{"id":66901,"date":"2026-10-13T10:10:12","date_gmt":"2026-10-13T07:10:12","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66901"},"modified":"2026-10-13T10:10:12","modified_gmt":"2026-10-13T07:10:12","slug":"rydberg-equation-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/chemistry\/rydberg-equation-calculator\/","title":{"rendered":"Rydberg Equation Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Find Hydrogen Line Wavelength from Quantum Numbers<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66901\" data-favorite-title=\"Rydberg Equation\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/chemistry\/rydberg-equation-calculator\/\" data-favorite-excerpt=\"Enter the initial and final principal quantum numbers of a hydrogen electron transition and get the wavelength, frequency and photon energy of the spectral line, plus its series (Lyman, Balmer, Paschen\u2026).\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-spectrum\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/chemistry\/rydberg-equation-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66901\" data-post=\"66901\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Initial Level (n\u1d62)<\/label><div class=\"term\">n\u1d62<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"initial_level_n\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 3\" min=\"1\" max=\"100\" value=\"3\"\/><small id=\"calc-info-1\">Principal quantum number of the starting level (higher n for an emission line).<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Final Level (n_f)<\/label><div class=\"term\">n_f<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"final_level_nf\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 2\" min=\"1\" max=\"100\" value=\"2\"\/><small id=\"calc-info-2\">Principal quantum number of the ending level (lower n for an emission line).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66901\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Hydrogen Transition Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Wavelength<\/span><span class=\"result-term\">\u03bb<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"nm\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Frequency<\/span><span class=\"result-term\">\u03bd<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"THz\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Photon Energy<\/span><span class=\"result-term\">E<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"eV\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Photon Energy (SI)<\/span><span class=\"result-term\">E<\/span><\/span><span class=\"field-value\" id=\"result-4\" data-unit=\"J\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-5\"><span class=\"field-title\"><span>Transition Type and Spectral Series<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-6\"><span class=\"field-title\"><span>Calculation Details<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-7\"><span class=\"field-title\"><span>About the Rydberg Equation<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"66901\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Rydberg Equation Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\frac{1}{\\lambda} = R_H \\left| \\frac{1}{n_f^2} - \\frac{1}{n_i^2} \\right| \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(\\lambda\\) \u2014 wavelength of the emitted or absorbed photon (m)<\/li>\n<li>\\(R_H = 1.0967758 \\times 10^7\\ \\text{m}^{-1}\\) \u2014 Rydberg constant for hydrogen (corrected for the reduced mass of the atom)<\/li>\n<li>\\(n_i\\) \u2014 initial principal quantum number of the electron<\/li>\n<li>\\(n_f\\) \u2014 final principal quantum number of the electron<\/li>\n<\/ul>\n<p>The formula gives the inverse wavelength (wavenumber) of the photon produced when an electron in a hydrogen atom jumps between two energy levels. The companion relations \\(\\nu = c\/\\lambda\\) and \\(E = h\\nu = hc\/\\lambda\\) convert the wavelength into frequency and photon energy.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Rydberg Equation \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> an electron in a hydrogen atom falls from level \\(n_i = 3\\) to level \\(n_f = 2\\) (the transition that produces the famous red H\u03b1 line).<\/p>\n<p>\\[ \\frac{1}{\\lambda} = 1.0967758 \\times 10^7 \\times \\left| \\frac{1}{2^2} - \\frac{1}{3^2} \\right| = 1.0967758 \\times 10^7 \\times \\frac{5}{36} = 1.5233 \\times 10^6\\ \\text{m}^{-1} \\]<\/p>\n<p>\\[ \\lambda = \\frac{1}{1.5233 \\times 10^6} \\approx 6.5647 \\times 10^{-7}\\ \\text{m} = 656.47\\ \\text{nm} \\]<\/p>\n<p>\\[ \\nu = \\frac{c}{\\lambda} = \\frac{2.9979 \\times 10^8}{6.5647 \\times 10^{-7}} \\approx 4.568 \\times 10^{14}\\ \\text{Hz} \\]<\/p>\n<p><strong>Result:<\/strong> \\(\\lambda \\approx 656.47\\) nm, \\(\\nu \\approx 456.81\\) THz, photon energy \\(E \\approx 1.89\\) eV \\((3.03 \\times 10^{-19}\\) J) \u2014 the red H\u03b1 line of the Balmer series.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Rydberg Equation Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Initial Level (n\u1d62)<\/strong> \u2014 the principal quantum number the electron starts from. For an emission line this is the higher level (e.g. 3).