Calculating Vibrational Frequency for Molecule XY

This question requires calculating the frequency of radiation needed for a vibrational transition (v = 0 to v = 1) in a diatomic molecule XY. This is determined using the harmonic oscillator model, relating the vibrational frequency to the bond's force constant ($k$) and the molecule's reduced mass ($\mu$).

Harmonic Oscillator Model Formula

The vibrational frequency ($\nu$) in Hertz (Hz) is given by:

$ \nu = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}} $

For the transition from v = 0 to v = 1, the energy difference is $\Delta E = h\nu$. The frequency of the absorbed radiation corresponds to this $\nu$. To obtain the frequency in wavenumbers ($\bar{\nu}$, in cm$^{-1}$), we use the formula:

$ \bar{\nu} = \frac{\nu}{c} $

where $c$ is the speed of light.

Input Values and Constants

• Reduced mass ($\mu$): $2 \times 10^{-26}$ kg
• Force constant ($k$): $8 \times 10^2$ N m$^{-1}$
• Speed of light ($c$): $3 \times 10^{10}$ cm s$^{-1}$

Note: The X-Y bond length is not required for this calculation.

Calculation Steps

1. Calculate vibrational frequency ($\nu$) in Hz:

   Substitute the given values into the formula:

   $ \nu = \frac{1}{2\pi} \sqrt{\frac{8 \times 10^2 \text{ N m}^{-1}}{2 \times 10^{-26} \text{ kg}}} $

   Simplify the expression under the square root:

   $ \nu = \frac{1}{2\pi} \sqrt{4 \times 10^{28} \text{ s}^{-2}} $

   Calculate the square root:

   $ \nu = \frac{1}{2\pi} \times (2 \times 10^{14} \text{ s}^{-1}) = \frac{10^{14}}{\pi} \text{ Hz} $

2. Convert frequency ($\nu$) to wavenumber ($\bar{\nu}$) in cm$^{-1}$:

   Use the relationship $\bar{\nu} = \frac{\nu}{c}$:

   $ \bar{\nu} = \frac{10^{14}/\pi \text{ Hz}}{3 \times 10^{10} \text{ cm s}^{-1}} $

   Simplify the expression:

   $ \bar{\nu} = \frac{10^{14}}{3\pi \times 10^{10}} \text{ cm}^{-1} = \frac{10^4}{3\pi} \text{ cm}^{-1} $

3. Calculate the final numerical value:

   Using $\pi \approx 3.14159$:

   $ \bar{\nu} \approx \frac{10000}{3 \times 3.14159} \approx \frac{10000}{9.42477} \approx 1061.03 \text{ cm}^{-1} $

   The calculated value is approximately 1061.03 cm$^{-1}$. This value closely matches option C.

   Final Answer: The final answer is $\boxed{\text{1061.6}}$