Bohr Model Calculator With Mass

Bohr Energy Levels with Mass:

\[ E_n = - \frac{\mu k^2 e^4}{2 \hbar^2 n^2} \]

Reduced Mass (μ):

Quantum Number (n):

(integer)

Coulomb Constant (k):

N·m²/C²

Electron Charge (e):

Energy Level (E_n):

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1. What is the Bohr Model with Mass?

The Bohr model with mass is a modification of the original Bohr model that accounts for the reduced mass (μ) of the electron-nucleus system. This provides more accurate energy levels for atoms where the nucleus isn't infinitely heavy compared to the electron.

2. How Does the Calculator Work?

The calculator uses the modified Bohr formula:

\[ E_n = - \frac{\mu k^2 e^4}{2 \hbar^2 n^2} \]

Where:

\( \mu \) — Reduced mass of the system (kg)
\( k \) — Coulomb constant (8.988×10⁹ N·m²/C²)
\( e \) — Electron charge (1.602×10^-19 C)
\( \hbar \) — Reduced Planck constant (1.055×10^-34 J·s)
\( n \) — Principal quantum number (integer ≥1)

Explanation: The equation gives the quantized energy levels of an electron in a hydrogen-like atom, accounting for the finite nuclear mass through the reduced mass term.

3. Importance of Energy Level Calculation

Details: Calculating accurate energy levels is essential for understanding atomic spectra, predicting emission/absorption lines, and modeling atomic behavior in different environments.

4. Using the Calculator

Tips: Enter reduced mass in kg (9.1044×10^-31 kg for hydrogen), quantum number (n ≥1), Coulomb constant (8.988×10⁹ N·m²/C²), and electron charge (1.602×10^-19 C).

5. Frequently Asked Questions (FAQ)

Q1: What is reduced mass?
A: Reduced mass (μ) accounts for the fact that both electron and nucleus move about their common center of mass: μ = (m_em_N)/(m_e+m_N).

Q2: How does this differ from standard Bohr model?
A: The standard model assumes infinite nuclear mass, while this version uses reduced mass for greater accuracy, especially with lighter nuclei.

Q3: What are typical energy level values?
A: For hydrogen (n=1), E₁ ≈ -2.18×10^-18 J (-13.6 eV). Values become less negative with increasing n.

Q4: Can this be used for multi-electron atoms?
A: No, the Bohr model only works accurately for hydrogen-like (single-electron) systems.

Q5: Why show results in both Joules and eV?
A: Joules are SI units, while electron-volts (eV) are more practical for atomic-scale energies (1 eV = 1.602×10^-19 J).