Explanation:
When an electron in a hydrogen atom moves from a higher energy level to a lower energy level, it emits energy in the form of electromagnetic radiation. This radiation has a specific wavelength, determined by the difference in energy between the two levels. The relationship between energy and wavelength is given by the equation:

E = hc/λ

where E is energy, h is Planck's constant, c is the speed of light, and λ is wavelength.

In this question, the electron in a hydrogen atom is in the nth excited state. It can move down to the second excited state by emitting ten different wavelengths. We need to find the value of n.

To solve this problem, we can use the formula for the energy levels of a hydrogen atom:

En = -13.6/n^2

where En is the energy of the nth level.

When the electron moves from the nth excited state to the second excited state, the change in energy is given by:

ΔE = E2 - En

where E2 is the energy of the second excited state.

We know that the electron can emit ten different wavelengths, which means that it undergoes ten different transitions from the nth excited state to the second excited state. Each transition corresponds to a different change in energy, and hence a different wavelength.

We can use the formula E = hc/λ to relate the change in energy to the wavelength. Rearranging the formula, we get:

λ = hc/ΔE

Substituting the values of h and c, we get:

λ = 1.24 x 10^-6 / ΔE

where λ is in meters and ΔE is in joules.

Since the electron can emit ten different wavelengths, we have ten different values of ΔE. We can use these values to find the value of n.

Let's consider the first transition. The change in energy is given by:

ΔE1 = E2 - En

Substituting the values, we get:

ΔE1 = -13.6/2^2 - (-13.6/n^2)

Simplifying, we get:

ΔE1 = 13.6(1/4 - 1/n^2)

We know that the wavelength corresponding to this transition is one of the ten wavelengths emitted by the electron. Let's call this wavelength λ1. Using the formula λ = hc/ΔE, we can relate the wavelength to the change in energy:

λ1 = 1.24 x 10^-6 / ΔE1

Substituting the value of ΔE1, we get:

λ1 = 1.24 x 10^-6 / 13.6(1/4 - 1/n^2)

Simplifying, we get:

λ1 = 0.0912(n^2 - 4)

We know that λ1 is one of ten different wavelengths, so we can write:

λ1 = a(n^2 - 4)

where a is a constant. We can use this equation to find the value of n.

Let's consider the second transition. Using the same method as before, we can write:

ΔE2 = 13.6(1/9 - 1/n^2)
λ2 = 0.0912(n^2 - 9)

We can continue this process for all ten transitions, and we will get ten different equations in