An electron drops from the second energy level to the first energy level within an excited hydrogen atom (a) determine the energy of the photon emitted (b) calculate the frequency of the photon emitted (c) calculate the wavelength of the photon emitted. (Answer: (a) 10.2 eV (b) 2.5 x 10¹⁵ Hz (c) 120 nm)
Given Data:
Atom: Hydrogen
Excited state = n = 2
Lower state = p = 1
To Find:
(a) Energy of the photon emitted = E = ?
(b) Frequency of the photon emitted = f = ?
(c) Wavelength of the photon emitted = λ = ?
Solution:
(a) Energy of the photon emitted = E = ?
The energy of the photon emitted is given by:
Where En is the energy of an electron in an excited state and Ep is the energy of an electron in the lowest state.
En = -E0n2
Where E0 = constant = 13.6 eV and n = 2 So,
En = - 13.6eV22
En = - 13.6eV4
En = - 3.4 eV
and
Ep = -E0p2
Where E0 = constant = 13.6 eV and p = 1 So,
Ep = - 13.6eV12
Ep = - 13.6eV1
Ep = - 13.6 eV
Now putting values in Equation (1)
E = En - Ep = - 3.4 eV - (- 13.6 eV ) = 10.2 eV
E = 10.2 eV -------------------Ans.1
(b) Frequency of the photon emitted = f = ?
We know that
E = hf
or
f = Eh
By putting values of E = 10.2 eV = 10.2 x 1.6 x 10⁻¹⁹ J and Plank's Constant = h = 6.62 x 10⁻³⁴ m² kg s⁻¹
f = 10.2x1.6x10⁻¹⁹J6.62x10⁻³⁴m²kgs⁻¹
By simplifying we get
f = 2.46 x 10¹⁵ Hz -------------------Ans.2
(c) Wavelength of the photon emitted = λ = ?
we know that v = f λ, where v = c = 3 x 10⁸ m s⁻¹
or
λ = 3x10⁸ms⁻¹2.46x10¹⁵Hz
λ = 1.219 x 10⁻⁷ m
or
λ = 121.9 x 10⁻⁹ m
λ = 121.9 nm ------------------Ans.3
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