Answer :
Final answer:
To find the wavelength for the electron transition in a hydrogen atom from n_i = 4 to n_f = 7, the Rydberg formula is applied. After recalculating, a wavelength of 2165 nm was found, which does not match the listed options, indicating an error in the initial calculation.
Explanation:
To calculate the wavelength of the spectral line corresponding to the electron transition from ni = 4 to nf = 7 in a hydrogen atom, we can use the Rydberg formula which is given by:
1/λ = R (1/nf2 - 1/ni2),
where λ is the wavelength, R is the Rydberg constant (approximately 1.097 x 107 m-1), nf is the final energy level, and ni is the initial energy level.
Plugging the values into the equation:
1/λ = 1.097 x 107 (1/72 - 1/42)
1/λ = 1.097 x 107 (1/49 - 1/16)
1/λ = 1.097 x 107 (0.020408 - 0.0625)
1/λ = 1.097 x 107 x -0.042092
1/λ = -461710.4 m-1
λ = -1 / -461710.4 m-1
λ = 2.165 x 10-6 m, or 2165 nm
Since this value does not match any of the answer choices, our calculation might have gone wrong. Let's reconsider the Rydberg formula and the calculation process and remember that energy levels are whole numbers and the transition should be from a higher to a lower energy state.