Transition from n = 2 to n = 1 energy level resulting in emission of photon

To determine the atomic number at which a transition from n = 2 to n = 1 energy level results in the emission of a photon with a wavelength of λ = 3 × 10–8 m, we need to consider the energy levels and transitions involved.

1. Energy levels and transitions
- In the Bohr model of the atom, electrons occupy specific energy levels or orbits around the nucleus.
- The energy levels are labeled by quantum numbers, with the lowest energy level being n = 1, followed by n = 2, n = 3, and so on.
- When an electron transitions from a higher energy level to a lower energy level, it emits a photon with a specific wavelength corresponding to the energy difference between the two levels.

2. Energy of a photon
- The energy of a photon can be calculated using the formula E = hc/λ, where E is the energy, h is Planck's constant (6.626 × 10^(-34) J·s), c is the speed of light (3 × 10^8 m/s), and λ is the wavelength of the photon.
- Rearranging the equation, we get λ = hc/E.

3. Energy difference between energy levels
- The energy difference between two energy levels can be calculated using the formula ΔE = E2 - E1, where ΔE is the energy difference, E2 is the energy of the final level, and E1 is the energy of the initial level.
- In the Bohr model, the energy of an electron in a particular energy level is given by the equation E = -13.6 eV/n^2, where E is the energy in electron volts (eV) and n is the quantum number of the energy level.

4. Calculating the atomic number
- Let's assume the atomic number we are looking for is Z.
- For hydrogen (Z = 1), the energy difference between n = 2 and n = 1 is ΔE = -13.6 eV/2^2 - (-13.6 eV/1^2) = -10.2 eV.
- Converting -10.2 eV to joules, we get ΔE = -10.2 eV × 1.602 × 10^(-19) J/eV = -1.634 × 10^(-18) J.
- Using the equation ΔE = hc/λ, we can solve for λ: λ = hc/ΔE = (6.626 × 10^(-34) J·s × 3 × 10^8 m/s) / (-1.634 × 10^(-18) J) = 3.84 × 10^(-7) m.
- This calculated wavelength is not equal to the given wavelength of λ = 3 × 10^(-8) m, indicating that the atomic number is not 1.

Conclusion:
- The atomic number at which a transition from n = 2 to n = 1 energy level results in the emission of a photon with a wavelength of λ = 3 × 10^(-8) m is not 1.
- Further

Λ-¹ = Z²R(1/n1² - 1/n2²)