The radius of the innermost electron orbit of a hydrogen atom is $5.3 imes 10^{-11}\ ext{m}$. What are the radii of the $n = 2$ and $n = 3$ orbits?

Question

The radius of the innermost electron orbit of a hydrogen atom is $5.3 	imes 10^{-11}\ 	ext{m}$. What are the radii of the $n = 2$ and $n = 3$ orbits?

Accepted Answer

> [!TLDR]
> In the Bohr model, the radius of an electron orbit scales as $n^2$ times the Bohr radius $r_1$. Given $r_1 = 5.3 	imes 10^{-11}\ 	ext{m}$, the radii for $n=2$ and $n=3$ are $2.12 	imes 10^{-10}\ 	ext{m}$ and $4.77 	imes 10^{-10}\ 	ext{m}$, respectively.

The Bohr model of the hydrogen atom is built on a simple but powerful idea: the electron can only occupy certain allowed orbits where its angular momentum is an integer multiple of $\frac{h}{2\pi}$. This quantization condition leads directly to a neat scaling law for orbit radii.

For a hydrogen-like atom (one electron orbiting a nucleus of charge $+Ze$), the centripetal force comes from the Coulomb attraction:
$$\frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{r^2}$$
Combine this with Bohr’s quantization of angular momentum:
$$mvr = n\frac{h}{2\pi}$$
Solving these two equations eliminates $v$ and gives the radius of the $n$th orbit:
$$r_n = \frac{4\pi\varepsilon_0 h^2}{4\pi^2 m e^2} \cdot \frac{n^2}{Z} = a_0 \frac{n^2}{Z}$$
where $a_0 = 5.3 	imes 10^{-11}\ 	ext{m}$ is the Bohr radius — the radius of the innermost orbit ($n=1$) for hydrogen ($Z=1$).

> [!FORMULA]
> $$r_n = n^2 \cdot r_1$$

This is the key: the radius grows as the square of the principal quantum number $n$. No need to re-derive constants each time — once you know $r_1$, all higher orbits follow immediately.

Now let’s apply it step by step.

1. **Identify the given data.**  
   The innermost orbit radius $r_1 = 5.3 	imes 10^{-11}\ 	ext{m}$. This is the Bohr radius for hydrogen.

2. **Write the general relation.**  
   For hydrogen ($Z=1$):
   $$r_n = n^2 \cdot r_1$$

3. **Find $r_2$ (for $n=2$).** …

Q.The radius of the innermost electron orbit of a hydrogen atom is $5.3 \times 10^{-11}\ \text{m}$ . What are the radii of the $n = 2$ and $n = 3$ orbits?

Concept understanding — Bohr Model Quantization

Why does an electron not spiral into the nucleus?

The radical idea: allowed orbits only

What this means physically

What it predicts

The key insight for exams

Why this formula?

Why Angular Momentum is Quantised in the Bohr Model

The core problem Bohr faced

The de Broglie wavelength argument (the cleanest derivation)

Why this fixes the energy levels

Method: Direct Application of Bohr's Radius Formula

Q.The radius of the innermost electron orbit of a hydrogen atom is 5.3×10−11 m5.3 \times 10^{-11}\ \text{m}5.3×10−11 m. What are the radii of the n=2n = 2n=2 and n=3n = 3n=3 orbits?

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Q.The radius of the innermost electron orbit of a hydrogen atom is $5.3 \times 10^{-11}\ \text{m}$ . What are the radii of the $n = 2$ and $n = 3$ orbits?

Unlock everything free for 14 days