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Chapter 2: Problem 151

Calculate, to four significant figures, the longest and shortest wavelengths of light emitted by electrons in the hydrogen atom that begin in the \(n=5\) state and then fall to states with smaller values of \(n\).

Short Answer

Expert verified
The longest wavelength of light emitted by electrons in the hydrogen atom when transitioning from the \(n=5\) state to a smaller value of \(n\) is approximately \(6.564 \times 10^{-7}\ m\), and the shortest wavelength is approximately \(4.342 \times 10^{-7}\ m\).

Step by step solution

01

Recall the Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of light emitted during electron transitions in hydrogen. The formula is given by: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_j^2} - \frac{1}{n_i^2}\right)\] Here, \(\lambda\) is the wavelength of light emitted, \(R_H\) is the Rydberg constant for hydrogen (\(R_H \approx 1.097\times10^7 m^{-1}\)), \(n_i\) is the initial quantum state, and \(n_j\) is the final quantum state.
02

Calculate the longest wavelength

To calculate the longest wavelength, we'll use the values \(n_i = 5\) and \(n_j = 4\) in the Rydberg formula: \[\frac{1}{\lambda} = R_H \left(\frac{1}{4^2} - \frac{1}{5^2}\right)\] Now, solve for \(\lambda\): \[\lambda = \frac{1}{R_H \left(\frac{1}{4^2} - \frac{1}{5^2}\right)}\] Plug in the value of \(R_H\) and solve for \(\lambda\): \[\lambda \approx \frac{1}{1.097\times10^7 \left(\frac{1}{16} - \frac{1}{25}\right)} \approx 6.564\times10^{-7} m\] The longest wavelength the electron can emit, when falling from \(n=5\) to \(n=4\), is approximately \(6.564\times10^{-7} m\).
03

Calculate the shortest wavelength

To calculate the shortest wavelength, we'll use the values \(n_i = 5\) and \(n_j = 1\) in the Rydberg formula: \[\frac{1}{\lambda} = R_H \left(\frac{1}{1^2} - \frac{1}{5^2}\right)\] Now, solve for \(\lambda\): \[\lambda = \frac{1}{R_H \left(\frac{1}{1^2} - \frac{1}{5^2}\right)}\] Plug in the value of \(R_H\) and solve for \(\lambda\): \[\lambda \approx \frac{1}{1.097\times10^7 \left(\frac{1}{1} - \frac{1}{25}\right)} \approx 4.342\times10^{-7} m\] The shortest wavelength the electron can emit, when falling from \(n=5\) to \(n=1\), is approximately \(4.342\times10^{-7} m\).
04

Express the results to four significant figures

Express the results of the longest and shortest wavelengths to four significant figures: \[\lambda_{longest} \approx 6.564\times10^{-7} m \approx 6.564 \times 10^{-7}\ m\] \[\lambda_{shortest} \approx 4.342\times10^{-7} m \approx 4.342 \times 10^{-7}\ m\] Thus, the longest wavelength of light emitted is approximately \(6.564 \times 10^{-7}\ m\) and the shortest wavelength is approximately \(4.342 \times 10^{-7}\ m\).

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Most popular questions from this chapter

Consider an electron for a hydrogen atom in an excited state. The maximum wavelength of electromagnetic radiation that can completely remove (ionize) the electron from the H atom is \(1460 \mathrm{nm}\). What is the initial excited state for the electron \((n=?) ?\)

Write the expected electron configurations for each of the following atoms: \(\mathrm{Cl}, \mathrm{Sb}, \mathrm{Sr}, \mathrm{W}, \mathrm{Pb}, \mathrm{Cf}\).

Which of the following sets of quantum numbers are not allowed in the hydrogen atom? For the sets of quantum numbers that are incorrect, state what is wrong in each set. a. \(n=3, \ell=2, m_{c}=2\) b. \(n=4, \ell=3, m_{\ell}=4\) c. \(n=0, \ell=0, m_{\ell}=0\) d. \(n=2, \ell=-1, m_{c}=1\)

The first-row transition metals from chromium through zinc all have some biologic function in the human body. How many unpaired electrons are present in each of these first-row transition metals in the ground state?

X rays have wavelengths on the order of \(1 \times 10^{-10} \mathrm{m}\). Calculate the energy of \(1.0 \times 10^{-10} \mathrm{m}\) X rays in units of kilojoules per mole of X rays. (1 mol X rays \(=6.022 \times 10^{23}\) X rays.) AM radio waves have wavelengths on the order of \(1 \times 10^{4} \mathrm{m}\). Calculate the energy of \(1.0 \times 10^{4} \mathrm{m}\) radio waves in units of kilojoules per mole of radio waves. Consider that the bond energy of a carbon- carbon single bond found in organic compounds is 347 kJ/mol. Would X rays and/or radio waves be able to disrupt organic compounds by breaking carbon- carbon single bonds?
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