help me with this question
2. A hydrogen electron drops from the n=4 level to the n=1 level.
A. What is the energy of an emitted photon?
B. Calculate the frequency of the photon.
C. Calculate the wavelength of the photon.
Sure! Let's break down the problem step by step:
Step 1: Calculate the energy difference
To find the energy of the emitted photon, we need to calculate the energy difference between the two levels (n=4 and n=1).
The energy difference can be found using the formula:
ΔE = E_final - E_initial
where E_final is the energy of the final level and E_initial is the energy of the initial level.
For a hydrogen atom, the energy levels can be calculated using the formula:
E = -13.6 eV / n^2
where n is the principal quantum number.
Using this formula, we can find the energy levels for n=4 and n=1:
E_final = -13.6 eV / (4^2)
E_initial = -13.6 eV / (1^2)
Step 2: Calculate the energy of the emitted photon
Now that we have the energy difference (ΔE), we can find the energy of the emitted photon using the formula:
E_photon = ΔE
Step 3: Convert the energy of the photon to frequency
The frequency of a photon can be calculated using the formula:
E_photon = h * f
where E_photon is the energy of the photon, h is Planck's constant (6.626 x 10^-34 J s), and f is the frequency of the photon.
Step 4: Calculate the wavelength of the photon
The wavelength of a photon can be calculated using the formula:
λ = c / f
where λ is the wavelength of the photon, c is the speed of light (3 x 10^8 m/s), and f is the frequency of the photon.
Let's calculate each part:
A. What is the energy of the emitted photon?
Step 1: Calculate the energy difference
E_final = -13.6 eV / (4^2) = -13.6 eV / 16 = -0.85 eV
E_initial = -13.6 eV / (1^2) = -13.6 eV
ΔE = E_final - E_initial = -0.85 eV - (-13.6 eV) = 12.75 eV
Step 2: Calculate the energy of the emitted photon
E_photon = ΔE = 12.75 eV
B. Calculate the frequency of the photon
Step 3: Convert the energy of the photon to frequency
E_photon = h * f
12.75 eV = (6.626 x 10^-34 J s) * f
Convert eV to Joules: 1 eV = 1.6 x 10^-19 J
12.75 eV = 12.75 * (1.6 x 10^-19 J/eV) = 2.04 x 10^-18 J
2.04 x 10^-18 J = (6.626 x 10^-34 J s) * f
f = (2.04 x 10^-18 J) / (6.626 x 10^-34 J s) = 3.08 x 10^15 Hz
Therefore, the frequency of the photon is 3.08 x 10^15 Hz.
C. Calculate the wavelength of the photon
Step 4: Calculate the wavelength of the photon
λ = c / f
λ = (3 x 10^8 m/s) / (3.08 x 10^15 Hz) = 9.74 x 10^-8 m
Therefore, the wavelength of the photon is 9.74 x 10^-8 meters.
To answer these questions, we need to use the formulas related to energy, frequency, and wavelength of a photon.
A. The energy of a photon can be calculated using the formula:
E = hc/λ
Where:
E - energy of a photon
h - Planck's constant (6.626 x 10^-34 J·s)
c - speed of light (2.998 x 10^8 m/s)
λ - wavelength of the photon
Since the electron is transitioning from n=4 to n=1, the emitted photon's wavelength will be determined by the difference in energy levels:
ΔE = E_final - E_initial
Assuming initial energy level (n=4) is higher than the final energy level (n=1), ΔE will be positive.
Substituting ΔE into the energy equation, we can solve for E.
B. The frequency of a photon can be calculated using the equation:
f = c/λ
Where:
f - frequency of the photon
c - speed of light
λ - wavelength of the photon
By rearranging the equation, we can solve for the frequency.
C. The wavelength of a photon can be calculated using the equation:
λ = c/f
Where:
λ - wavelength of the photon
c - speed of light
f - frequency of the photon
Now, let's calculate the answers using these formulas.