{"id":2016,"date":"2026-10-07T06:47:58","date_gmt":"2026-10-07T05:47:58","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/chemia\/kalkulator-rownania-van-der-waalsa\/"},"modified":"2026-10-07T06:47:58","modified_gmt":"2026-10-07T05:47:58","slug":"kalkulator-rownania-van-der-waalsa","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/chemia\/kalkulator-rownania-van-der-waalsa\/","title":{"rendered":"Kalkulator r\u00f3wnania van der Waalsa"},"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\">Obliczanie ci\u015bnienia gazu rzeczywistego z r\u00f3wnania van der Waalsa<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2016\" data-favorite-title=\"R\u00f3wnanie van der Waalsa\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/chemia\/kalkulator-rownania-van-der-waalsa\/\" data-favorite-excerpt=\"Oblicz ci\u015bnienie gazu rzeczywistego z r\u00f3wnania van der Waalsa. Wprowad\u017a ilo\u015b\u0107 gazu, obj\u0119to\u015b\u0107, temperatur\u0119 oraz sta\u0142e a i b, aby otrzyma\u0107 ci\u015bnienie, wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci Z i odchylenie od prawa gaz\u00f3w doskona\u0142ych.\" aria-label=\"Dodaj do ulubionych\" 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\/pl\/chemia\/kalkulator-rownania-van-der-waalsa\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2016\" data-post=\"2016\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Ilo\u015b\u0107 gazu (n)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"ilo_gazu_n\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 1\" min=\"0\" value=\"1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"mol\" selected>mol<\/option><option value=\"mmol\">mmol<\/option><option value=\"kmol\">kmol<\/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><small id=\"calc-info-1\">Liczba moli gazu<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Obj\u0119to\u015b\u0107 (V)<\/label><div class=\"term\">V<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"objto_v\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"L\" selected>L<\/option><option value=\"mL\">mL<\/option><option value=\"m3\">m\u00b3<\/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><small id=\"calc-info-3\">Obj\u0119to\u015b\u0107 naczynia zawieraj\u0105cego gaz<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-5\">Temperatura (T)<\/label><div class=\"term\">T<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"temperatura_t\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 300\" min=\"0\" value=\"300\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"K\" selected>K<\/option><option value=\"C\">\u00b0C<\/option><option value=\"F\">\u00b0F<\/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><small id=\"calc-info-5\">Temperatura bezwzgl\u0119dna gazu<\/small><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-7\">Sta\u0142a przyci\u0105gania (a)<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"staa_przycigania_a\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 1.3708\" min=\"0\" value=\"1.3708\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"barL2\" selected>L\u00b2\u00b7bar\/mol\u00b2<\/option><option value=\"atmL2\">L\u00b2\u00b7atm\/mol\u00b2<\/option><option