An electron in a hydrogen atom relaxes to the n= 4 level, emitting light of 138 THz.
What is the value of n for the level in which the electron originated?
a. figure energy for the photon, for the given frequency. E=hf
b. then, E= constant(1/n^2-1/16)
figure n
I can check your work if necessary.
Well, the electron must have come from a pretty relaxed place! I bet it was sipping on a martini on a beach somewhere... but I digress. To find the value of n for the level the electron originated, we can use the Rydberg formula.
The Rydberg formula states that the frequency of light emitted or absorbed is given by the difference in energy levels, so we can use it to find n. The formula is:
1/λ = R(1/n1^2 - 1/n2^2)
Where λ is the wavelength of light, R is the Rydberg constant, and n1 and n2 are the initial and final energy levels, respectively.
Since we know the frequency (138 THz), we can convert it to wavelength (λ) using the speed of light (c = 3 x 10^8 m/s):
λ = c/f
= (3 x 10^8 m/s) / (138 x 10^12 Hz)
≈ 2.17 x 10^-6 m
Now, we can plug in the values into the Rydberg formula and solve for n1:
2.17 x 10^-6 m = R(1/n1^2 - 1/4^2)
We know that R is a constant, so we can rearrange the equation and solve for n1:
1/n1^2 = (2.17 x 10^-6 m) / R + 1/4^2
≈ 0.333 + 0.0625
≈ 0.3955
Taking the reciprocal of both sides:
n1^2 ≈ 1 / 0.3955
≈ 2.5283
Taking the square root:
n1 ≈ √(2.5283)
≈ 1.589
Since n must be a positive integer value, we round the value of n1 to the nearest integer:
n ≈ 2
So, the electron must have originated from the n = 2 level before relaxing to n = 4. It must have wanted a change of scenery, you know, to mix things up!
To determine the value of n for the level in which the electron originated, we can use the relation between the frequency (ν) of light emitted and the energy difference (ΔE) between two energy levels of an electron in a hydrogen atom.
The frequency of light emitted (ν) can be related to the energy difference (ΔE) between two energy levels by the equation:
ΔE = hν
where h is the Planck's constant, which is approximately 6.626 x 10^-34 J·s.
In the case of the hydrogen atom, the energy difference between two energy levels is given by:
ΔE = (13.6 eV) * (1/n₁² - 1/n₂²)
Here, n₁ and n₂ represent the initial and final quantum numbers respectively.
Given that the frequency of light emitted is 138 THz, we can convert it to Hz by multiplying it by 10^12:
ν = 138 THz = 138 × 10^12 Hz
Now, let's find the energy difference (ΔE) using the relation ΔE = hν.
ΔE = (6.626 x 10^-34 J·s) * (138 × 10^12 Hz)
≈ 9.14 x 10^-19 J
Next, substitute the energy difference into the formula for hydrogen atom energy levels to solve for n₁:
9.14 x 10^-19 J = (13.6 eV) * (1/n₁² - 1/4²)
Simplifying the equation:
n₁² = 1/(1 - (9.14 x 10^-19 J) / (13.6 eV))
≈ 1/(1 - 1.35 x 10^-5)
Taking the reciprocal of both sides:
1/n₁² ≈ 1.35 x 10^-5
Simplifying:
n₁² ≈ 1/(1.35 x 10^-5)
≈ 7.41 x 10⁴
Taking the square root of both sides:
n₁ ≈ √(7.41 x 10⁴)
≈ 272
Therefore, the electron originated from the n=272 level.