For hydrogen atom, λ₁ and λ₂ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of λ₁ and λ₂ is x/32. The value of x is __________.

Solution

Physics sir · 4 steps

Bigger energy jump means shorter wavelength, so flip the energy ratio to get the wavelength ratio.

1. Read the figure. Transition 1 goes from n = 3 to n = 1. Transition 2 goes from n = 2 to n = 1. Both end in the ground state, so both are Lyman lines.

   1/λ = R(1/n₁² - 1/n₂²)

2. Each transition. For transition 1, 1/λ₁ = R(1 - 1/9) = 8R/9. For transition 2, 1/λ₂ = R(1 - 1/4) = 3R/4.
3. Take the ratio. Wavelength is the inverse of these, so λ₁/λ₂ = (3/4)/(8/9) = (3 × 9)/(4 × 8) = 27/32.
4. So. Comparing with x/32 gives x = 27. As expected, λ₁ < λ₂ because the 3 → 1 jump releases more energy.

The idea to keep. The Rydberg formula gives 1/λ, not λ. When you compare two lines, take the ratio of the brackets and turn it upside down.

Answer from the NTA final answer key. Solution by PrepWiser.