Use the Rydberg Equation to calculate the energy in Joules of the transition between n = 7 and n = 3 for the hydrogen atom. Find the frequency in Hz of this transition if the wavelength is 1000nm.

The energy of the transition between n = 7 and n = 3  is 1.96 × 10^-19 J while the frequency is 3 × 10^14 Hz.

Using the Rydberg Equation for energy;

ΔE = -RH(1/n^2final - 1/n^2initial)

Given that;

nfinal = 3

ninitial = 7

RH = 2.18 × 10^-18 J

ΔE = - 2.18 × 10^-18(1/3^2 - 1/7^2)

ΔE = - 2.18 × 10^-18(0.11 - 0.02)

ΔE = - 1.96 × 10^-19 J

For the second part;

Since the wavelength is 1000nm, we have;

λ = 1000nm

c = 3 × 10^8 m/s

f = ?

c = λf

f = c/λ

f = 3 × 10^8 m/s/1000 × 10^-9 m

f = 3 × 10^8 m/s/ 1 × 10^-6 m

f = 3 × 10^14 Hz

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The energy of each transition is approximately .

The frequency of photons released in such transitions is approximately .

Explanation:

The Rydberg Equation gives the wavelength (in vacuum) of photons released when the electron of a hydrogen atom transitions from one main energy level to a lower one.

The Rydberg Equation gives the following relation:

.

Rearrange to obtain and expression for :

.

In this question, while . Therefore:

.

Note, that is equivalent to . That is: .

Look up the speed of light in vacuum: . Calculate the frequency of this photon:

.

Let represent Planck constant. The energy of a photon of wavelength would be .

Look up the Planck constant: . With a frequency of (,) the energy of each photon released in this transition would be:

.

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