value=\"Pam6\">Pa\u00b7m\u2076\/mol\u00b2<\/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><small id=\"calc-info-7\">Sta\u0142a a van der Waalsa \u2014 1,3708 L\u00b2\u00b7bar\/mol\u00b2 to azot (N\u2082)<\/small><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-9\">Sta\u0142a kowolumenu (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"staa_kowolumenu_b\" id=\"field-9\" aria-describedby=\"calc-info-9\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.0387\" min=\"0\" value=\"0.0387\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"Lmol\" selected>L\/mol<\/option><option value=\"cm3mol\">cm\u00b3\/mol<\/option><option value=\"m3mol\">m\u00b3\/mol<\/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><small id=\"calc-info-9\">Sta\u0142a b van der Waalsa \u2014 0,0387 L\/mol to azot (N\u2082)<\/small><\/div><div class=\"field is-single\" id=\"input-6\"><label for=\"field-11\">Jednostka ci\u015bnienia dla wynik\u00f3w<\/label><div class=\"grid grid-cols-1\"><select id=\"field-11\" name=\"jednostka_cinienia_dla_wynikw\"><option value=\"atm\" selected>atm<\/option><option value=\"bar\">bar<\/option><option value=\"kPa\">kPa<\/option><option value=\"MPa\">MPa<\/option><option value=\"psi\">psi<\/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><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2016\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/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> Wyniki ci\u015bnienia<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Ci\u015bnienie gazu rzeczywistego<\/span><span class=\"result-term\">P<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci Z<\/span><span class=\"result-term\">Z<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Odchylenie od ci\u015bnienia gazu doskona\u0142ego<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Obj\u0119to\u015b\u0107 molowa<\/span><span class=\"result-term\">V\/n<\/span><\/span><span class=\"field-value\" id=\"result-4\" data-unit=\"L\/mol\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Szczeg\u00f3\u0142y oblicze\u0144<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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\">Kopiuj link do tego obliczenia<\/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>Zostaw komentarz<\/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=\"2016\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/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\">Lubi\u0119 to <\/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\">Pomocne <\/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\">Niejasne <\/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\">Rozczarowanie <\/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\">Niedok\u0142adne <\/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>Wz\u00f3r r\u00f3wnania van der Waalsa<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\"><\/div>\r\n\t<\/figure>\r\n\t{formula}<br \/>\n\\[ \\left(P + a\\frac{n^2}{V^2}\\right)(V - nb) = nRT \\quad\\Longleftrightarrow\\quad P = \\frac{nRT}{V - nb} - a\\left(\\frac{n}{V}\\right)^2 \\\\\\ Z = \\frac{PV}{nRT} \\\\[\/formula]<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(P\\) \u2014 ci\u015bnienie gazu rzeczywistego<\/li>\n<li>\\(n\\) \u2014 