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Question: The energy of a photon of radiation having wavelength \({\text{300 nm}}\) is: A) \(6.63 \times {10...

The energy of a photon of radiation having wavelength 300 nm{\text{300 nm}} is:
A) 6.63×10−29 J6.63 \times {10^{ - 29}}{\text{ J}}
B) 6.63×10−19 J6.63 \times {10^{ - 19}}{\text{ J}}
C) 6.63×10−28 J6.63 \times {10^{ - 28}}{\text{ J}}
D) 6.63×10−17 J6.63 \times {10^{ - 17}}{\text{ J}}

Explanation

Solution

The energy of the photon is inversely proportional to the wavelength and thus, directly proportional to the frequency. This is because wavelength and frequency are inversely proportional to each other. As the wavelength increases, the energy of the photon decreases.

Formula Used: E=hcλE = \dfrac{{hc}}{\lambda }

Complete step by step answer:
Convert the units of wavelength from nm{\text{nm}} to m{\text{m}} using the relation as follows:
1 nm=1×10−9 m1{\text{ nm}} = 1 \times {10^{ - 9}}{\text{ m}}
Thus,
λ=300 nm×1×10−9 m1 nm\lambda = 300{\text{ }}{{{\text{nm}}}} \times \dfrac{{1 \times {{10}^{ - 9}}{\text{ m}}}}{{1{\text{ }}{{{\text{nm}}}}}}
λ=300×10−9 m\lambda = 300 \times {10^{ - 9}}{\text{ m}}
Thus, the wavelength is 300×10−9 m300 \times {10^{ - 9}}{\text{ m}}.
Calculate the energy of the photon using the Planck-Einstein equation as follows:
E=hcλE = \dfrac{{hc}}{\lambda }
Where, E is the energy of the photon,
h is the Planck’s constant,
c is the velocity of light in vacuum,
λ\lambda is the wavelength.
Substitute 6.626×10−34 J s6.626 \times {10^{ - 34}}{\text{ J s}} for the Planck’s constant, 3×108 m s−13 \times {10^8}{\text{ m }}{{\text{s}}^{ - 1}} for the velocity of light in vacuum, 300×10−9 m300 \times {10^{ - 9}}{\text{ m}} for the wavelength. Thus,
E=6.626×10−34 J s×3×108 m s−1300×10−9 mE = \dfrac{{6.626 \times {{10}^{ - 34}}{\text{ J }}{{\text{s}}} \times 3 \times {{10}^8}{\text{ }}{{\text{m}}}{\text{ }}{{{{\text{s}}^{ - 1}}}}}}{{300 \times {{10}^{ - 9}}{\text{ }}{{\text{m}}}}}
E=6.63×10−19 JE = 6.63 \times {10^{ - 19}}{\text{ J}}
Thus, energy of a photon of radiation having wavelength 300 nm{\text{300 nm}} is 6.63×10−19 J6.63 \times {10^{ - 19}}{\text{ J}}.

Thus, the correct option is (B) 6.63×10−19 J6.63 \times {10^{ - 19}}{\text{ J}}.

Note: Tiny particles having no charge and no resting mass are known as photons. The photons are emitted by charged particles, radioactive decay, etc. photons always move at the speed of light in vacuum.
Photons can be destroyed as well as created. When electromagnetic waves are emitted by the source, photons are created. When photons hit with matter, they either absorb or transfer the energy to the atoms and molecules. The creation and destruction of photons conserves energy and momentum.