Engineering Chemistry S Datta

INSTRUMENTAL METHODS OF ANALYSIS 521
Two balls joined by a spring
For example to stretch spring requires more energy than to bend it; thus stretching
energy of a bond is greater than the bending energy and stretching absorptions of a bond
appear at higher frequencies in the infrared spectrum than the bending absorption of the
same bond.
Calculation of vibrational frequencies
We can calculate the vibrational frequency of a bond with a reasonable accuracy, in the
same way as we can calculate the vibrational frequency of a ball and spring system; the equation
of calculation is Hook’s law,
ν =
1
2.π
F
HG
k
mm / m + m
1 2 1 2
where ν = frequency, k = a constant related to the strength of the spring (the force constant of
the bond), m 1
, m 2
= the masses of two balls or atoms, m 1
m 2
/m 1
+ m 2
is known as reduced mass.
As an example, we can calculate the approximate frequency of C—H stretching vibration
from the following data,
I
KJ
1/2
k = 500 Nm –1 = 5 × 10 5 gm s –2 × 1 m
= 5 × 10 5 gs –2
1 N = 1 kg × m/sec 2 = 1000 gm s –2
m C
= mass of C-atom = 20 × 10 –24 gm
m N
= mass of H-atom = 1.6 × 10 –24 gm
7
ν =
2×
22
L
N
M
5×
10
5 −2
−24 −24 −24
( 20 × 10 g)( 16 . × 10 g)/( 20 + 16 . ) 10 g
To express this in wave numbers ( ν ) we use relationship,
gs
O
Q
P
1/2
= 9.3 × 10 13 s –1
13 −1
ν = ν c = 93 . × 10 s
3×
10 ms
8 −1
= 3.1 × 10 5 m –1 = 3100 cm –1
So, ν max.
for C—H (Str.) is 3100 cm –1 .
where c = velocity of light
The vibrational frequency of a bond is expected to increase when the bond strength
increases and also when the reduced mass of the system decreases. As for example,
Stretching : C = C > C—C Stretching : C=O > C—O
1600 cm –1 1700 cm –1 1100 – 1300 cm –1
Stretching : C—H > C—C Stretching : O—H > C—O
3100 cm –1 3500 cm –1 1100–1300 cm –1
12 × 1
13
<
12 × 12
16 × 1
24
17
<
12 × 16
28