ilo\u015b\u0107 gazu (mol)<\/li>\n<li>\\(V\\) \u2014 obj\u0119to\u015b\u0107 naczynia (L)<\/li>\n<li>\\(T\\) \u2014 temperatura bezwzgl\u0119dna (K)<\/li>\n<li>\\(R\\) \u2014 sta\u0142a gazowa, \\(0.082057\\ \\text{L\u00b7atm\/(mol\u00b7K)}\\)<\/li>\n<li>\\(a\\) \u2014 sta\u0142a przyci\u0105gania, uwzgl\u0119dniaj\u0105ca przyci\u0105ganie mi\u0119dzycz\u0105steczkowe<\/li>\n<li>\\(b\\) \u2014 sta\u0142a kowolumenu, uwzgl\u0119dniaj\u0105ca sko\u0144czone rozmiary cz\u0105steczek<\/li>\n<li>\\(Z\\) \u2014 wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci (\\(Z = 1\\) dla gazu doskona\u0142ego)<\/li>\n<\/ul>\n<p>R\u00f3wnanie uzupe\u0142nia prawo gaz\u00f3w doskona\u0142ych o dwa cz\u0142ony poprawkowe: cz\u0142on \\(a\\) obni\u017ca ci\u015bnienie, poniewa\u017c cz\u0105steczki przyci\u0105gaj\u0105 si\u0119 wzajemnie, a cz\u0142on \\(b\\) podnosi je, poniewa\u017c same cz\u0105steczki zajmuj\u0105 miejsce, kt\u00f3re nie jest dost\u0119pne do ruchu.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.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>R\u00f3wnanie van der Waalsa \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> 1 mol azotu (N\u2082) w naczyniu o obj\u0119to\u015bci 10 L w temperaturze 300 K. Dla azotu \\(a = 1.3708\\ \\text{L\u00b2\u00b7bar\/mol\u00b2}\\) oraz \\(b = 0.0387\\ \\text{L\/mol}\\). Przeliczenie \\(a\\) na L\u00b2\u00b7atm\/mol\u00b2 daje \\(a = 1.3708 \\times 0.986923 = 1.3529\\ \\text{L\u00b2\u00b7atm\/mol\u00b2}\\).<\/p>\n<p>\\[ P = \\frac{1 \\times 0.082057 \\times 300}{10 - 1 \\times 0.0387} - 1.3529 \\times \\left(\\frac{1}{10}\\right)^2 = \\frac{24.6171}{9.9613} - 0.01353 \\\\[\/p]<\/p>\n<p>\\[ P = 2.4713 - 0.0135 = 2.4578\\ \\text{atm} \\\\[\/p]<\/p>\n<p><strong>Wynik:<\/strong> ci\u015bnienie gazu rzeczywistego wynosi <strong>P = 2.4578 atm<\/strong> (oko\u0142o 2.4900 bar lub 248.98 kPa). Prawo gaz\u00f3w doskona\u0142ych da\u0142oby 2.4617 atm, wi\u0119c rzeczywiste ci\u015bnienie jest o 0.16% ni\u017csze. Wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci wynosi Z = 0.9984, a obj\u0119to\u015b\u0107 molowa 10 L\/mol.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wprowad\u017a <strong>ilo\u015b\u0107 gazu (n)<\/strong> i wybierz jednostk\u0119 (mol, mmol lub kmol).<\/li>\n<li>Wprowad\u017a <strong>obj\u0119to\u015b\u0107 (V)<\/strong> naczynia i wybierz L, mL lub m\u00b3.<\/li>\n<li>Wprowad\u017a <strong>temperatur\u0119 (T)<\/strong>; mo\u017cesz j\u0105 poda\u0107 w K, \u00b0C lub \u00b0F, a kalkulator przeliczy j\u0105 na kelwiny.<\/li>\n<li>Wprowad\u017a <strong>sta\u0142\u0105 przyci\u0105gania (a)<\/strong> dla swojego gazu. Warto\u015b\u0107 domy\u015blna 1.3708 L\u00b2\u00b7bar\/mol\u00b2 odpowiada azotowi; jednostk\u0105 mo\u017ce by\u0107 L\u00b2\u00b7bar\/mol\u00b2, L\u00b2\u00b7atm\/mol\u00b2 lub Pa\u00b7m\u2076\/mol\u00b2.<\/li>\n<li>Wprowad\u017a <strong>sta\u0142\u0105 kowolumenu (b)<\/strong> w L\/mol, cm\u00b3\/mol lub m\u00b3\/mol. Dla azotu wynosi ona 0.0387 L\/mol.<\/li>\n<li>Wybierz <strong>jednostk\u0119 ci\u015bnienia w wynikach<\/strong> \u2014 atm, bar, kPa, MPa lub psi \u2014 i odczytaj ci\u015bnienie, wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci Z, odchylenie procentowe od ci\u015bnienia gazu doskona\u0142ego, obj\u0119to\u015b\u0107 molow\u0105 oraz szczeg\u00f3\u0142owe obliczenia krok po kroku.