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Chapter 8: Problem 33

What electron transition in a hydrogen atom, starting from the orbit \(n=7,\) will produce light of wavelength \(410 \mathrm{nm} ?\)

Short Answer

Expert verified
The electron transition in a hydrogen atom that will produce light of wavelength \(410nm\) will end in the ground state with principal quantum number \(n_f=2\).

Step by step solution

01

Conversion of Wavelength to Energy

Convert the given wavelength \(410 \mathrm{nm}\) to energy. Energy is related to wavelength by the following relation, \[E = \frac{{hc}}{λ}\] where \(h = 6.62607015 \times 10^{-34} m^2kgs^{-1}\) is Planck’s constant, \(λ = 410nm = 410 \times 10^{-9}m\) is the wavelength of light, and \(c = 3 \times 10^8 ms^{-1}\) is the speed of light.
02

Calculation of Energy

Plugging in the values into the energy formula \[E = \frac{{(6.62607015 \times 10^{-34} m^2kgs^{-1}) \times (3 \times 10^8 ms^{-1})}}{{410 \times 10^{-9}m}} = 4.85 \times 10^{-19} joules\] This value represents the energy difference between two levels in the hydrogen atom.
03

Implementation of Rydberg's Formula

According to the Rydberg’s formula for the energy levels of hydrogen, we have \[∆E = 13.6eV \times \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\] where ∆E is the energy difference, \(n_i\) and \(n_f\) are the initial and final (unknown) quantum numbers.
04

Conversion of Energy Units of Energy Change

Before equating the energy change to the separated energy levels, convert the energy from joules to eV (electron volts). \(1 Joule = 6.242 \times 10^{18} eV\). So, the energy of photon in eV would be \[ E = 4.85 \times 10^{-19} Joules \times 6.242 \times 10^{18} eV/Joule = 3.02 eV \] This is the change in energy.
05

Calculation of Final Quantum Number

Substitute this value and the initial quantum number in the Rydberg’s formula for the hydrogen's energy levels and solve for final quantum level, \(n_f\). \[ 3.02eV = 13.6eV \times \left( \frac{1}{n_f^2} - \frac{1}{7^2} \right) \] Solving for \(n_f\) gives \(n_f=2\) as the principal quantum number for the final state.

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Most popular questions from this chapter

Use the basic rules for electron configurations to indicate the number of (a) unpaired electrons in an atom of \(\mathrm{P} ;\) (b) \(3 d\) electrons in an atom of \(\mathrm{Br} ;\) (c) \(4 p\) electrons in an atom of \(\mathrm{Ge} ;\) (d) \(6 \mathrm{s}\) electrons in an atom of \(\mathrm{Ba}\) (e) \(4 f\) electrons in an atom of Au.

Briefly describe each of the following ideas or phenomena: (a) atomic (line) spectrum; (b) photoelectric effect; (c) matter wave; (d) Heisenberg uncertainty principle; (e) electron spin; (f) Pauli exclusion principle; (g) Hund's rule; (h) orbital diagram; (i) electron charge density; (j) radial electron density.

The magnesium spectrum has a line at \(266.8 \mathrm{nm}\). Which of these statements about this radiation is (are) correct? Explain. (a) It has a higher frequency than radiation with wavelength \(402 \mathrm{nm}\) (b) It is visible to the eye. (c) It has a greater speed in a vacuum than does red light of wavelength \(652 \mathrm{nm}\) (d) Its wavelength is longer than that of X-rays.

Calculate the increase in (a) distance from the nucleus and (b) energy when an electron is excited from the first to the third Bohr orbit.

The Pfund series of the hydrogen spectrum has as its longest wavelength component a line at \(7400 \mathrm{nm}\) Describe the electron transitions that produce this series. That is, give a Bohr quantum number that is common to this series.
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