BIPM-ratified constants · v1.0

Converter

Rydberg, to joule converter calculator.

Convert energy between Rydbergs and Joules using the CODATA factor 2.1798723611035 × 10⁻¹⁸ J/Ry. Supports both Ry→J and J→Ry directions.

From

rydberg

ry_to_j

1 ry_to_j =2.18e-18Converted Energy

Equivalents

Precision: 6 dp · Notation: Decimal · 2 units

→ Joule

Rydbergry_to_j2.18e-18

→ Rydberg

Joulej_to_ry4.59e17

Common pairings

1 ry_to_jequals4.59e17 j_to_ry

1 j_to_ryequals2.18e-18 ry_to_j

The conversion

How the value
is computed.

Understanding the Rydberg Energy Unit and Its Relationship to Joules

The Rydberg (symbol: Ry) is a unit of energy used extensively in atomic and molecular physics. It represents exactly half the energy required to ionize a hydrogen atom from its ground state — a quantity that arises naturally from the mathematics of the Bohr model. Because atomic-scale energies are inconveniently small when expressed in Joules, physicists and chemists routinely work in Rydbergs and convert only when interfacing with SI-unit frameworks. The unit derives its name from Johannes Rydberg, the Swedish physicist who empirically discovered the fundamental frequency patterns in hydrogen's spectrum during the 1880s.

The Joule (J) is the SI unit of energy, defined as one kilogram times metre squared per second squared (kg·m²·s⁻²). While the Joule is indispensable in thermodynamics, engineering, and macroscopic physics, its magnitude makes it impractical for single-electron transitions, where energies fall in the range of 10⁻¹⁸ to 10⁻¹⁹ J. To illustrate the scale problem: the energy released when a single hydrogen atom transitions from the n=2 to n=1 state is approximately 1.6 × 10⁻¹⁸ J — a number so small that writing it routinely introduces transcription errors and obscures physical intuition.

The Rydberg-to-Joule Conversion Formula

The conversion rests on the Rydberg constant times hc (R∞hc), a CODATA-recommended fundamental physical constant. According to NIST CODATA (Rydberg constant times hc in J), the accepted value is:

1 Ry = 2.1798723611035 × 10⁻¹⁸ J

This yields the primary conversion formula:

E_J = E_Ry × 2.1798723611035 × 10⁻¹⁸

Where E_J is the energy in Joules and E_Ry is the energy in Rydbergs.

Reverse Conversion: Joules to Rydbergs

To convert from Joules back to Rydbergs, divide by the same constant:

E_Ry = E_J / 2.1798723611035 × 10⁻¹⁸

Derivation of the Conversion Factor

The Rydberg unit is formally defined as R∞hc, where R∞ is the Rydberg constant (1.0973731568539 × 10⁷ m⁻¹), h is Planck's constant (6.62607015 × 10⁻³⁴ J·s), and c is the speed of light in vacuum (2.99792458 × 10⁸ m/s). Multiplying these three constants together yields the energy of one Rydberg in Joules. The Rydberg constant itself encodes the binding energy of the hydrogen atom and emerges from solving the Schrödinger equation for a single electron orbiting a nucleus. As documented in the CODATA Recommended Values of the Fundamental Physical Constants, this product equals 2.1798723611035 × 10⁻¹⁸ J with a relative standard uncertainty of 1.9 × 10⁻¹², making it one of the most precisely determined values in all of physics. The high precision reflects decades of experimental refinement and the role this constant plays in defining the modern atomic unit system.

Variable Definitions

E_Ry (Energy Value in Rydbergs): The input energy quantity expressed in Rydberg units. One Rydberg equals approximately 13.6057 eV and corresponds to the ionization energy of the hydrogen atom from its n = 1 ground state. Negative Rydberg values indicate bound states below the ionization continuum.
E_J (Energy Value in Joules): The output energy quantity expressed in SI Joules. For reference, the ionization energy of hydrogen from the ground state is exactly 1 Ry = 2.17987 × 10⁻¹⁸ J. This SI representation allows comparison with measurements from calorimetry, spectroscopy, and thermodynamics.
Conversion Direction: Selects whether to apply the forward formula (Ry → J) by multiplying, or the inverse formula (J → Ry) by dividing. Selecting the correct direction prevents factor-of-10¹⁸ mistakes.

