Question

if the energy of the photon of light emitted from a hydrogen atom has a magnitude of 2.042 x 10-18 j, and the electron starts at level n = 4, what is the level (n) of the final state of the electron?

Answer 1

The final state of the electron is in level (n=2).

Step-by-step explanation:

The energy levels of electrons in a hydrogen atom are given by the formula
`\(E_n = -\frac{{R_H}}{{n^2}}\)`, where (R_H) is the Rydberg constant
`(\(2.18 * 10^(-18)\ J\))`, and (n) is the principal quantum number representing the energy level.

The change in energy ((Delta E)) between two energy levels is given by
`\(\Delta E = E_i - E_f\)`, where \(E_i\) is the initial energy level and \(E_f\) is the final energy level.

`(\(2.18 * 10^(-18)\ J\))`la to find the final energy level:

`2.042×10 −18 J​ =E i​ −E f​ =− 4 2 2.18×10 −18 J​ + n 2 2.18×10 −18 J​ ​`

Solving for (n), we find that the final state of the electron is in level (n=2).

In summary, the emitted photon's energy corresponds to the difference in energy levels of the electron. By applying the energy level formula and solving for (n), we determine that the electron ends up in level (n=2). This calculation is crucial for understanding the quantum behavior of electrons in atoms.

Answer 2

Given the energy of the photon emitted from a hydrogen atom, we determine the final energy level of the electron by using the Rydberg formula. For the problem provided, with an initial electron level at n=4 and an emitted photon energy of 2.042 x 10^-18 J, the final energy level of the electron is n=2.

Step-by-step explanation:

To find the final energy level (n) of an electron in a hydrogen atom that emits a photon with a given energy, we can use the Rydberg formula for energy levels in a hydrogen atom. This formula relates the photon's energy to the energy levels the electron transitions between:

`E = E_1 - E_2 = -R_H\left((1)/(n_1^2) - (1)/(n_2^2)\right)`

Where E is the energy of the emitted photon, E₁ and E₂ are the energy levels of the initial and final states, R_H is the Rydberg constant (2.18 \times 10^{-18} J), and n₁ and n₂ are the principal quantum numbers of the initial and final states. We are given that E = 2.042 \times 10^{-18} J and n₁ = 4 (the starting level). Our task is to find n₂ (the final level).

First, we rearrange the Rydberg formula to solve for n₂^2:

`n_2^2 = (1)/(n_1^2 - (E)/(R_H))`

Then, plug in the given values:

`n_2^2 = (1)/(4^2 - (2.042 * 10^(-18) J)/(2.18 * 10^(-18) J)) = (1)/(16 - 0.937)`

Calculate n₂^2:

`n_2^2 \approx (1)/(15.063) \approx 0.0664`

Find n₂ by taking the square root:

`n_2 \approx √(0.0664) \approx 0.2578`

Since n₂ must be an integer, we can infer that the actual final state is n = 2, which is the closest integer value below n₂ calculated (you cannot have a fractional principal quantum number). Therefore, the final level (n) of the electron is n = 2.