<\/li>\n<\/ol>\n<h2>Czym jest r\u00f3wnanie van der Waalsa?<\/h2>\n<p>Prawo gaz\u00f3w doskona\u0142ych, \\(PV = nRT\\), zak\u0142ada, \u017ce cz\u0105steczki gazu s\u0105 punktowymi obiektami, kt\u00f3re nie oddzia\u0142uj\u0105 ze sob\u0105. Cz\u0105steczki rzeczywiste zajmuj\u0105 jednak sko\u0144czon\u0105 obj\u0119to\u015b\u0107 i przyci\u0105gaj\u0105 si\u0119 nawzajem. W 1873 roku Johannes Diderik van der Waals zaproponowa\u0142 zmodyfikowane r\u00f3wnanie stanu uwzgl\u0119dniaj\u0105ce oba efekty \u2014 osi\u0105gni\u0119cie to zosta\u0142o docenione Nagrod\u0105 Nobla w dziedzinie fizyki w 1910 roku. R\u00f3wnanie jako pierwsze wyja\u015bnia\u0142o, dlaczego gazy skraplaj\u0105 si\u0119 w ciecze, i pozostaje klasycznym podr\u0119cznikowym modelem zachowania gaz\u00f3w rzeczywistych.<\/p>\n<p>Poprawka \\(a(n\/V)^2\\) przedstawia ci\u015bnienie wewn\u0119trzne: cz\u0105steczki, kt\u00f3re maj\u0105 uderzy\u0107 w \u015bciank\u0119, s\u0105 \u015bci\u0105gane z powrotem przez s\u0105siad\u00f3w, wi\u0119c obserwowane ci\u015bnienie jest ni\u017csze od warto\u015bci idealnej. Sta\u0142a \\(a\\) ro\u015bnie wraz z si\u0142\u0105 przyci\u0105gania mi\u0119dzycz\u0105steczkowego i jest najwi\u0119ksza dla gaz\u00f3w polarnych, \u0142atwo ulegaj\u0105cych skropleniu, takich jak woda i amoniak. Cz\u0142on \\(nb\\) odejmuje obj\u0119to\u015b\u0107 zajmowan\u0105 przez same cz\u0105steczki, faktycznie zmniejszaj\u0105c woln\u0105 przestrze\u0144, w kt\u00f3rej mog\u0105 si\u0119 one porusza\u0107; \\(b\\) jest w przybli\u017ceniu cztery razy wi\u0119ksze od rzeczywistej obj\u0119to\u015bci cz\u0105steczek na mol.<\/p>\n<p>To, kt\u00f3ra poprawka dominuje, zale\u017cy od warunk\u00f3w. Przy umiarkowanych g\u0119sto\u015bciach zwykle wygrywaj\u0105 przyci\u0105gania i rzeczywiste ci\u015bnienie jest <em>ni\u017csze<\/em> ni\u017c przewidywane dla gazu doskona\u0142ego. Przy bardzo du\u017cych g\u0119sto\u015bciach g\u00f3r\u0119 bierze wykluczona obj\u0119to\u015b\u0107 i rzeczywiste ci\u015bnienie staje si\u0119 <em>wy\u017csze<\/em>. Podsumowuje to wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci \\(Z = PV\/nRT\\): \\(Z  1\\) oznacza, \u017ce dominuje sko\u0144czona obj\u0119to\u015b\u0107 cz\u0105steczek, a \\(Z = 1\\) odpowiada zachowaniu idealnemu.<\/p>\n<h2>Sta\u0142e dla typowych gaz\u00f3w<\/h2>\n<table>\n<thead>\n<tr>\n<th>Gaz<\/th>\n<th>a (L\u00b2\u00b7bar\/mol\u00b2)<\/th>\n<th>b (L\/mol)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Hel (He)<\/td>\n<td>0.0346<\/td>\n<td>0.0238<\/td>\n<\/tr>\n<tr>\n<td>Azot (N\u2082)<\/td>\n<td>1.3708<\/td>\n<td>0.0387<\/td>\n<\/tr>\n<tr>\n<td>Tlen (O\u2082)<\/td>\n<td>1.382<\/td>\n<td>0.0319<\/td>\n<\/tr>\n<tr>\n<td>Metan (CH\u2084)<\/td>\n<td>2.300<\/td>\n<td>0.0430<\/td>\n<\/tr>\n<tr>\n<td>Dwutlenek w\u0119gla (CO\u2082)<\/td>\n<td>3.657<\/td>\n<td>0.0429<\/td>\n<\/tr>\n<tr>\n<td>Amoniak (NH\u2083)<\/td>\n<td>4.225<\/td>\n<td>0.0371<\/td>\n<\/tr>\n<tr>\n<td>Woda (H\u2082O)<\/td>\n<td>5.537<\/td>\n<td>0.0305<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Zwr\u00f3\u0107 uwag\u0119, \u017ce tabela zawiera sta\u0142e w barach; korzystaj\u0105c z tych warto\u015bci, prze\u0142\u0105cz jednostk\u0119 dla <strong>a<\/strong> na L\u00b2\u00b7bar\/mol\u00b2.