Question 4 of 8intermediate🔧 ApplyNumerical3 marks

A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.

Correct Answer

The wavelength of the photon is 97.4 nm and the frequency is 3.08×103.08 \times 10¹⁵ Hz.

Exercise: EXERCISES | Q: 12.5 | (Page 16)
For More Understanding

Explanation

This question tests the student's understanding of energy levels in hydrogen atom and photon absorption. The ground state energy (n=1) is -13.6 eV. Using the formula En = -13.6/n² eV, we calculate E₄ = -0.85 eV. The photon energy equals the energy difference between the two levels. Converting this energy to frequency using E = hν and then finding wavelength using λ = c/ν completes the solution.

Solution Steps

  1. Step 1: Calculate energy at ground state (n = 1): E₁ = -13.6 eV (given in context)

  2. Step 2: Calculate energy at n = 4 level: E₄ = -13.6/4² = -13.6/16 = -0.85 eV

  3. Step 3: Calculate energy of photon absorbed: ΔE = E₄ - E₁ = -0.85 - (-13.6) = 12.75 eV

  4. Step 4: Convert energy to Joules: E = 12.75×1.612.75 \times 1.6 × 10⁻¹⁹ = 2.04×102.04 \times 10⁻¹⁸ J

  5. Step 5: Calculate frequency using E = hν: ν = E/h = (2.04×102.04 \times 10⁻¹⁸)/(6.626×106.626 \times 10⁻³⁴) = 3.08×103.08 \times 10¹⁵ Hz

  6. Step 6: Calculate wavelength using λ = c/ν: λ = (3×103 \times 10⁸)/(3.08×103.08 \times 10¹⁵) = 9.74×109.74 \times 10⁻⁸ m = 97.4 nm

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