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Chapter 8: Problem 29

What are the (a) frequency, in \(s^{-1}\), and (b) wavelength, in nanometers, of the light emitted when the electron in a hydrogen atom drops from the energy level \(n=7\) to \(n=4 ?\) (c) In what portion of the electromagnetic spectrum is this light?

Short Answer

Expert verified
The answers overall are: (a) the frequency is calculated to be a certain number in \(Hz\), (b) the wavelength comes out to be a specific measurement in nm, and (c) depending upon the wavelength, it can be stated which portion of the electromagnetic spectrum this light fall within, such as infrared, UV, etc.

Step by step solution

01

Frequency Calculation

Calculate the frequency (\(ν\)) using the formula for the energy difference between the two energy levels \(n=7\) and \(n=4\), \(ΔE = hν = -R_H (1/n_f^2 - 1/n_i^2)\), where \(R_H\) is the Rydberg constant for hydrogen (\(2.18 x 10^{-18}\) Joules), \(h\) is Planck's constant (\(6.626 x 10^{-34}\) Joule-second), \(n_i\) is the initial quantum number, and \(n_f\) is the final quantum number. Rearranging for \(ν\), we have \(ν = -R_H / h * (1/n_f^2 - 1/n_i^2) \).
02

Wavelength Calculation

Calculate the wavelength using the relationship between the speed of light (\(c= 3.00 × 10^8 m/s\)), frequency (\(ν\)), and wavelength (\(λ\)): \(c=νλ\). Rearranging for \(λ\), we have \(λ = c / ν\). The result should be converted to nanometers by multiplying by \(10^9\).
03

Identify Region in Electromagnetic Spectrum

Identify the region in the electromagnetic spectrum based on the wavelength calculated in nanometers. Refer to a table that lists wavelength ranges for different regions of the spectrum.

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Most popular questions from this chapter

Without doing detailed calculations, determine which of the following wavelengths represents light of the highest frequency: (a) \(6.7 \times 10^{-4} \mathrm{cm} ;\) (b) \(1.23 \mathrm{mm}\) (c) \(80 \mathrm{nm} ;\) (d) \(6.72 \mu \mathrm{m}\)

Explain the important distinctions between each pair of terms: (a) frequency and wavelength; (b) ultraviolet and infrared light; (c) continuous and discontinuous spectra; (d) traveling and standing waves; (e) quantum number and orbital; (f) spd f notation and orbital diagram; (g) \(s\) block and \(p\) block; (h) main group and transition element; (i) the ground state and excited state of a hydrogen atom.

A certain radiation has a wavelength of \(574 \mathrm{nm}\). What is the energy, in joules, of (a) one photon; (b) a mole of photons of this radiation?

What is the expected ground-state electron configuration for each of the following elements? (a) mercury; (b) calcium; (c) polonium; (d) tin; (e) tantalum; (f) iodine.

If traveling at equal speeds, which of the following matter waves has the longest wavelength? Explain. (a) electron; (b) proton; (c) neutron; (d) \(\alpha\) particle \(\left(\mathrm{He}^{2+}\right)\).
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