Bohr Radius & Energy Level Calculator Physics

r = ---, E = ---
Radius (rn): -- Energy (En): --

What is the Bohr Radius & Energy Level Calculator?

Quick Answer: An interactive tool that calculates the radius and energy of an electron in any Bohr orbit for hydrogen-like species using Bohr's model. Input atomic number (Z) and principal quantum number (n) to get radius (rn) and energy (En) instantly. Visualizes the electron orbit.

Theory of Bohr's Model

According to Bohr's theory for hydrogen-like atoms (one electron), the radius of the nth orbit is:
rn = n2 × (a0 / Z)
where a0 = 0.529 Å (Bohr radius).
Energy of the electron in the nth orbit:
En = -13.6 × Z2 / n2 eV
The negative sign indicates bound states. The visual draws a nucleus and a circular orbit to scale (radius proportional to n2/Z).

Step-by-Step Examples

Example 1: Hydrogen Ground State

  1. Set Z=1, n=1. Click Calculate. r1 = 0.529 Å (Bohr radius). E1 = -13.6 eV.
  2. The canvas shows a small orbit around a single proton.

Example 2: Helium ion He+

  1. Z=2, n=1. r1 = 0.529/2 = 0.2645 Å. E1 = -13.6 × 4 = -54.4 eV.
  2. Orbit radius shrinks, energy becomes more negative (more tightly bound).

Example 3: Hydrogen n=3

  1. Z=1, n=3. r3 = 9 × 0.529 = 4.761 Å. E3 = -13.6 / 9 = -1.51 eV.
  2. The orbit is much larger.

Frequently Asked Questions

What is the Bohr radius?

It is the radius of the first orbit (n=1) in hydrogen (Z=1), equal to 0.529 Å (52.9 pm). Denoted by a0.

Why is energy negative?

Negative energy indicates that the electron is bound to the nucleus. Energy zero corresponds to a free electron (n = ∞).

Can this calculator be used for multi-electron atoms?

No, Bohr's model is accurate only for hydrogen-like species (single electron). For other atoms, screening effects make it inaccurate.

How does Z affect the orbit radius?

Radius decreases as Z increases (r ∝ 1/Z), because stronger nuclear attraction pulls the electron closer.

What units are used?

Radius in angstroms (Å), where 1 Å = 10-10 m. Energy in electronvolts (eV).