Determine which H atom transition(iinitital, nfinal) was responsible for the absorption of a photon with a frequency of 4.57 x 10^14 Hz. [RH = 2.18 x 10^-18 J; h = 6.63 x 10^-34 J s]
(a) First find the transition energy from the frequency by using the Planck equation:
deltaE = (6.63x10^-34Js)(frequency)
(b) Use the equation below:
deltaE = (-2.18x10^-18J)(1/b^2 - 1/a^2)
a and b are small integers with b>a
Calculate deltaE using trial values for b and a and repeat this calculation until the value you get matches the value you got for part (a). Start with b = 3 and a = 2. If it doesn't work try other combinations.
E = hf. You have h and frequency. Calculate E (in Joules) for the transition.
I think the easiest way to go about this is to calculate E for the energy levels of the H atom. It's a little work.
E = 2.180 x 10^-18/N^2.
Now substitute 1 for N and calculate E.
Substitute 2 for N and calculate E.
Substitute 3 for N and calculate E.
Do this for N = 4, 5, 6, and 7 (probably that's enough). Now subtract E2-E1, E3-E1, E3-E2, etc. You get the idea. Continue until you find a difference between energy levels that equals the energy you calculated in the first step for the radiation of 4.57 x 10^14 Hz.
Why did the photon go to therapy? Because it had some frequency issues!
To determine the initial and final energy levels involved in the hydrogen atom transition responsible for the absorption of the photon with a frequency of 4.57 x 10^14 Hz, we can use the equation:
E = hf
Where:
E is the energy of the photon,
h is Planck's constant (6.63 x 10^-34 J s),
f is the frequency of the photon.
Rearranging the equation to solve for the energy:
E = hf
E = (6.63 x 10^-34 J s) * (4.57 x 10^14 Hz)
Calculating the energy:
E = 3.02 x 10^-19 J
Now, we can use the formula for the energy of a hydrogen atom transition:
ΔE = -RH [(1/ni^2) - (1/nf^2)]
Where:
ΔE is the difference in energy between the initial and final states,
RH is the Rydberg constant (2.18 x 10^-18 J),
ni is the initial energy level,
nf is the final energy level.
Rearranging the equation to solve for ni and nf:
ΔE = -RH [(1/ni^2) - (1/nf^2)]
ΔE / -RH = (1/ni^2) - (1/nf^2)
(1/ni^2) = ΔE / -RH + (1/nf^2)
ni^2 = 1 / (ΔE / -RH + (1/nf^2))
ni = sqrt(1 / (ΔE / -RH + (1/nf^2)))
Substituting the values:
ni = sqrt(1 / (3.02 x 10^-19 J / -2.18 x 10^-18 J + (1/nf^2)))
ni = sqrt(1 / (-0.1388 + (1/nf^2)))
Now, let's calculate the possible initial energy levels, starting with nf = 2:
ni = sqrt(1 / (-0.1388 + (1/2^2)))
ni ≈ 1.66
So, for the absorption of a photon with a frequency of 4.57 x 10^14 Hz, the initial energy level was approximately n = 1.66 (rounded to two decimal places).
Remember, though, that in a hydrogen atom, energy levels are quantized, so our result suggests a fractional value for the initial energy level. In reality, the most likely initial energy level would be 2 (ni = 2) since the hydrogen atom has discrete, whole-number energy levels.
To determine the initial and final energy levels of the hydrogen atom transition responsible for the absorption of a photon, we can use the equation:
∆E = hf
Where ∆E is the change in energy, h is Planck's constant (6.63 x 10^-34 J s), and f is the frequency of the photon.
First, let's calculate the change in energy (∆E):
∆E = hf
∆E = (6.63 x 10^-34 J s) * (4.57 x 10^14 Hz)
∆E = 3.03 x 10^-19 J
Next, we can use the Rydberg formula to find the initial and final energy levels (n):
∆E = RH * (1 / n_final^2 - 1 / n_initial^2)
where RH is the Rydberg constant (2.18 x 10^-18 J), and n is the principal quantum number.
Rearranging the formula, we get:
1 / n_initial^2 = 1 / n_final^2 + ∆E / RH
Substituting the values:
1 / n_initial^2 = 1 / n_final^2 + (3.03 x 10^-19 J) / (2.18 x 10^-18 J)
Calculating:
1 / n_initial^2 = 1 / n_final^2 + (1/7)
To solve for the values of n_initial and n_final, we can try different combinations of principal quantum numbers until we find a valid solution. We start from the lowest values of n and work our way up until we obtain a solution.
Let's start with n_initial = 1. We can calculate the corresponding value of n_final using the equation:
1 / n_final^2 = 1 / n_initial^2 + (1/7)
1 / n_final^2 = 1 / 1^2 + (1/7)
1 / n_final^2 = 1 + (1/7)
1 / n_final^2 = 8/7
Taking the reciprocal of both sides:
n_final^2 = 7/8
n_final ≈ 0.94
Since n_final should be an integer, this combination is not valid.
Let's try n_initial = 2:
1 / n_final^2 = 1 / n_initial^2 + (1/7)
1 / n_final^2 = 1 / 2^2 + (1/7)
1 / n_final^2 = 1/4 + (1/7)
1 / n_final^2 = 11/28
Taking the reciprocal:
n_final^2 = 28/11
n_final ≈ 2.13
Again, the value of n_final is not an integer.
Continuing this process, we find that for n_initial = 3:
1 / n_final^2 = 1 / n_initial^2 + (1/7)
1 / n_final^2 = 1 / 3^2 + (1/7)
1 / n_final^2 = 1/9 + (1/7)
1 / n_final^2 = 16/63
Taking the reciprocal:
n_final^2 = 63/16
n_final ≈ 1.99
Finally, we have a valid solution with n_initial = 3 and n_final = 2. Therefore, the hydrogen atom transition responsible for the absorption of the given photon has an initial energy level of 3 and a final energy level of 2.