Energy Levels, Quantization of Angular Momentum, and the Bohr Model
Energy Levels in Hydrogen
In the early 20th century, Niels Bohr proposed that the energy of an electron in a hydrogen atom is quantized, it can only take on specific, discrete values.
NoteThis was a radical departure from classical physics, which suggested that electrons could have any energy.
The Energy Formula
Bohr discovered that the energy of an electron in the $n$-th energy level of a hydrogen atom is given by:
$$E = -\frac{13.6}{n^2} \, \text{eV}$$
where:
- $E$ is the energy of the electron,
- $n$ is the principal quantum number ($n = 1, 2, 3, \dots$).
The negative sign indicates that the electron is bound to the nucleus, meaning energy must be added to free it.
Energy Levels and Transitions
Each value of $n$ corresponds to a specific energy level:
- $n = 1$: Ground state ($E = -13.6 \, \text{eV}$),
- $n = 2$: First excited state ($E = -3.40 \, \text{eV}$),
- $n = 3$: Second excited state ($E = -1.51 \, \text{eV}$),
- and so on...
When an electron transitions from a higher energy level ($n_{\text{high}}$) to a lower one ($n_{\text{low}}$), it emits a photon with energy equal to the difference between the two levels:
$$E_{\text{photon}} = E_{n_{\text{high}}} - E_{n_{\text{low}}}$$
For instance, a transition from $n = 3$ to $n = 2$ releases a photon with energy:
$$E_{\text{photon}} = -1.51 \, \text{eV} - (-3.40 \, \text{eV}) = 1.89 \, \text{eV}$$
Common Mistake- The energy levels in hydrogen are given in electron volts (eV), but when using equations like $E = hf$, energy must be in joules (J) since Planck’s constant ($h$) is in J·s.
- Always convert energy from eV to J using $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$ before applying the formula.
Using the relationship $E_{\text{photon}} = \frac{hc}{\lambda}$, you can calculate the wavelength of the emitted photon.
A hydrogen atom transitions from $n = 3$ to $n = 2$. What is the wavelength of the emitted photon?
Solution
- Energy difference: $$E_{\text{photon}} = 1.89 \, \text{eV} = 1.89 \times 1.6 \times 10^{-19} \, \text{J}$$ $$ = 3.02 \times 10^{-19} \, \text{J}$$
- Wavelength: $$\lambda = \frac{hc}{E_{\text{photon}}} = \frac{(6.63 \times 10^{-34}) (3.0 \times 10^8)}{3.02 \times 10^{-19}} $$ $$\approx 6.58 \times 10^{-7} \, \text{m}$$
- The wavelength is approximately $658 \, \text{nm}$, corresponding to red light in hydrogen's emission spectrum.
- Energy levels are unique to each element.
- For hydrogen, the formula $E = -\frac{13.6}{n^2} \, \text{eV}$ applies because it has only one electron.
Quantization of Angular Momentum
- Bohr's model also introduced a bold idea: the angular momentum of the electron is quantized.
- This means the electron can only occupy specific orbits around the nucleus, each corresponding to a fixed angular momentum.
The Angular Momentum Condition
The angular momentum of the electron is given by:
$$L = mvr = \frac{nh}{2\pi}$$
where:
