What’s the rearranged Rydberg(Bohr) equation applied to find the ni or nf?
The correct Answer and Explanation is:
The Rydberg equation, also known as the Bohr equation, describes the wavelengths of spectral lines in many chemical elements. It is commonly expressed as:
[
\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} – \frac{1}{n_i^2} \right)
]
where:
- (\lambda) is the wavelength of the emitted or absorbed light,
- (R) is the Rydberg constant ((R \approx 1.097 \times 10^7 \, \text{m}^{-1})),
- (n_i) is the principal quantum number of the initial energy level,
- (n_f) is the principal quantum number of the final energy level.
To rearrange the Rydberg equation to solve for either (n_i) or (n_f), we can start with the original equation:
- Finding (n_f):
Rearranging to solve for (n_f):
[
\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} – \frac{1}{n_i^2} \right)
]
This can be rewritten as:
[
\frac{1}{n_f^2} = \frac{1}{\lambda R} + \frac{1}{n_i^2}
]
Now, isolating (n_f):
[
n_f^2 = \frac{1}{\frac{1}{\lambda R} + \frac{1}{n_i^2}}
]
Therefore:
[
n_f = \sqrt{\frac{1}{\frac{1}{\lambda R} + \frac{1}{n_i^2}}}
]
- Finding (n_i):
Similarly, to solve for (n_i), we can manipulate the equation:
[
\frac{1}{n_i^2} = \frac{1}{\lambda R} – \frac{1}{n_f^2}
]
This gives us:
[
n_i^2 = \frac{1}{\frac{1}{\lambda R} – \frac{1}{n_f^2}}
]
Thus:
[
n_i = \sqrt{\frac{1}{\frac{1}{\lambda R} – \frac{1}{n_f^2}}}
]
Explanation
The Rydberg equation is pivotal in atomic physics, especially when examining the hydrogen atom’s spectral lines. It illustrates the quantized nature of electron transitions between energy levels. When an electron in a hydrogen atom transitions from a higher energy level ((n_i)) to a lower one ((n_f)), it emits light at a specific wavelength. The Rydberg formula quantifies this relationship, allowing us to predict the wavelengths of light emitted during these transitions.
By rearranging the equation, we can isolate either (n_i) or (n_f), depending on which energy level is unknown. This flexibility is crucial in various applications, including spectroscopy and quantum mechanics, where understanding energy transitions and the resulting electromagnetic radiation is essential.