Show that the transition corresponding to the line at 434 nm arises from a transition from the  = 5 state. Clearly show all the steps in your working.

The question seems to be related to the Balmer series of the hydrogen spectrum, which describes the emission of light when an electron in a hydrogen atom transitions from a higher energy level (n > 2) to the second energy level (n = 2).

The wavelength of the emitted light (λ) can be calculated using the Rydberg formula:

1/λ = R * (1/2² - 1/n²)

where R is the Rydberg constant (approximately 1.097373 x 10^7 m^-1), n is the principal quantum number of the initial energy level, and λ is the wavelength of the emitted light.

Given that the wavelength of the line is 434 nm (or 434 x 10^-9 m), we can rearrange the Rydberg formula to solve for n:

1/n² = 1/λR + 1/2²

Substituting the given values:

1/n² = 1/(434 x 10^-9 m * 1.097373 x 10^7 m^-1) + 1/4

1/n² = 2.09968 + 0.25

1/n² = 2.34968

Taking the reciprocal and square root of both sides to solve for n:

n = sqrt(1/2.34968)

n ≈ 5

Therefore, the transition corresponding to the line at 434 nm arises from the n = 5 state.