Calculate the wavelength (in nanometers) of a photon emitted by a hydrogen atom

Question

Calculate the wavelength (in nanometers) of a photon emitted by a hydrogen atom when its electron drops from the n = 4 to n = 2 state.

Consider the following energy levels of a hypothetical atom:
E4 1.61 × 1019 J

E3 7.51 × 1019 J

E2 1.35 × 1018 J

E1 1.45 × 1018 J

(a) What is the wavelength of the photon needed to excite an electron from E1 to E4?
____ ×10m

(b) What is the energy (in joules) a photon must have in order to excite an electron from E2 to E3?
___× 10J

(c) When an electron drops from the E3 level to the E1 level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.
___×10 m

Explanation / Answer

a) E =E4 - E1 = 1.61 × 1019 J - (1.45 × 1018 J ) = 1.289x1018 J

E = hc/ = ( 6.625x10-34Js x 3x108ms-1) / (1.289x1018 J) = 15.41 x 10-8 m = 154 nm

The photon needed to excite an electron from E1 to E4 is 154 nm

b) E =E2 - E3 = 1.35 × 1018 J - (7.51 × 1019 J ) = 0.599x1018 J = 5.99x1019 J

The energy (in joules) a photon must have in order to excite an electron from E2 to E3 is 5.99x1019 J

c) E =E1 - E3 = 1.45 × 1018 J - (7.51 × 1019 J ) = 0.699x1018 J

E = hc/ = (6.625x10-34Js x 3x108ms-1) / (0.699x1018 J) = 28.4 x 10-8 m = 284 nm

The wavelength of the photon emitted in this process is 284 nm

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