Question

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

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

Step-by-step solution

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.

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: \(\frac{1}{397.2 * 10^{-9} \,m} = 1.097373 * 10^7 m^{-1} \left(\frac{1}{2^2} - \frac{1}{n_i^2}\right)\)

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: \(\frac{1}{0.1875} = n_i^2\)

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.

Key concepts

Principal Quantum Number

In the world of quantum mechanics, the principal quantum number, often symbolized as \( n \), is a key player in determining the electron's position within an atom. It is an integer that signifies the main energy level of an electron. The larger the value of \( n \), the more energetic the electron is, and the farther away it is from the nucleus.
Understanding this concept is crucial for analyzing electron transitions and the resulting emission spectrum. When an electron in an atom transitions from a higher energy level (initial \( n_i \)) to a lower energy level (final \( n_f \)), it releases energy. This energy is emitted in the form of light, sometimes visible, and can be measured in terms of its wavelength, thanks to the Rydberg formula.
In cases like the one we handled in the problem, finding the initial principal quantum number helps us trace back where the electron journeyed from before dropping to a lower energy state. By calculating \( n_i \), we gain insights into the energy levels and the behavior of the atom.

Wavelength Conversion

Converting wavelengths from nanometers to meters is a small but essential step when applying the Rydberg formula. Wavelengths in the electromagnetic spectrum can be very short, making measurements in meters unwieldy. Hence, wavelengths are often expressed in nanometers (nm), where 1 nm is equivalent to \( 10^{-9} \) meters.
This conversion simplifies the equation application in physics, especially because physical constants like the Rydberg constant are usually given in terms of meters. So, to maintain consistency and accuracy in computations, we convert the given wavelength from nanometers to meters. For example, \( 397.2 \, nm \) converts to \( 397.2 \times 10^{-9} \, m \), to directly plug into the Rydberg formula.
Precision in these calculations is paramount to obtaining accurate results, as even slight differences in units can lead to incorrect quantum number determination, impacting our understanding of the atomic structure.

Emission Spectrum

The emission spectrum is a collection of wavelengths emitted by electrons of an atom as they transition from higher to lower energy levels. Each element has a unique emission spectrum, acting like its fingerprint in the universe. These spectra are typically observed as distinct lines, known as spectral lines, on a dark background.
When the electron in a hydrogen atom descends from one energy level to another lower-lying level, it releases energy as electromagnetic radiation. This released light, characterized by a specific wavelength, composes part of the atom's emission spectrum.
The emissions depend on the changes in the principal quantum number during these transitions. The details, like whether an electron falls from \( n = 3 \) to \( n = 2 \), determine the wavelength of light emitted. Through phenomena like this, we not only understand individual atomic behaviors but also gather data crucial for fields such as astrophysics and quantum mechanics.
By analyzing these emission spectra, scientists can determine the composition of distant stars and galaxies, contributing invaluable information to our understanding of the universe.