GATE 2025 CY – Question 34

Physical Chemistry · Spectroscopy: Atomic spectroscopy; Russell-Saunders coupling; Term symbols and spectral details; origin of selection rules. Rotational, Vibrational, Electronic and Raman spectroscopy of diatomic and simple polyatomic molecules. Line broadening and line widths; simple properties of Gaussian and Lorentzian line shapes. Molecular spectroscopy: Absorbance, Beer- Lambert’s law, Einstein’s coefficient, Jablonski diagram. Relationship of transition moment integral with molar extinction coefficient and oscillator strength. Basic principles of Nuclear Magnetic Resonance: Gyromagnetic ratio; Chemical shift, nuclear coupling. · 1 mark · Numerical answer

The ratio of the fundamental vibrational frequencies $(\nu_{^{13}C^{16}O}/\nu_{^{12}C^{16}O})$ of two diatomic molecules $^{13}\mathrm C^{16}\mathrm O$ and $^{12}\mathrm C^{16}\mathrm O$, considering their force constants to be the same, is _______ (rounded off to two decimal places).

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Correct answer: 0.97 to 0.99

Explanation

For a diatomic molecule treated as a harmonic oscillator,
$$\nu=\frac1{2\pi}\sqrt{\frac k\mu},$$
so with the same force constant $k$, $\nu\propto\mu^{-1/2}$, where $\mu$ is the reduced mass.

**Reduced masses** (in atomic mass units, using integer masses):
$$\mu(^{12}\text{C}^{16}\text{O})=\frac{12\times16}{28}=6.857,\qquad\mu(^{13}\text{C}^{16}\text{O})=\frac{13\times16}{29}=7.172 .$$

**Ratio:**
$$\frac{\nu_{13}}{\nu_{12}}=\sqrt{\frac{\mu_{12}}{\mu_{13}}}=\sqrt{\frac{6.857}{7.172}}=\sqrt{0.9560}=\mathbf{0.978}\approx0.98.$$

(Heavier isotopes vibrate at lower frequency.)