m

=+

=+

+ = +

( ) ( )

)

( ) ( . ) ( )

. .]

1

2

1

2

2 1 6605 10 160 10

1

1 008

1

126 90

1 1

2 2

22

27 12 2

1 2

h h

=l(l +1)x(1.055x10

[1

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8/3/2019 Chem 373- Assigned question for the angular momentum

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the

E

Therefore

E l l x x l l x GHz

energies may be expressed in terms of the equivalent frequencies with

=E

h= 1.509x10 33

221 1 31 10 1 198

,

( ) ( . ) ( ) ( )= + = +

Hence , the energies and the equivalent frequencies are :

I = 0 1 2 3

10 E/J 0 2.62 7.86 15.72

/GHz 0 396 1188 2376

22