What is the wavelength of a photon with an energy of 3.38 $x 10^{-19} J$?
A. 588 nm
B. 510 nm
C. 460 nm
D. 416 nm
Answer (2)
The wavelength of a photon with an energy of 3.38 x 10^{-19} J is approximately 588 nm, which corresponds to option A.
;
Relate the energy of a photon to its wavelength using the formula E = h f and c = λ f .
Solve for the wavelength: λ = E h c .
Substitute the given values: h = 6.626 × 1 0 − 34 J ⋅ s , c = 3.00 × 1 0 8 m / s , and E = 3.38 × 1 0 − 19 J .
Calculate the wavelength and convert it to nanometers: λ ≈ 588 nm . The answer is 588 nm .
Explanation
Problem Analysis We are given the energy of a photon and asked to find its wavelength. We will use the formula relating energy, wavelength, Planck's constant, and the speed of light to solve this problem.
Relevant Equations The energy of a photon is given by the equation: E = h f
where E is the energy of the photon, h is Planck's constant, and f is the frequency of the photon. We also know that the speed of light is related to the wavelength and frequency by the equation: c = λ f
where c is the speed of light and λ is the wavelength.
Solving for Wavelength We want to find the wavelength λ , so we can rearrange the equations to solve for λ . First, we can solve the first equation for f :
f = h E
Then, we can substitute this into the second equation: c = λ h E
Finally, we can solve for λ :
λ = E h c
Plugging in Values Now, we can plug in the given values: E = 3.38 × 1 0 − 19 J , h = 6.626 × 1 0 − 34 J ⋅ s , and c = 3.00 × 1 0 8 m / s :
λ = 3.38 × 1 0 − 19 J ( 6.626 × 1 0 − 34 J ⋅ s ) ( 3.00 × 1 0 8 m / s )
Calculating Wavelength Calculating the wavelength: λ = 3.38 × 1 0 − 19 J 1.9878 × 1 0 − 25 J ⋅ m = 5.881065 × 1 0 − 7 m
Converting to Nanometers Converting the wavelength from meters to nanometers: λ = 5.881065 × 1 0 − 7 m × 1 m 1 0 9 nm = 588.1065 nm
Final Answer The calculated wavelength is approximately 588 nm, which matches option A.
Examples
Understanding the relationship between a photon's energy and its wavelength is crucial in many applications, such as designing solar panels. Solar panels work by absorbing photons from sunlight. The efficiency of a solar panel depends on its ability to absorb photons of different wavelengths. By knowing the wavelength of light that corresponds to a particular energy, engineers can select materials that are most effective at capturing solar energy. This ensures that the solar panel converts sunlight into electricity as efficiently as possible, maximizing energy production and reducing reliance on other energy sources.
