Question

For hydrogen iodide, 1 H^127 I, force constant is 314 N m-1. Calculate the
fundamental frequency in cm^-1 unit for
i) 1H127I
ii) 2H127I
Hint: Assume that force constant does not change with isotopic substitution.

Answer 1

Answer on Question #44197-Physics-Mechanics-Kinematics-Dynamics

For hydrogen iodide, 1 H^127 I, force constant is 314 N m-1. Calculate the fundamental frequency in cm^-1 unit for

i) 1H127I

ii) 2H127I

Hint: Assume that force constant does not change with isotopic substitution.

Solution

In the harmonic approximation, the stationary vibrational energy levels for a diatomic molecule A–B are given by

$$E _ {\mathrm {v}} = (\mathrm {v} + \frac {1}{2}) \hbar \omega = (\mathrm {v} + \frac {1}{2}) \hbar \sqrt {\frac {k}{\mu}}, \qquad \mathrm {v} = 0, 1, 2, \dots$$

where $k$ is the "force constant" for the chemical bond between the atoms A and B, $\mu$ is the reduced mass, $\mu = m_{\mathrm{A}}m_{\mathrm{B}} / (m_{\mathrm{A}} + m_{\mathrm{B}})$.

i)

$$\omega_ {1} = \frac {1}{2 \pi c} \sqrt {\frac {k}{\mu_ {1}}} = \frac {1}{2 \pi \cdot 3 \cdot 1 0 ^ {8}} \sqrt {\frac {3 1 4}{\frac {1 . 6 7 \cdot 1 0 ^ {- 2 7} \cdot 2 1 2 \cdot 1 0 ^ {- 2 7}}{1 . 6 7 \cdot 1 0 ^ {- 2 7} + 2 1 2 \cdot 1 0 ^ {- 2 7}}}} = 0. 2 3 1 \cdot 1 0 ^ {6} m ^ {- 1} = 2 3 1 0 c m ^ {- 1}.$$

ii)

$$\omega_ {2} = \frac {1}{2 \pi c} \sqrt {\frac {k}{\mu_ {2}}} = \frac {1}{2 \pi \cdot 3 \cdot 1 0 ^ {8}} \sqrt {\frac {3 1 4}{\frac {3 . 3 4 \cdot 1 0 ^ {- 2 7} \cdot 2 1 2 \cdot 1 0 ^ {- 2 7}}{3 . 3 4 \cdot 1 0 ^ {- 2 7} + 2 1 2 \cdot 1 0 ^ {- 2 7}}}} = 0. 1 6 4 \cdot 1 0 ^ {6} m ^ {- 1} = 1 6 4 0 c m ^ {- 1}.$$

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