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Chapter 7: Problem 68

An excited hydrogen atom emits light with a wavelength of \(397.2 \mathrm{~nm}\) to reach the energy level for which \(n=2\). In which principal quantum level did the electron begin?

Short Answer

Expert verified
The electron began in the \(n_i = 3\) energy level.

Step by step solution

01

1. Define the Rydberg formula for the hydrogen atom

The Rydberg formula for hydrogen atom is given as: \(\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\) where \(R_H\) is Rydberg constant for hydrogen which is approximately 1.097373 * 10^7 m^-1, \(n_i\) is the initial principal quantum number, \(n_f\) the final principal quantum number, and \(\lambda\) is the wavelength of the emitted light.
02

2. Convert wavelength from nm to meters and plug in the given values

The given wavelength of light is 397.2 nm. We need to convert it into meters: \(397.2\, nm = 397.2 * 10^{-9}\, m\) Now, we can plug given values into the Rydberg formula: \(\frac{1}{397.2 * 10^{-9} \,m} = 1.097373 * 10^7 m^{-1} \left(\frac{1}{2^2} - \frac{1}{n_i^2}\right)\)
03

3. Solve the equation for \(n_i\)

Now, we need to solve the equation for \(n_i\), the initial principal quantum number. First, simplify by multiplying out the constant term: \(\frac{1}{397.2 * 10^{-9}\, m} = 1.097373 * 10^7 m^{-1} \left(\frac{1}{4} - \frac{1}{n_i^2}\right)\) Next, divide both sides by the Rydberg constant for hydrogen: \(\frac{1}{4 * 397.2 * 10^{-9}\, m} - \frac{1}{397.2 * 10^{-9}\, m} = -\frac{1}{n_i^2}\) Now, we calculate the numeric value on the left side: \(\frac{1}{4 * 397.2 * 10^{-9}\, m} - \frac{1}{397.2 * 10^{-9}\, m} \approx -0.1875\) So, we have: \(-0.1875 = -\frac{1}{n_i^2}\) Now, multiply both sides by -1 and invert both sides: \(\frac{1}{0.1875} = n_i^2\)
04

4. Calculate the initial principal quantum number \(n_i\)

Now, we can find \(n_i\) by taking the square root of both sides: \(n_i = \sqrt{\frac{1}{0.1875}} \approx 2.3\) Since the initial principal quantum number must be an integer greater than the final principal quantum number (which is 2), the electron began in the \(n_i = 3\) energy level.

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