Q.Calculate the radius of the third orbit of the hydrogen atom. (a0 = 52.9 pm)

Bohr Orbit Radius and Energy — From Intuition to Formula

Imagine an electron trapped near a nucleus. It wants to fall in because of the electric attraction, but if it moved like a classical planet it would spiral inward, radiating energy and crashing. Bohr’s genius was to say: no, the electron can only live in certain allowed orbits — and in those orbits it does not radiate. That’s the starting point.

Why would orbits be fixed? Bohr borrowed an idea from waves. If you tie a string in a loop, only certain wavelengths fit smoothly around it. Similarly, the electron’s wave nature means its orbit must contain an integer number of wavelengths. That integer is $n$, the principal quantum number ($n = 1, 2, 3, \dots$). The smallest orbit ($n=1$) is the ground state; larger $n$ means higher energy, farther out.

Radius of the $n$th orbit

For a hydrogen-like atom (one electron, nuclear charge $+Ze$), the Coulomb force provides the centripetal force:

where $k = \frac{1}{4\pi\varepsilon_0}$ and $m$ is the electron mass. Bohr’s quantization condition says the angular momentum is an integer multiple of $\hbar = h/2\pi$:

Solve these two equations together. From the second, $v = n\hbar/(mr)$. Substitute into the first:

Multiply both sides by $r^3$ and rearrange:

Define the Bohr radius $a_0 = \frac{\hbar^2}{k m e^2} \approx 0.529 \, \text{Å}$ (the radius of the $n=1$ orbit for hydrogen, $Z=1$). Then:

The radius grows as $n^2$ — the $n=2$ orbit is four times larger than $n=1$, $n=3$ is nine times larger, and so on. For higher $Z$, the stronger pull shrinks all orbits by $1/Z$.

Energy of the $n$th orbit

Total energy = kinetic + potential. For a Coulomb force, $K = \frac12 m v^2$ and $U = -\frac{k Z e^2}{r}$ (negative because the force is attractive; zero at infinity). Using the force equation $kZe^2/r^2 = mv^2/r$, we get $mv^2 = kZe^2/r$, so:

Hence:

Now substitute $r_n$ from above:

The constant $\frac{k e^2}{2 a_0}$ is the Rydberg energy $R_y \approx 13.6 \, \text{eV}$. So:

For hydrogen ($Z=1$): $E_1 = -13.6$ eV, $E_2 = -3.4$ eV, $E_3 = -1.51$ eV, … approaching zero as $n \to \infty$.

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