GATE 2026 CY – Question 33

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

At magnetic field 14.1 T, proton NMR resonances occur at 1.25 and 5.75 ppm. Their separation in Hz (one decimal place) is _____. Given $\gamma_H=2.675\times10^8$ T−1 s−1.

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Correct answer: 2687.79 to 2714.81

Explanation

In NMR, the resonance frequency of a proton is $\nu_0=\dfrac{\gamma B_0}{2\pi}$, and a chemical-shift difference of $\Delta\delta$ ppm corresponds to a frequency difference
$$\Delta\nu=\Delta\delta\times10^{-6}\times\nu_0 .$$

**Step 1: spectrometer frequency.**
$$\nu_0=\frac{2.675\times10^8\times14.1}{2\pi}=6.0033\times10^8\text{ Hz}\ (\approx600\text{ MHz}).$$

**Step 2: separation.** The shifts are 1.25 and 5.75 ppm, so $\Delta\delta=4.50$ ppm:
$$\Delta\nu=4.50\times10^{-6}\times6.0033\times10^8=\mathbf{2701.3\ Hz}.$$