The Rydberg formula is a mathematical expression used in atomic physics to predict the wavelength of light resulting from an electron moving between energy levels of an atom. Originally formulated by Swedish physicist Johannes Rydberg in 1888, it remains a fundamental tool for understanding the emission and absorption spectra of hydrogen and hydrogen-like elements.
1/λ = R * (1/n₁² – 1/n₂²)
Where:
λ (Lambda): The wavelength of the emitted or absorbed photon.
R: The Rydberg constant (approximately 1.097 x 10⁷ m⁻¹).
n₁: The lower principal quantum number (the destination level for emission).
n₂: The higher principal quantum number (the starting level for emission).
Common Spectral Series
Depending on the value of n₁, different "series" of spectral lines are formed:
Lyman Series (n₁ = 1): Ultraviolet region.
Balmer Series (n₁ = 2): Visible light region.
Paschen Series (n₁ = 3): Infrared region.
Brackett Series (n₁ = 4): Far infrared region.
Practical Example
If an electron in a hydrogen atom drops from the third energy level (n₂ = 3) to the second energy level (n₁ = 2), what is the wavelength of the light emitted?
Using the calculator:
Set n₁ to 2.
Set n₂ to 3.
The result will be approximately 656.3 nm, which corresponds to the H-alpha red line in the visible spectrum.
Why Energy Levels Matter
The Rydberg formula proved that energy is "quantized." Electrons cannot exist between levels; they must jump between them, releasing or absorbing a specific "packet" of energy (a photon). This discovery was crucial in the development of the Bohr model of the atom and modern quantum mechanics.
function calculateRydberg() {
var n1 = parseFloat(document.getElementById('n1Value').value);
var n2 = parseFloat(document.getElementById('n2Value').value);
var R = parseFloat(document.getElementById('rydbergConst').value);
// Constants
var c = 299792458; // Speed of light in m/s
var h = 6.62607015e-34; // Planck's constant in J·s
var evConv = 1.602176634e-19; // 1 eV in Joules
if (isNaN(n1) || isNaN(n2) || isNaN(R)) {
alert("Please enter valid numerical values for all fields.");
return;
}
if (n1 <= 0 || n2 = n2) {
alert("For emission calculation, n2 (Starting Level) must be greater than n1 (Final Level).");
return;
}
// Calculation logic
var factor = (1 / (n1 * n1)) – (1 / (n2 * n2));
var invLambda = R * factor;
var lambdaMeters = 1 / invLambda;
var lambdaNm = lambdaMeters * 1e9;
// Frequency (f = c/λ)
var frequency = c / lambdaMeters;
// Energy (E = hf or E = hc/λ)
var energyJoules = h * frequency;
var energyEV = energyJoules / evConv;
// Display results
document.getElementById('resWavelength').innerText = lambdaNm.toFixed(2);
document.getElementById('resFreq').innerText = frequency.toExponential(4);
document.getElementById('resEnergy').innerText = energyEV.toFixed(4);
document.getElementById('resJoules').innerText = energyJoules.toExponential(4);
document.getElementById('rydbergResult').style.display = 'block';
}