Worked Examples

Example 1: Hydrogen Lyman-Alpha Transition

The Lyman-alpha photon corresponds to the n = 2 → n = 1 transition in hydrogen. Its energy is 0.75 Ry (since E = 1 − 1/4 = 0.75 Ry by the Rydberg formula). Converting to Joules:

E_J = 0.75 × 2.1798723611035 × 10⁻¹⁸ = 1.6349 × 10⁻¹⁸ J

Example 2: Helium Ground-State Energy

The non-relativistic ground-state energy of helium is approximately −5.8074 Ry. Converting the magnitude to Joules:

E_J = 5.8074 × 2.1798723611035 × 10⁻¹⁸ = 1.2659 × 10⁻¹⁷ J

Example 3: Reverse Conversion

A photon carries 4.359 × 10⁻¹⁸ J. Converting to Rydbergs:

E_Ry = 4.359 × 10⁻¹⁸ / 2.1798723611035 × 10⁻¹⁸ = 2.000 Ry

Practical Applications

Atomic spectroscopy: Calculating photon energies for hydrogen-like emission and absorption lines, as described in Essential Radio Astronomy (NRAO, Chapter 7). Converting Rydberg-derived transition energies to Joules enables direct comparison with laboratory spectroscopic measurements.
Quantum chemistry: Comparing ab initio electronic structure energies against SI-unit results from calorimetry or thermodynamics experiments. Computational codes like Gaussian and Molpro output energies in Hartrees (2 Ry); conversion to Joules bridges theory and experiment.
Astrophysics: Converting stellar atmospheric energy levels between unit systems for cross-disciplinary spectral analysis. Stellar spectra encode atomic transitions; Rydberg-to-Joule conversion facilitates modeling and data interpretation.
Undergraduate education: Teaching the Bohr model and hydrogen atom quantization, as covered in MIT OpenCourseWare 5.111 Principles of Chemical Science. Using familiar SI units reinforces the connection between atomic theory and SI mechanics.

Reference

Frequently asked questions

What is 1 Rydberg in Joules?

One Rydberg equals exactly 2.1798723611035 × 10⁻¹⁸ Joules, per the CODATA 2018 recommended value published by NIST. This corresponds to the ionization energy of a hydrogen atom from its ground state (n = 1), equivalent to approximately 13.6057 electron-volts. The uncertainty on this value is just 1.9 × 10⁻¹², so it is effectively exact for all practical calculations.

How do you convert Rydbergs to Joules?

Multiply the energy value in Rydbergs by the conversion factor 2.1798723611035 × 10⁻¹⁸ to get the result in Joules. For example, an energy of 3 Ry becomes 3 × 2.1798723611035 × 10⁻¹⁸ = 6.5396 × 10⁻¹⁸ J. This Rydberg constant to Joule converter automates that multiplication, eliminating manual exponent arithmetic and transcription errors.

Why do atomic physicists use Rydbergs instead of Joules?

Single-electron atomic energies are on the order of 10⁻¹⁸ to 10⁻¹⁹ Joules, making the SI unit awkward for day-to-day calculations. The Rydberg unit scales those energies to numbers near 1 — hydrogen's ground state is exactly −1 Ry — which simplifies spectroscopic formulas, reduces the risk of powers-of-ten errors, and aligns directly with the Rydberg formula for hydrogen-like spectra.

What is the difference between a Rydberg and a Hartree?

One Hartree (the atomic unit of energy) equals exactly 2 Rydbergs, or 4.3597447222071 × 10⁻¹⁸ J. Rydbergs dominate spectroscopic literature derived from the Bohr model, while Hartrees are the native energy unit of the atomic units system used in quantum chemistry codes such as Gaussian, ORCA, and MOLPRO. Converting either to Joules requires only a single multiplication by the respective CODATA constant.

How accurate is the Rydberg-to-Joule conversion factor used in this calculator?

The calculator uses the CODATA 2018 value of 2.1798723611035 × 10⁻¹⁸ J per Rydberg, which carries a relative standard uncertainty of 1.9 × 10⁻¹². This precision far exceeds any laboratory measurement of atomic energies, meaning the converter itself introduces no meaningful rounding error. Results are accurate to at least 12 significant figures, sufficient for the most demanding spectroscopic or computational chemistry applications.

Can this converter handle energies for multi-electron atoms or molecules?

Yes. The Rydberg-to-Joule conversion factor is a universal physical constant, not restricted to hydrogen. Computational quantum chemistry codes routinely report total electronic energies for multi-electron systems — helium, benzene, proteins — in Rydbergs or Hartrees. Entering any such value and multiplying by 2.1798723611035 × 10⁻¹⁸ correctly converts it to Joules for comparison with thermochemical measurements or spectroscopic data.