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Sta\u0142e nie s\u0105 w pe\u0142ni sta\u0142e.<\/strong> Tabelaryczne warto\u015bci \\(a\\) i \\(b\\) s\u0105 dopasowane do danych do\u015bwiadczalnych i s\u0105 najbardziej wiarygodne przy umiarkowanych ci\u015bnieniach (mniej wi\u0119cej do kilkuset bar\u00f3w). Przy ekstremalnych g\u0119sto\u015bciach r\u00f3wnanie ma raczej charakter jako\u015bciowy ni\u017c ilo\u015bciowy.<\/li>\n<li><strong>Niezawodno\u015b\u0107 blisko skraplania.<\/strong> W pobli\u017cu punktu krytycznego lub w obszarze dwufazowym r\u00f3wnanie mo\u017ce nawet przewidywa\u0107 ci\u015bnienie ujemne lub kilka obj\u0119to\u015bci dla jednego ci\u015bnienia. Kalkulator sygnalizuje takie przypadki komunikatem o b\u0142\u0119dzie zamiast liczby.<\/li>\n<li><strong>Najlepsze dla ma\u0142ych cz\u0105steczek niepolarnych.<\/strong> Gazy szlachetne, azot i tlen s\u0105 opisywane dobrze; gazy silnie asocjuj\u0105ce, takie jak para wodna, odbiegaj\u0105 bardziej, zw\u0142aszcza blisko nasycenia.<\/li>\n<li><strong>Obj\u0119to\u015b\u0107 musi przekracza\u0107 kowolumen.<\/strong> Cz\u0142on \\(V - nb\\) wymaga \\(V &gt; nb\\); w przeciwnym razie model nie ma rozwi\u0105zania, co kalkulator wyra\u017anie zg\u0142asza.<\/li>\n<li><strong>Gdy a = b = 0<\/strong>, r\u00f3wnanie sprowadza si\u0119 dok\u0142adnie do prawa gaz\u00f3w doskona\u0142ych, co jest szybkim sposobem na sprawdzenie poprawno\u015bci danych wej\u015bciowych.<\/li>\n<\/ul>\n<p>Je\u015bli interesuj\u0105 Ci\u0119 inne zjawiska w fazie gazowej, przydatne mog\u0105 by\u0107 tak\u017ce: <a href=\"https:\/\/wpcalc.com\/pl\/chemia\/kalkulator-rownania-van-t-hoffa\/\">kalkulator r\u00f3wnania van 't Hoffa<\/a> do zale\u017cno\u015bci sta\u0142ych r\u00f3wnowagi od temperatury, <a href=\"https:\/\/wpcalc.com\/pl\/chemia\/kalkulator-prawa-grahama\/\">kalkulator prawa Grahama<\/a> do szybko\u015bci dyfuzji oraz <a href=\"https:\/\/wpcalc.com\/pl\/chemia\/kalkulator-clausiusa-clapeyrona\/\">kalkulator r\u00f3wnania Clausiusa\u2013Clapeyrona<\/a> do ci\u015bnienia par.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Oblicz ci\u015bnienie gazu rzeczywistego z r\u00f3wnania van der Waalsa. Wprowad\u017a ilo\u015b\u0107 gazu, obj\u0119to\u015b\u0107, temperatur\u0119 oraz sta\u0142e a i b, aby otrzyma\u0107 ci\u015bnienie, wsp\u00f3\u0142czynnik \u015bci\u015bliwo\u015bci Z i odchylenie od prawa gaz\u00f3w doskona\u0142ych.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[39],"class_list":["post-2016","post","type-post","status-publish","format-standard","hentry","category-chemia","tag-gazy"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2016","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=2016"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2016\/revisions"}],"predecessor-version":[{"id":2057,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2016\/revisions\/2057"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2016"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2016"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2016"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}