{"id":66494,"date":"2026-10-06T17:30:59","date_gmt":"2026-10-06T14:30:59","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66494"},"modified":"2026-10-06T17:30:59","modified_gmt":"2026-10-06T14:30:59","slug":"grahams-law-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/chemistry\/grahams-law-calculator\/","title":{"rendered":"Graham's Law 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\">Compare the Diffusion or Effusion Rates of Two Gases<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66494\" data-favorite-title=\"Graham&#039;s Law\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/chemistry\/grahams-law-calculator\/\" data-favorite-excerpt=\"Compare how fast two gases diffuse or effuse. Enter the molar masses of both gases (and a known rate if you have one) to get the rate ratio or the unknown rate from Graham&#039;s law.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wind-2\">\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\/grahams-law-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66494\" data-post=\"66494\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">What to Calculate<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"what_to_calculate\"><option value=\"mode_ratio\" selected>Rate ratio only<\/option><option value=\"mode_r1\">Rate of gas 1 (rate of gas 2 is known)<\/option><option value=\"mode_r2\">Rate of gas 2 (rate of gas 1 is known)<\/option><option value=\"mode_both\">Compare two measured rates<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Molar Mass of Gas 1<\/label><div class=\"term\">M\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"molar_mass_of_gas_1\" id=\"field-2\" aria-describedby=\"calc-unit-2 calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 2.016\" min=\"0.0001\" value=\"2.016\"\/><div class=\"unit\" id=\"calc-unit-2\">g\/mol<\/div><small id=\"calc-info-2\">Molar mass of the lighter\/faster gas (default: hydrogen H\u2082)<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Molar Mass of Gas 2<\/label><div class=\"term\">M\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"molar_mass_of_gas_2\" id=\"field-3\" aria-describedby=\"calc-unit-3 calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 32\" min=\"0.0001\" value=\"32\"\/><div class=\"unit\" id=\"calc-unit-3\">g\/mol<\/div><small id=\"calc-info-3\">Molar mass of the other gas (default: oxygen O\u2082)<\/small><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-4\">Rate of Gas 1<\/label><div class=\"term\">r\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"rate_of_gas_1\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 10\" min=\"0.0001\" value=\"10\"\/><small id=\"calc-info-4\">Diffusion or effusion rate of gas 1 in any unit (mL\/s, mL\/min, mol\/s\u2026) \u2014 the answer uses the same unit<\/small><\/div><div class=\"field has-term\" id=\"input-5\" hidden><label for=\"field-5\">Rate of Gas 2<\/label><div class=\"term\">r\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"rate_of_gas_2\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 10\" min=\"0.0001\" value=\"10\"\/><small id=\"calc-info-5\">Diffusion or effusion rate of gas 2 in any unit (mL\/s, mL\/min, mol\/s\u2026) \u2014 the answer uses the same unit<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66494\">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> Graham&#039;s Law Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Rate Ratio<\/span><span class=\"result-term\">r\u2081\/r\u2082<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/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>Inverse Ratio<\/span><span class=\"result-term\">r\u2082\/r\u2081<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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-3\"><span class=\"field-title\"><span>Details<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/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=\"66494\" 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>Graham's Law 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{r_1}{r_2} = \\sqrt{\\frac{M_2}{M_1}} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(r_1\\), \\(r_2\\) \u2014 diffusion or effusion rates of gas 1 and gas 2 (any consistent unit, e.g. mL\/s or mol\/s)<\/li>\n<li>\\(M_1\\) \u2014 molar mass of gas 1 (g\/mol)<\/li>\n<li>\\(M_2\\) \u2014 molar mass of gas 2 (g\/mol)<\/li>\n<\/ul>\n<p>The formula states that the ratio of the rates of two gases equals the square root of the inverse ratio of their molar masses: the lighter gas always moves faster.<\/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>Graham's Law \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> compare the effusion rates of hydrogen (H\u2082, \\(M_1 = 2.016\\) g\/mol) and oxygen (O\u2082, \\(M_2 = 32\\) g\/mol).<\/p>\n<p>\\[ \\frac{r_1}{r_2} = \\sqrt{\\frac{32}{2.016}} = \\sqrt{15.873} \\approx 3.984 \\]<\/p>\n<p>If oxygen effuses at 10 mL\/s, then hydrogen effuses at \\(r_1 = 10 \\times 3.984 \\approx 39.84\\) mL\/s.<\/p>\n<p><strong>Result:<\/strong> \\(r_1\/r_2 \\approx 3.984\\) and \\(r_2\/r_1 \\approx 0.251\\) \u2014 hydrogen effuses about 3.98 times faster than oxygen.