AP Physics 2: Bohr Model of Hydrogen

Before the 20th century, atoms were imagined as tiny, featureless spheres. The discovery of the electron and the atomic nucleus created a puzzle: why didn’t electrons simply spiral into the nucleus, causing all matter to collapse? The emission of light from excited gases, which produced distinct colors rather than a continuous rainbow, presented another mystery. Niels Bohr’s 1913 model of the hydrogen atom solved these puzzles by introducing a revolutionary idea—quantization—where certain physical properties, like an electron’s energy, can only take on specific, discrete values. This model successfully predicted the spectrum of hydrogen and laid the groundwork for modern quantum mechanics.

Historical Context and Bohr's Postulates

The Rutherford nuclear model established the atom as mostly empty space with a dense, positively charged nucleus surrounded by electrons. Classical electromagnetism, however, predicted that an accelerating electron (like one orbiting a nucleus) would continuously emit radiation, lose energy, and crash into the nucleus in a fraction of a second. This contradicted the observed stability of atoms. Furthermore, when hydrogen gas was energized, it emitted light at only specific wavelengths, known as a line spectrum, not a continuous band.

Bohr proposed three radical postulates to reconcile these facts:

1. Quantized Orbits: Electrons orbit the nucleus only in certain stable, stationary states (orbits) without radiating energy.
2. Angular Momentum Quantization: The electron's orbital angular momentum is quantized in integer multiples of the reduced Planck constant: $m_{e}vr=n\hslash$, where $n$ is the principal quantum number (1, 2, 3,...), $m_e$ is the electron mass, $v$ is its speed, $r$ is the orbital radius, and $\hslash =h/(2\pi )$.
3. Photon Emission/Absorption: An electron jumps between stationary states by absorbing or emitting a photon. The photon's energy equals the absolute difference in energy between the two states: $E_{\text{photon}}=|E_f-E_i|=hf$.

These postulates directly explain atomic stability and the origin of line spectra. The electron can only lose energy in discrete "packets" when it jumps to a lower orbit, emitting a photon of a very specific energy and wavelength.

Deriving Orbital Radii and Energy Levels

Bohr combined his quantization condition with classical physics to derive specific predictions for hydrogen. For an electron in a circular orbit around a single proton, two classical equations apply. First, Coulomb's law provides the centripetal force: $$\frac{k{e}^2}{r^2}=\frac{m_e{v}^2}{r}$$ where $k$ is Coulomb's constant and $e$ is the fundamental charge.

Second, Bohr's angular momentum quantization postulate states: $$m_{e}vr=n\hslash$$

We can solve these two equations simultaneously for the orbital radius $r$. Solving the force equation for $v^2$ gives $v^2=k{e}^2/(m_{e}r)$. Substituting into the squared quantization condition ($m_e^2{v}^2{r}^2=n^2{\hslash }^2$) allows us to isolate $r$: $$r_n=\frac{n^2{\hslash }^2}{k{e}^2{m}_e}$$ This is the Bohr radius for the orbit with quantum number $n$. The ground state radius ($n=1$) is a fundamental constant: $$r_1=a_0\approx 5.29\times 10^{-11}\text{ m}$$ Thus, radii increase with the square of $n$: $r_n=n^2{a}_0$.

To find the energy, we calculate the total mechanical energy (kinetic + potential) for an orbit. Kinetic energy is $K=\frac{1}{2}{m}_e{v}^2=\frac{k{e}^2}{2r}$. Potential energy is $U=-\frac{k{e}^2}{r}$. Total energy is therefore: $$E=K+U=-\frac{k{e}^2}{2r}$$

Substituting our expression for $r_n$ gives the quantized energy levels: $$E_n=-\frac{k{e}^2}{2a_0}\cdot \frac{1}{n^2}$$ Evaluating the constants yields the famous formula: $$E_n=\frac{-13.6\text{ eV}}{n^2}$$ where electron-volt (eV) is a convenient energy unit for atomic physics (1 eV = $1.602\times 10^{-19}$ J). The negative sign indicates the electron is bound to the nucleus; zero energy corresponds to being free. The ground state energy ($n=1$) is -13.6 eV.

Predicting Spectral Lines: The Lyman, Balmer, and Paschen Series

When an electron transitions from a higher energy level $n_i$ to a lower level $n_f$, it emits a photon. Using $E_n=-13.6/n^2$ eV, the photon energy is: $$E_{\text{photon}}=13.6\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)\text{ eV}$$ Using the relation between photon energy and wavelength, $E=hc/\lambda$, we can solve for the wavelength of the emitted light: $$\frac{1}{\lambda }=R\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$$ where $R$ is the Rydberg constant, approximately $1.097\times 10^7{\text{ m}}^{-1}$.

