Hydrogen-Like Atom Calculator

What Is a Hydrogen-Like Atom?

A hydrogen-like atom is a single electron bound to a nucleus with charge $+Ze$, where $Z$ is the atomic number. Examples include H, He$^+$, Li$^{2+}$, and other one-electron ions. The Bohr model gives simple closed-form energy levels and orbital radii that students and spectroscopists use as first approximations.

Bohr Model Energy and Radius

The energy of level $n$ is:

$$E_n = -\frac{13.6 \cdot Z^2}{n^2} \text{ eV}$$

The Bohr radius for level $n$ is:

$$r_n = \frac{0.0529 \cdot n^2}{Z} \text{ nm}$$

Negative energy values mean the electron is bound to the nucleus. More negative energy indicates a more tightly bound state. Larger $n$ means a farther, higher-energy orbital.

Frequently Asked Questions

What is Z in the Bohr model?

Z is the atomic number, equal to the number of protons in the nucleus. For He$^+$, Z = 2. For Li$^{2+}$, Z = 3. The model assumes only one electron remains.

Why is the energy negative?

Energy is measured relative to a free electron at rest (zero energy). Bound states sit below that reference, so their energies are negative. Ionization requires adding energy equal to the absolute value of $E_n$.

What does principal quantum number n mean?

n labels the main electron shell. n = 1 is the ground state closest to the nucleus. Higher n values correspond to excited states with larger radius and less negative energy.

How accurate is the Bohr model?

It works well for one-electron ions and gives correct order-of-magnitude energies. Multi-electron atoms need quantum mechanical models because electron-electron interactions are significant.

What is the ground state of hydrogen?

For hydrogen (Z = 1) in the ground state (n = 1), $E_1 = -13.6$ eV and $r_1 = 0.0529$ nm, also called the Bohr radius $a_0$.

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What Is a Hydrogen-Like Atom?

Bohr Model Energy and Radius

Frequently Asked Questions

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