Atomic hydrogen produces a well-known series of spectral lines in several regions of the electromagnetic spectrum. each series fits the rydberg equation with its own particular n1 value. calculate the value of n1 that would produce a series of lines in which the highest energy line has a wavelength of 821 nm.

n₁ = 3

Explanation:

The energy of the states in the hydrogen atom is explained by the Bohr model, the transitions heal when an electron passes from a state of higher energy to another of lower energy,

       ΔE = - E₀ = - k²e² / 2m (1 / ²2 - 1 / n₀²)

The energy of this transition is given by the Planck equation

       E = h f = h c / λ

       h c / λ = -k²e² / 2m (1 / no ²- 1 / no²)

       1 / λ = Ry (1/ ² - 1 / n₀²)

Let's apply these equations to our case

     λ = 821 nm = 821 10⁻⁹ m

     E = h c / λ

     E = 6.63 10⁻³⁴ 3 10⁸/821 10⁻⁹

     E = 2.423 10⁻¹⁹ J

Now we can use the Bohr equation

Let's reduce to eV

       E = 2,423 10⁻¹⁹ J (1eV / 1.6 10⁻¹⁹) = 1,514 eV

        - E₀ = -13.606 (1 /  ² - 1 / n₀²)   [eV]

Let's look for the energy of some levels

n         (eV)           - E (eV)

1         -13,606           E₂-E₁ = 10.20

2        -3.4015           E₃-E₂ = 1.89

3        -1.512              E₄- E₃ = 0.662

4        -0.850375

We see the lines of greatest energy for each possible series, the closest to our transition is n₁ = 3 in which a transition from infinity (n = inf) to this level has an energy of 1,512 eV that is very close to the given value

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