Textbook Question
Order the following transitions in the hydrogen atom from smallest to largest frequency of light absorbed: n = 3 to n = 6, n = 4 to n = 9, n = 2 to n = 3, and n = 1 to n = 2.
9. Quantum Mechanics
Bohr Equation
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Using the Bohr equation, calculate the wavelength of light emitted when an electron in a hydrogen atom transitions from n=3 to n=1.
A
486 nm
B
656 nm
C
122 nm
D
103 nm
Verified step by step guidance1
Understand the Bohr model: The Bohr model describes the electron transitions between energy levels in a hydrogen atom. When an electron moves from a higher energy level (n=3) to a lower energy level (n=1), it emits energy in the form of light.
Use the Bohr equation to calculate the energy difference between the two levels: The energy difference \( \Delta E \) can be calculated using the formula \( \Delta E = -R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R_H \) is the Rydberg constant (approximately \( 2.18 \times 10^{-18} \) J), \( n_1 \) is the final energy level, and \( n_2 \) is the initial energy level.
Calculate the energy difference: Substitute \( n_1 = 1 \) and \( n_2 = 3 \) into the Bohr equation to find \( \Delta E \).
Relate energy to wavelength: Use the equation \( E = \frac{hc}{\lambda} \) to relate the energy difference to the wavelength of the emitted light, where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Js) and \( c \) is the speed of light (\( 3.00 \times 10^8 \) m/s).
Solve for wavelength: Rearrange the equation to solve for \( \lambda \), the wavelength of the emitted light, using \( \lambda = \frac{hc}{\Delta E} \). Substitute the values for \( h \), \( c \), and \( \Delta E \) to find the wavelength.
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