Position of Second Line in Microwave Spectrum

HPSC Assistant Professor Exam 2018

The bond length in CN⁺ is 0.129nm. The position of second line in microwave spectrum is

• (A) 9.396 cm⁻¹
• (B) 6.264 cm⁻¹
• (C) 12.528 cm⁻¹
• (D) 3.132 cm⁻¹

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Answer: (B) 6.264 cm⁻¹

Explanation: The second line (J = 1 to 2) is at 4B cm⁻¹.

1. Calculate reduced mass μ for CN⁺ (~6.46 amu).
   μ = (12 × 14) / (12 + 14) = 6.4615 amu
   or, μ = 1.0726 × 10⁻²⁶ Kg (as 1amu = 1.66 × 10⁻²⁷ Kg)


2. Calculate Moment of Inertia I = μr²
   r = 0.129 nm = 1.29 × 10⁻¹⁰.
   I = 1.0726 × 10⁻²⁶ × (1.29 × 10⁻¹⁰)²
   or, I = 1.785 × 10⁻⁴⁶ kg m²




3. Find B = h / (8π²Ic) ≈ 1.566 cm^-1.
   B = 6.626 × 10⁻³⁴ / 8 × π² × (1.785 × 10⁻⁴⁶) × (3 × 10¹⁰)
   or, B = 1.566 cm⁻¹


4. The second line in a microwave spectrum occurs at 4B.
   4 × 1.566 = 6.264 cm^-1.