Quick answer
The Rydberg equation gives hydrogen's spectral lines: 1/λ = R_H (1/n₁² − 1/n₂²) with R_H = 1.0973731568160 × 10⁷ m⁻¹. For n₁ = 2 and n₂ = 3, λ = 656.112 nm (Balmer series, red), and the photon energy is 1.8897 eV.
Enter the lower and upper levels. The calculator returns wavelength, series and photon energy.
Related tools: Circle Equation Calculator: Equation of a circle from its centre and radius. Linear Equation Calculator: Solve ax + b = c for x. Net Ionic Equation Calculator: Balanced molecular, complete ionic and net ionic equations for common double-displacement reactions in water.
How to use the Rydberg Equation Calculator
- Enter lower level n₁ (a whole number, between 1 and 100).
- Enter upper level n₂ (a whole number, between 2 and 200).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Lower level n₁ | A whole number | ≥ 1, ≤ 100 |
| Upper level n₂ | A whole number | ≥ 2, ≤ 200 |
Formula and method
Formula
1 ÷ λ = R_H (1 ÷ n₁² − 1 ÷ n₂²), R_H = 1.0973731568160×10⁷ m⁻¹
IEEE-754 double precision; shown to the decimals indicated.
Hydrogen-like transitions with n₂ greater than n₁.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Lower level n₁ = 2; Upper level n₂ = 3.
| Result | Value |
|---|---|
| Wavelength | 656.112 nm |
| Series | Balmer |
| Photon energy | 1.8897 eV |
Example 2
Inputs: Lower level n₁ = 1; Upper level n₂ = 2.
| Result | Value |
|---|---|
| Wavelength | 121.502 nm |
| Series | Lyman |
| Photon energy | 10.2043 eV |
Example 3
Inputs: Lower level n₁ = 3; Upper level n₂ = 7.
| Result | Value |
|---|---|
| Wavelength | 1,004.672 nm |
| Series | Paschen |
| Photon energy | 1.2341 eV |
The first line is the headline result (wavelength). Below it the rydberg Equation Calculator lists series and photon energy, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Series
n₁ = 1 is the Lyman series (ultraviolet), n₁ = 2 Balmer (visible), n₁ = 3 Paschen (infrared). The 3→2 line at 656 nm is the red line H-alpha.
Energy
The photon energy is E = hc ÷ λ. For 656.112 nm that is 1.8897 eV. The equation works exactly for hydrogen and approximately for hydrogen-like ions with a Z² factor.
How the result changes with upper level n₂
Holding every other input at the first worked example, this table changes only upper level n₂. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Upper level n₂ | Wavelength | Series | Photon energy |
|---|---|---|---|
| 3 | 656.112 nm | Balmer | 1.8897 eV |
| 6 | 410.07 nm | Balmer | 3.0235 eV |
| 12 | 374.921 nm | Balmer | 3.3069 eV |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Upper level n₂: entering "2" shows "The upper level must be greater than the lower level." beside that field and no result.
- Lower level n₁: entering "0" shows "Lower level n₁ must be at least 1." beside that field and no result.
Choosing between the Rydberg Equation Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Circle Equation Calculator](/math/circle-equation/) | centre x, centre y and radius | Standard form |
| [Linear Equation Calculator](/math/linear-equation-calculator/) | a, b and c | x |
| [Net Ionic Equation Calculator](/math/net-ionic-equation/) | two aqueous reactants | Net ionic equation |
| [Dew Point Calculator](/science/dew-point/) | air temperature and relative humidity | Dew point |
| [Acceleration Calculator](/science/acceleration-calculator/) | initial velocity, final velocity and time | Acceleration |
Assumptions and limitations
- n₂ must exceed n₁
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for upper level n₂ and lower level n₁ to confirm this.
Frequently asked questions
What is the Balmer series?
Transitions down to n = 2.
Why is the Rydberg constant positive?
It is defined as a positive wavenumber; the sign of the energy comes from the order of the levels.
Can I use it for helium?
Only for He⁺ (Z = 2).
What does the Rydberg Equation Calculator calculate?
Wavelength of a hydrogen spectral line between two energy levels. It returns Wavelength, Series and Photon energy.
What formula does the Rydberg Equation Calculator use?
1 ÷ λ = R_H (1 ÷ n₁² − 1 ÷ n₂²), R_H = 1.0973731568160×10⁷ m⁻¹. Hydrogen-like transitions with n₂ greater than n₁.
What do I need to enter in the rydberg Equation Calculator?
You enter Lower level n₁ and Upper level n₂.
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. n₂ must exceed n₁.
What is an example of a rydberg Equation result?
With lower level n₁ 2 and upper level n₂ 3, the wavelength is 656.112 nm.
Where does the method come from?
The method follows: CODATA 2018 Rydberg constant.
Related calculators
- Circle Equation Calculator — Equation of a circle from its centre and radius.
- Linear Equation Calculator — Solve ax + b = c for x.
- Net Ionic Equation Calculator — Balanced molecular, complete ionic and net ionic equations for common double-displacement reactions in water.
- Dew Point Calculator — Dew point from temperature and relative humidity.
- Acceleration Calculator — Average acceleration from change in velocity over time.
- Mole Calculator — Convert between mass, moles and number of particles.
Sources
- CODATA 2018 Rydberg constant
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.