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Open <strong>What to Calculate<\/strong> and choose a mode: <em>Rate ratio only<\/em> (just the ratio), <em>Rate of gas 1<\/em> (you know the rate of gas 2), <em>Rate of gas 2<\/em> (you know the rate of gas 1), or <em>Compare two measured rates<\/em> (checks how well your data match the law).<\/li>\n<li>Enter the <strong>Molar Mass of Gas 1<\/strong> in g\/mol \u2014 this is typically the lighter, faster gas.<\/li>\n<li>Enter the <strong>Molar Mass of Gas 2<\/strong> in g\/mol.<\/li>\n<li>If your mode needs it, enter the known <strong>Rate of Gas 1<\/strong> or <strong>Rate of Gas 2<\/strong> in any unit you like; the answer comes back in the same unit.<\/li>\n<li>Read the results: the rate ratio \\(r_1\/r_2\\), the inverse ratio \\(r_2\/r_1\\), and a line telling you which gas is faster and by how much.<\/li>\n<\/ol>\n<h2>What Is Graham's Law?<\/h2>\n<p>Graham's law, formulated by the Scottish chemist Thomas Graham in 1848, states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. In practical terms: lighter gases spread out and escape through small openings faster than heavier gases.<\/p>\n<p><strong>Diffusion<\/strong> is the spontaneous mixing of a gas with another gas through random molecular motion \u2014 for example, the smell of perfume spreading through a room. <strong>Effusion<\/strong> is the escape of a gas through a tiny hole into a vacuum, such as air slowly leaking from a balloon. Graham's law applies to both processes because both are governed by how fast the molecules travel.<\/p>\n<p>The law follows directly from the kinetic theory of gases. At a given temperature, all gases have the same average kinetic energy, so the average molecular speed is \\(v \\propto 1\/\\sqrt{M}\\). A molecule of hydrogen (2.016 g\/mol) therefore moves about four times faster than a molecule of oxygen (32 g\/mol) at the same temperature \u2014 exactly what the example above shows.<\/p>\n<p>Interpreting the result is simple: if \\(r_1\/r_2 = 3.98\\), gas 1 covers distance or escapes through an opening 3.98 times faster than gas 2. If the ratio is below 1, gas 1 is the slower one, and the inverse ratio \\(r_2\/r_1\\) tells you the speed advantage of gas 2. One classic application was the Manhattan Project, where Graham's law of effusion guided the separation of uranium isotopes by effusing gaseous UF\u2086 through porous membranes.<\/p>\n<h2>Molar Masses of Common Gases<\/h2>\n<table>\n<thead>\n<tr>\n<th>Gas<\/th>\n<th>Formula<\/th>\n<th>Molar mass (g\/mol)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Hydrogen<\/td>\n<td>H\u2082<\/td>\n<td>2.016<\/td>\n<\/tr>\n<tr>\n<td>Helium<\/td>\n<td>He<\/td>\n<td>4.003<\/td>\n<\/tr>\n<tr>\n<td>Methane<\/td>\n<td>CH\u2084<\/td>\n<td>16.04<\/td>\n<\/tr>\n<tr>\n<td>Ammonia<\/td>\n<td>NH\u2083<\/td>\n<td>17.03<\/td>\n<\/tr>\n<tr>\n<td>Nitrogen<\/td>\n<td>N\u2082<\/td>\n<td>28.01<\/td>\n<\/tr>\n<tr>\n<td>Oxygen<\/td>\n<td>O\u2082<\/td>\n<td>32.00<\/td>\n<\/tr>\n<tr>\n<td>Carbon dioxide<\/td>\n<td>CO\u2082<\/td>\n<td>44.01<\/td>\n<\/tr>\n<tr>\n<td>Chlorine<\/td>\n<td>Cl\u2082<\/td>\n<td>70.90<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Both gases must be at the same temperature and pressure.<\/strong> The rates only depend on molar mass when temperature is identical, since speed scales with \\(\\sqrt{T}\\).<\/li>\n<li><strong>The law assumes ideal-gas behaviour.<\/strong> Real gases deviate at high pressure, low temperature, or when molecules interact strongly (polar gases, hydrogen bonding). Deviations of a few percent are normal.<\/li>\n<li><strong>The hole must be tiny for effusion.<\/strong> If the opening is large compared to the mean free path, gas flows as a bulk fluid and the simple law no longer holds.<\/li>\n<li><strong>Use molar mass, not density or molecular diameter.<\/strong> If you compare a gas with air, use the effective molar mass of air, about 28.96 g\/mol.<\/li>\n<li>For gas-law problems involving pressure and volume, see the <a href=\"https:\/\/wpcalc.com\/en\/chemistry\/daltons-law\/\">Dalton's Law calculator<\/a>, the <a href=\"https:\/\/wpcalc.com\/en\/chemistry\/combined-gas-law\/\">Combined Gas Law calculator<\/a>, and the <a href=\"https:\/\/wpcalc.com\/en\/chemistry\/charles-law\/\">Charles' Law calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Compare how fast two gases diffuse or effuse. Enter the molar masses of both gases (and a known rate if you have one) to get the rate ratio or the unknown rate from Graham's law.<\/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":[55],"class_list":["post-66494","post","type-post","status-publish","format-standard","hentry","category-chemistry","tag-gases"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66494","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=66494"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66494\/revisions"}],"predecessor-version":[{"id":66495,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66494\/revisions\/66495"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}