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Consider the two waves shown here (figure 1), which we will consider to represent two electromagnetic radiations.
9. Quantum Mechanics
Wavelength and Frequency
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What is the wavelength necessary to completely remove an electron from the second shell (n = 2) of a hydrogen atom, given the Rydberg constant R∞ = 1.097 × 10^7 m^-1?
A
364 nm
B
656 nm
C
91 nm
D
486 nm
Verified step by step guidance1
Understand that the problem involves calculating the energy required to remove an electron from the second shell (n = 2) of a hydrogen atom, which is known as the ionization energy for that level.
Use the Rydberg formula for hydrogen to calculate the energy difference between the n = 2 level and n = ∞ (ionization): \( E = R_∞ \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( n_1 = 2 \) and \( n_2 = ∞ \).
Since \( \frac{1}{∞^2} = 0 \), the formula simplifies to \( E = R_∞ \left( \frac{1}{2^2} \right) = R_∞ \left( \frac{1}{4} \right) \).
Convert the energy calculated to wavelength using the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \) and \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \). Rearrange to find \( \lambda = \frac{hc}{E} \).
Substitute the values for \( h \), \( c \), and the energy \( E \) calculated from the Rydberg formula into the equation for \( \lambda \) to find the wavelength necessary to ionize the electron from the n = 2 level.
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