E.1.4 Bohr model and quantized energy levels (HL only)

Energy Levels, Quantization of Angular Momentum, and the Bohr Model

1. While standing in a dark room with a prism and a glowing tube of hydrogen gas, you pass the light from the tube through the prism.
2. It splits into distinct, sharp lines of color—red, green, blue, and violet.
3. These lines are not random; they are the "fingerprints" of hydrogen, revealing something profound about the atom's structure.

Why are these lines discrete? What secrets do they hold about the atom?

Energy Levels in Hydrogen

1. 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.

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 ($n=1,2,3,\dots$).

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:

$$E_{\text{photon}}=-1.51\text{eV}-(-3.40\text{eV})=1.89\text{eV}$$

Quantization of Angular Momentum

1. Bohr's model also introduced a bold idea: the angular momentum of the electron is quantized.
2. 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:

• $L$ is the angular momentum,
• $m$ is the mass of the electron,
• $v$ is the speed of the electron,
• $r$ is the radius of the orbit,
• $n$ is the principal quantum number ($n=1,2,3,\dots$),
• $h$ is Planck's constant.

Spectral Predictions

Bohr's model accurately predicts the emission spectrum of hydrogen, including the Balmer series (visible light) and other spectral series in ultraviolet and infrared regions.

How Does the Bohr Model Explain Spectra?

1. Discrete Energy Levels:
   • Electrons can only occupy specific energy levels.
2. Transitions:
   • When an electron transitions between levels, it emits or absorbs a photon with energy equal to the difference between the levels.
3. Unique Wavelengths:
   • Each transition corresponds to a photon of a specific wavelength, creating the discrete lines observed in the hydrogen spectrum.

Limitations of the Bohr Model

While the Bohr model was revolutionary, it has significant limitations:

1. Inapplicability to Multi-Electron Atoms:
   • The model only works for hydrogen and hydrogen-like ions (e.g., $H{e}^{+}$, where there is a single electron.
2. Lack of Explanation for Angular Momentum Quantization:
   • Bohr provided no theoretical basis for why angular momentum is quantized.
3. Fails to Account for Fine Structure:
   • The model cannot explain the subtle splitting of spectral lines observed in high-resolution spectra.
4. Incompatibility with Quantum Mechanics:
   • The Bohr model is a semi-classical theory and does not incorporate the wave-particle duality of electrons or the probabilistic nature of quantum mechanics.

Reflection and Connections

The Bohr model laid the foundation for modern quantum mechanics, even though it was eventually replaced by the more sophisticated Schrödinger model. It introduced the idea of quantization, a concept central to understanding atomic and subatomic systems.