Use the Rydberg equation to find the wavelength (in \(\mathrm{A}\) ) of the photon absorbed when an electron in an \(\mathrm{H}\) atom undergoes a transition from \(n=1\) to \(n=3\).

Calculating the wavelength of a photon when an electron undergoes a transition in a hydrogen atom is a key concept in quantum mechanics.
The Rydberg equation is instrumental for this purpose.
Here's how you use it:
First, recall that the Rydberg equation for the wavelength \(\text{λ}\) is given by:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Here, \(\text{R}\_H = 1.097 \times 10^7 \ \mathrm{m}^{-1}\) (Rydberg constant), and \(n_1\) and \(n_2\) represent the principal quantum numbers of the energy levels involved in the transition.
Second, substitute the principal quantum numbers into the equation.
For instance, if an electron moves from \(n=1\) to \(n=3\), we get:
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = 1.097 \times 10^7 \left( \frac{8}{9} \right) \]
Simplify and solve for \(\lambda\).
Finally, to find the wavelength in meters, invert the resulting number, and convert to appropriate units if necessary. In this case, the answer is 1025 Å.
Understanding each step can help demystify the process.

In a hydrogen atom, electron transitions between different energy levels play a crucial role in photon emission and absorption.
These energy levels are defined by the principal quantum number \(n\).
The energy required for an electron to transition between these levels can be calculated using the Rydberg formula.
During a transition:

• If an electron moves to a higher energy level \(n_2 \), energy is absorbed.
• If it falls to a lower energy level \(n_1 \), energy is emitted as a photon.

The energy difference between two levels, given by \( \frac{1}{n_1^2} - \frac{1}{n_2^2} \), determines the photon's properties.
For example, a transition from \(n=1\) to \(n=3\) in the hydrogen atom involves a higher absorption of energy which relates to a specific wavelength, as calculational steps demonstrated.

Energy levels in a hydrogen atom are quantized and denoted by the principal quantum number \(n\).
Each level corresponds to a specific energy, decreasing as \(n\) increases.
Specifically, the energy \(E_n\) for each level \(n\) is calculated using:
\[ E_n = - \frac{R_H h c}{n^2} \] where \(h\) is Planck's constant, \(c\) is the speed of light, and \(R_H\) is the Rydberg constant.
An electron at higher \(n\) values is less tightly bound to the nucleus and has higher energy.
When it transitions to a lower \(n\), it releases energy as a photon. Conversely, moving to a higher \(n\) requires energy absorption.
The difference in energy levels explains the observed spectral lines in hydrogen spectral emission or absorption. Understanding these levels is foundational to grasping broader quantum mechanics principles.
Remember, simple transitions such as from \(n=1\) to \(n=3\) already offer deep insights into the nature of atomic energy and photon interactions.