In a hydrogen atom, from what energy level must an electron fall to the n=3 level to give a line at 1.28 um (micrometers) in the infrared region?
1/wavelength = R(1/3^2 - 1/n^2)
R is the Rydberg constant = 1.09737E7.
wavelength must be converted to m.
1.28 um = 1.28E-6m
To determine the energy level from which an electron must fall to the n=3 level in a hydrogen atom, we can use the formula for calculating the energy difference between energy levels in hydrogen:
ΔE = 13.6 * (1/n1² - 1/n2²)
Given that the wavelength of the line is 1.28 μm (micrometers) in the infrared region, we can use the equation c = λν, where λ is the wavelength, c is the speed of light, and ν is the frequency.
First, let's convert the wavelength from μm to meters:
1.28 μm = 1.28 * 10^(-6) m
The speed of light, c, is approximately 3.00 * 10^8 m/s.
Using the equation c = λν, we can rearrange it to find the frequency:
ν = c / λ = (3.00 * 10^8 m/s) / (1.28 * 10^(-6) m)
ν ≈ 2.34 * 10^14 Hz
Now, we can calculate the energy difference, ΔE, between the n=3 level and the higher energy level, by inserting the given information into the formula:
ΔE = 13.6 * (1/n1² - 1/n2²)
By comparing the calculated frequency, ν, to the known formula for the energy of a photon, E = hν, where h is Planck's constant (6.626 * 10^(-34) J·s), we can solve for the energy difference, ΔE:
ΔE = hν = (6.626 * 10^(-34) J·s) * (2.34 * 10^14 Hz)
ΔE ≈ 1.55 * 10^(-19) J
Now, we can equate the energy difference, ΔE, to 13.6 * (1/n1² - 1/n2²) and solve for n1:
1.55 * 10^(-19) J = 13.6 * (1/n1² - 1/3²)
1/n1² - 1/9 = (1.55 * 10^(-19) J) / 13.6
Let's solve for n1:
1/n1² = (1.55 * 10^(-19) J) / 13.6 + 1/9
1/n1² ≈ 1.1408 * 10^(-20)
n1² ≈ 1 / (1.1408 * 10^(-20))
n1² ≈ 8.7504 * 10^19
n1 ≈ sqrt(8.7504 * 10^19)
n1 ≈ 9.353
Therefore, the electron must fall from approximately the n = 9 level to the n = 3 level in a hydrogen atom to produce a line at 1.28 μm in the infrared region.
To determine from what energy level an electron must fall to reach the n=3 level and emit light at a specific wavelength, we can use the formula for the energy levels of the hydrogen atom, known as the Rydberg formula:
1/λ = R * (1/n₁² - 1/n₂²)
Where:
- λ is the wavelength of light emitted or absorbed
- R is the Rydberg constant (approximately 1.097 * 10^7 m⁻¹)
- n₁ and n₂ represent the initial and final energy levels, respectively
In this case, we are given the wavelength (1.28 μm) and the final energy level (n=3). We need to determine the initial energy level (n₁).
First, we convert the given wavelength from micrometers to meters:
1.28 μm = 1.28 * 10⁻⁶ m
Next, we substitute the values into the Rydberg formula and solve for n₁:
1 / (1.28 * 10⁻⁶ m) = (1.097 * 10^7 m⁻¹) * (1/n₁² - 1/3²)
The first term on the right side of the equation represents the difference between the reciprocal of the squared initial energy level (1/n₁²) and the reciprocal of the squared final energy level (1/3²).
Simplifying the equation, we have:
1.28 * 10⁶ / 1 = (1.097 * 10^7) * (9/n₁² - 1/9)
Multiplying both sides by n₁² and simplifying further:
n₁² = (9 * (1.097 * 10^7) / (1.28 * 10⁶) + 1/9)
n₁² = 81 + 0.0079844
n₁² ≈ 81.0079844
Taking the square root of both sides, we find:
n₁ ≈ √81.0079844
n₁ ≈ 9.001
Since energy levels are integers, we round n₁ to the nearest whole number:
n₁ ≈ 9
Therefore, the electron must fall from the n=9 energy level to the n=3 energy level to emit light with a wavelength of 1.28 μm in the infrared region.