Radii & Velocity Derivations

Consider an electron of mass $m_e$ and charge $-e$ moving in a circular orbit of radius $r$ around a nucleus of charge $+Ze$. Two conditions must be satisfied simultaneously. First, the electrostatic (Coulomb) attraction provides exactly the centripetal force needed to keep the electron in circular motion: $\dfrac{Ze^2}{4\pi\varepsilon_0 r^2} = \dfrac{m_e v^2}{r}$. Rearranging gives $m_e v^2 r = \dfrac{Ze^2}{4\pi\varepsilon_0}$, which links the electron's speed $v$ and orbit radius $r$.

Bohr postulated that only orbits where the electron's angular momentum is an integer multiple of the reduced Planck constant $\hbar = h/(2\pi)$ are allowed: $L = m_e v r = n\hbar$, where $n = 1, 2, 3, \ldots$ is the principal quantum number. This is the radical quantum postulate that breaks from classical mechanics. Solving for $v$: $v = \dfrac{n\hbar}{m_e r}$.

Substitute $v = \dfrac{n\hbar}{m_e r}$ into the force-balance equation $m_e v^2 r = \dfrac{Ze^2}{4\pi\varepsilon_0}$. This gives $m_e \left(\dfrac{n\hbar}{m_e r}\right)^2 r = \dfrac{Ze^2}{4\pi\varepsilon_0}$, which simplifies to $\dfrac{n^2\hbar^2}{m_e r} = \dfrac{Ze^2}{4\pi\varepsilon_0}$. Solving for $r$: $r_n = \dfrac{4\pi\varepsilon_0 \hbar^2}{m_e e^2} \cdot \dfrac{n^2}{Z} = a_0 \dfrac{n^2}{Z}$. The constant $a_0 = \dfrac{4\pi\varepsilon_0\hbar^2}{m_e e^2} \approx 0.529$ Å is known as the Bohr radius. This immediately reveals $r_n \propto n^2/Z$: orbits grow as the square of $n$ and shrink for heavier nuclei.

Now substitute $r_n = a_0 n^2/Z$ back into the angular momentum condition $v = n\hbar/(m_e r)$: $v_n = \dfrac{n\hbar}{m_e \cdot a_0 n^2/Z} = \dfrac{\hbar Z}{m_e a_0 n}$. Since $a_0 = \hbar^2/(m_e k e^2)$ where $k = 1/(4\pi\varepsilon_0)$, the combination $\hbar/(m_e a_0) = ke^2/\hbar \equiv v_1 \approx 2.187 \times 10^6$ m/s, the ground-state speed in hydrogen. Thus $v_n = v_1 \dfrac{Z}{n}$, confirming $v_n \propto Z/n$. Electrons move faster in higher-$Z$ atoms and slower in higher shells.

The two derived proportionalities have clear physical meanings. $r_n \propto n^2/Z$: increasing $n$ spreads the orbit further from the nucleus (electron cloud expands), while increasing $Z$ pulls the orbit inward (stronger nuclear attraction). $v_n \propto Z/n$: a more highly charged nucleus accelerates the electron to higher speed, while moving to a higher shell requires the electron to slow down to satisfy the angular momentum condition. For hydrogen ($Z=1$) in the ground state ($n=1$): $r_1 = 0.529$ Å and $v_1 = 2.187 \times 10^6$ m/s, which is about 0.73% the speed of light — fast but still non-relativistic, validating the classical mechanical approach.

• r_n — Orbital radius of nth shell (metres (m) or ångströms (Å)): Distance from the nucleus to the electron in the nth allowed orbit for a nucleus of charge Z.
• a_0 — Bohr radius (Å (1 Å = 10⁻¹⁰ m)): The smallest possible orbit radius in hydrogen; $a_0 = 4\pi\varepsilon_0\hbar^2/(m_e e^2) \approx 0.529$ Å.
• n — Principal quantum number (dimensionless): A positive integer (1, 2, 3, …) labelling the allowed orbital shells; n=1 is the ground state.
• Z — Atomic number (dimensionless): Number of protons in the nucleus; determines the nuclear charge $+Ze$ experienced by the electron.
• v_n — Orbital velocity of nth shell (m/s): The speed of the electron in the nth allowed orbit; it depends on nuclear charge and shell number.
• v_1 — Ground-state orbital speed (hydrogen) (m/s): $v_1 = e^2/(4\pi\varepsilon_0\hbar) \approx 2.187 \times 10^6$ m/s; also equals $\alpha c$ where $\alpha \approx 1/137$ is the fine-structure constant.

Problem. Calculate (a) the radius and (b) the orbital speed of the electron in the He⁺ ion (Z = 2) in its n = 3 shell. Express the radius in ångströms and the speed as a fraction of the speed of light.

Answer. Orbital radius r₃ ≈ 2.38 Å; orbital speed v₃ ≈ 1.46 × 10⁶ m/s (≈ 0.49% of the speed of light).

• Bohr Energy Levels: $E_n = -13.6 \text{ eV} \cdot Z^2/n^2$ — The orbital radius and velocity derivations feed directly into calculating the total energy of each Bohr orbit, which explains the hydrogen emission spectrum.
• Quantization of Angular Momentum ($L = n\hbar$) — Bohr's angular momentum postulate is the key step that selects allowed orbits and is used in every step of the radius and velocity derivations.
• Hydrogen Emission Spectrum and the Rydberg Equation — The quantized radii and energies directly explain why hydrogen emits light only at specific wavelengths, giving series like Lyman, Balmer, and Paschen.
• Quantum Numbers and Electron Configuration — The principal quantum number n introduced by Bohr is the foundation for the full set of quantum numbers used in modern atomic orbital theory.
• Coulomb's Law and Electrostatic Force — The Coulomb force between nucleus and electron is the centripetal force in Bohr's model; understanding its form is essential for Step 1 of the derivation.