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Chapter 8: Problem 96

Between which two orbits of the Bohr hydrogen atom must an electron fall to produce light of wavelength \(1876 \mathrm{nm} ?\)

Short Answer

Expert verified
The electron must fall from the \(n_i = 3\) orbit to the \(n_f = 2\) orbit to produce light of the given wavelength.

Step by step solution

01

Convert wavelength to meters

Since \(\lambda\) is given in \(\mathrm{nm}\) (nanometers), it needs to be converted to meters for use in the formula. \(1 \mathrm{nm} = 1 \times 10^{-9} m\). Thus, \(\lambda = 1876 \times 10^{-9} m \approx 1.876 \times 10^{-6} m\).
02

Solve the Rydberg formula for \(n_i\)

Rearrange the Rydberg formula to solve for \(n_i\): \(\frac{1}{n_i^2} = \frac{1}{\lambda R} + \frac{1}{n_f^2}\).
03

Substitute the known values into the formula and solve for \(n_i\) for each \(n_f\)

As we are only dealing with hydrogen, \(n_f\) will start from 1 (lowest possible orbit) and increment until we find \(n_i\). Plugging in the value of \(\lambda = 1.876 \times 10^{-6} m\) and \(R = 1.097 \times 10^7 m^{-1}\), we calculate the value of \(n_i\) for each \(n_f\). Since \(n_i\) and \(n_f\) are quantum numbers, they can only be positive integers, so we round \(n_i\) to the nearest integer. The correct \(n_f\) is the one that gives us a value of \(n_i\) greater than \(n_f\). Repeat this until we identify the correct values.

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Most popular questions from this chapter

Select the correct answer and explain your reasoning. An electron having \(n=3\) and \(m_{\ell}=0\) (a) must have \(m_{s}=+\frac{1}{2} ;(\mathbf{b})\) must have \(\ell=1 ;(\mathbf{c})\) may have \(\ell=0,1\) or \(2 ;\) (d) must have \(\ell=2\).

Show that the probability of finding a \(3 d_{x z}\) electron in the \(x y\) plane is zero.

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.

A standing wave in a string \(42 \mathrm{cm}\) long has a total of six nodes (including those at the ends). What is the wavelength, in centimeters, of this standing wave?

Use the Balmer equation (8.2) to determine (a) the frequency, in \(s^{-1}\), of the radiation corresponding to \(n=5\) (b) the wavelength, in nanometers, of the line in the Balmer series corresponding to \(n=7\) (c) the value of \(n\) corresponding to the Balmer series line at \(380 \mathrm{nm}\)
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