<\/li>\n<li>Enter the <strong>Final Level (n_f)<\/strong> \u2014 the level the electron ends on. For an emission line this is the lower level (e.g. 2).<\/li>\n<li>Both numbers must be whole numbers from 1 to 100, and they must differ \u2014 identical levels produce no photon.<\/li>\n<li>Read the results: the <strong>Wavelength<\/strong> in nanometres, the <strong>Frequency<\/strong> in terahertz, and the <strong>Photon Energy<\/strong> in both eV and joules, together with the transition type, spectral series and wavelength region.<\/li>\n<\/ol>\n<p>If \\(n_i &gt; n_f\\) the calculator describes an <em>emission<\/em> line (the electron falls and releases a photon); if \\(n_i &lt; n_f\\) it describes an <em>absorption<\/em> line (the electron is lifted up by absorbing a photon of exactly the same wavelength).<\/p>\n<h2>What Is the Rydberg Equation?<\/h2>\n<p>The Rydberg equation, formulated by Johannes Rydberg in 1888, predicts the wavelengths of all the spectral lines of atomic hydrogen. It generalised Balmer's empirical formula for the visible lines into a single expression covering the entire spectrum. In 1913 Niels Bohr showed why it works: the electron in a hydrogen atom can occupy only the discrete levels \\(n = 1, 2, 3, \\ldots\\), and a photon is emitted or absorbed when the electron jumps between them, carrying exactly the energy difference of the two levels.<\/p>\n<p>Because each level has a fixed energy, every pair of levels corresponds to one specific wavelength. This is why hydrogen's spectrum is not a continuous rainbow but a set of sharp lines \u2014 and why those lines act as a fingerprint of hydrogen in laboratory analysis and astronomy alike.<\/p>\n<p>The lines group into <strong>series<\/strong> named after their lower level: the <em>Lyman series<\/em> (to \\(n = 1\\), ultraviolet), the <em>Balmer series<\/em> (to \\(n = 2\\), the visible lines H\u03b1 656 nm, H\u03b2 486 nm, H\u03b3 434 nm and H\u03b4 410 nm), the <em>Paschen<\/em> (to \\(n = 3\\)), <em>Brackett<\/em> (to \\(n = 4\\)) and <em>Pfund<\/em> (to \\(n = 5\\)) series, all in the infrared. As the upper level grows, the lines crowd together and converge to the series limit, beyond which the atom is ionised. The calculator identifies the series automatically and recognises several famous lines such as H\u03b1 and Lyman-alpha.<\/p>\n<p>The photon energy is worth interpreting physically: \\(E \\approx 1.89\\) eV for H\u03b1 corresponds to the gap between the \\(n = 2\\) and \\(n = 3\\) levels, and 13.6 eV \u2014 the series limit of the Lyman series \u2014 is the ionisation energy of hydrogen from the ground state.<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li>The formula applies to the hydrogen atom only (\\(Z = 1\\)) in the gas phase. For hydrogen-like ions such as He\u207a or Li\u00b2\u207a, multiply the right-hand side by \\(Z^2\\).<\/li>\n<li>Wavelengths are vacuum values computed with the reduced-mass constant \\(R_H = 1.0967758 \\times 10^7\\ \\text{m}^{-1}\\). In air the lines shift by roughly 0.1\u20130.3 nm (H\u03b1 appears at 656.28 nm instead of 656.47 nm).<\/li>\n<li>Fine structure, the Lamb shift and isotope effects (deuterium, tritium) are neglected; these cause sub-nanometre splittings that matter only in high-resolution spectroscopy.<\/li>\n<li>The model treats the atom as a single electron around a single nucleus \u2014 it does not apply to multi-electron atoms, molecules or solids.<\/li>\n<\/ul>\n<p>If you are analysing other types of spectra or energy conversions, you may also find our <a href=\"https:\/\/wpcalc.com\/en\/chemistry\/braggs-law-calculator\/\">Bragg's Law calculator<\/a> useful for diffraction-based wavelength analysis, or the <a href=\"https:\/\/wpcalc.com\/en\/chemistry\/standard-cell-potential-calculator\/\">Standard Cell Potential calculator<\/a> for electrochemical energy calculations. For the kinetic behaviour of gas-phase atoms, see the <a href=\"https:\/\/wpcalc.com\/en\/chemistry\/molecular-speed-of-a-gas-calculator\/\">Molecular Speed of a Gas calculator<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter the initial and final principal quantum numbers of a hydrogen electron transition and get the wavelength, frequency and photon energy of the spectral line, plus its series (Lyman, Balmer, Paschen\u2026).<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[54],"tags":[59],"class_list":["post-66901","post","type-post","status-publish","format-standard","hentry","category-chemistry","tag-spectrophotometry"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66901","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66901"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66901\/revisions"}],"predecessor-version":[{"id":66902,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66901\/revisions\/66902"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66901"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66901"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}