This equation predicts all the spectral lines of hydrogen. They are grouped into series based on the final energy level $n_f$:

• Lyman Series: $n_f=1$, $n_i=2,3,4,...$ These transitions involve the largest energy changes, producing photons in the ultraviolet part of the spectrum.
• Balmer Series: $n_f=2$, $n_i=3,4,5,...$ These moderate-energy transitions produce photons in the visible spectrum. The red line at 656 nm (from $n=3$ to $n=2$) is a classic example.
• Paschen Series: $n_f=3$, $n_i=4,5,6,...$ These lower-energy transitions produce photons in the infrared.

Worked Example: Calculate the wavelength of the photon emitted when an electron in a hydrogen atom drops from the $n=4$ state to the $n=2$ state.

1. Identify the series: Final level is $n_f=2$, so this is a Balmer series line.
2. Apply the formula:

$$\frac{1}{\lambda }=R\left(\frac{1}{2^2}-\frac{1}{4^2}\right)=1.097\times 10^7\left(\frac{1}{4}-\frac{1}{16}\right)$$ $$\frac{1}{\lambda }=1.097\times 10^7\left(0.25-0.0625\right)=1.097\times 10^7\times 0.1875$$ $$\frac{1}{\lambda }=2.057\times 10^6{\text{ m}}^{-1}$$

1. Invert to find wavelength:

$$\lambda =\frac{1}{2.057\times 10^6}\approx 4.86\times 10^{-7}\text{ m}=486\text{ nm}$$ This is a blue-green visible light photon, consistent with the Balmer series.

Limitations and Legacy of the Bohr Model

Despite its success with hydrogen, the Bohr model has significant limitations that reveal it is not a complete theory.

• Only Works for Hydrogen-like Atoms: It fails to predict spectra for atoms with more than one electron (like helium). The model cannot account for electron-electron interactions.
• No Explanation for Fine Structure: Higher-resolution spectrometers reveal that individual spectral lines are actually closely spaced doublets. The Bohr model cannot explain this fine structure, which arises from relativistic effects and electron spin—concepts unknown in 1913.
• Contradicts the Uncertainty Principle: The model specifies an electron's exact path (a well-defined orbit with known radius and momentum). Modern quantum mechanics shows this is impossible; we can only describe an electron's location probabilistically as a "cloud" or orbital.
• Hybrid Theory: Bohr's model is a mix of classical and quantum rules. It arbitrarily imposes quantization on a classical framework rather than deriving it from fundamental principles.

The Bohr model's true legacy is as a crucial stepping stone. It correctly introduced the concepts of quantized energy states and quantum jumps, which are central to the fully developed quantum mechanical model that followed. In that model, the electron is described by a wavefunction, and the Bohr energy levels emerge naturally from solving the Schrödinger equation for the hydrogen atom.

Common Pitfalls

1. Misunderstanding the Negative Energy Sign: Students often think negative energy is unphysical. In this context, the negative sign is essential: it indicates a bound system. The ground state ($n=1$) has the lowest (most negative) energy, meaning it is the most tightly bound and stable state. An electron with zero or positive energy is free from the atom.
2. Confusing Energy Absorption and Emission: Remember, an electron absorbs a photon to jump up to a higher $n$ (gaining energy). It emits a photon when it falls down to a lower $n$ (losing energy). A common exam trap is to ask for the wavelength of light absorbed to go from $n=1$ to $n=3$; you must use the same formula but ensure the energy difference is positive.
3. Incorrect Wavelength Calculations: The formula $1/\lambda =R(1/n_f^2-1/n_i^2)$ requires that $n_i>n_f$ for emission. Swapping these numbers will give you a negative reciprocal wavelength, which is a clear sign you've reversed the transition order. Always double-check that your calculated $\lambda$ is positive.
4. Over-applying the Model: The biggest conceptual error is assuming the Bohr model is universally true. It is specifically a model for hydrogen and hydrogen-like ions (e.g., ${\text{He}}^{+}$). Applying its simple formulas to multi-electron atoms will lead to incorrect answers. On the AP exam, if a question mentions an atom other than hydrogen, the Bohr model likely does not apply.

Summary

• The Bohr model postulates stable, quantized orbits where electrons do not radiate, with angular momentum restricted to integer multiples of $\hslash$.
• It successfully derives the hydrogen atom energy levels: $E_n=-13.6/n^2$ eV, and orbital radii: $r_n=n^2{a}_0$, where $a_0$ is the Bohr radius.
• Spectral lines arise from photon emission during electron transitions between levels. The wavelength is given by the Rydberg formula: $1/\lambda =R(1/n_f^2-1/n_i^2)$, defining the Lyman (UV), Balmer (visible), and Paschen (IR) series.
• While groundbreaking, the model is limited: it fails for multi-electron atoms, ignores electron spin and wave nature, and is superseded by the more robust quantum mechanical model.