A monochromatic laser is exciting hydrogen atoms from the n=2 state to the n=5 state.
(A) What is the wavelength λ of the laser? Express your answer to three significant digits in nanometers.
(B) Eventually, all of the excited hydrogen atoms will emit photons until they fall back to the ground state. How many different wavelengths can be observed in this process?
(C) What is the longest wavelength λ_max that is observed? Express your answer to three significant digits in nanometers.
(D) What is the shortest wavelength λ_min observed? Express your answer to three significant digits in nanometers.

a)  λ = 435 nm , c) c) λ = 4052 nm, d) λ= 95 nm

Explanation:

A) To carry out this excitation, the energy of the laser must be greater than or equal to the energy of the transition of the hydrogen atom, whose states of energy are described by the Bohr model.

        En = -13,606 / n²    [eV]

therefore the energy of the transition is

          ΔE = E₅ -E₂

          ΔE = 13.606 (1 / n₂² - 1 / n₅²)

         ΔE = 13.606 (1/2² - 1/5²)

         ΔE = 2,85726 eV

now let's use Planck's equation

          E = h f

the speed of light is related to wavelength and frequencies

        c = λ f

        f = c /λ

       E = h c /λ

       λ = h c / E

let's reduce the energy to the SI system

      E = 2,85726 eV (1.6 10⁻¹⁹ J / 1 eV) = 4.5716 10⁻¹⁹ J

let's calculate

       λ = 6,626 10⁻³⁴ 3 10⁸ / 4,5716 10⁻¹⁹

       λ = 4.348 10⁺⁷ m (10⁹ nm / 1 m)

       λ = 435 nm

B) photon emission processes from this state with n = 5 to the base state n = 1, can give transition

    initial state n = 5

    final state   n = 4

      ΔE = 13.606 (1/4² - 1/5²)

      ΔE = 0.306 eV

      λ = h c / E

      λ = 4052  nm

      n = 5

 final        ΔE (eV)     λ (nm)

 level   

   4             0.306      4052

   3             0.9675     1281

   2             2,857       435

   1            13.06           95

  n = 4

  3              0.661      1876

  2              2,551       486

  1              11,905       104

n = 3

  2              1.89         656

  1             12.09         102.5

n = 2

 1               10.20         121.6

c) λ = 4052 nm

d)  λ= 95 nm

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