{"id":30065,"date":"2021-03-10T13:00:00","date_gmt":"2021-03-10T13:00:00","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30065"},"modified":"2025-10-20T07:24:29","modified_gmt":"2025-10-20T07:24:29","slug":"the-ideal-gas-equation","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/the-ideal-gas-equation\/","title":{"rendered":"The Ideal Gas Equation"},"content":{"rendered":"<style>\np, ul, ol{\n  text-align: justify;\n}\nh1{\n  text-align:center;\n  text-transform: uppercase;\n}\nh2{\n  text-align:center;\n  text-transform: uppercase;\n  font-size:24pt;\n}\nh3 { \n  text-align: center;\n  text-transform: uppercase;\n  font-size: 24px !important;\n}\n.example{\n  background:#f6f8fa; \n  border-left:4px solid #d00000; \n  padding:12px 14px; \n  margin:14px 0;\n}\n.small{\n  font-size: 0.95em;\n  color:#333;\n}\n<\/style>\n<h1>Empirical Formulation of the Ideal Gas<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\">Have you ever wondered why a balloon expands when heated or why a tire\u2019s pressure changes with altitude? In this lesson, we will review the laws governing these behaviors and how they lead to the ideal gas equation, along with its considerations and key points.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><b>Learning Objectives<\/b><br \/>\nAt the end of this lesson, the student will be able to:<\/p>\n<ol>\n<li><b>Explain<\/b> the empirical laws of ideal gases (Boyle\u2013Mariotte, Charles, Gay-Lussac) and their synthesis in the equation of state (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV = nRT<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV = N k_B T<\/span><\/span>).<\/li>\n<li><b>Apply<\/b> the ideal gas equation and the relation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte<\/span><\/span> to solve state changes with consistent units.<\/li>\n<li><b>Analyze<\/b> isothermal, isobaric, and isochoric processes and their trajectories in <i>P\u2013V<\/i>, <i>V\u2013T<\/i>, and <i>P\u2013T<\/i> diagrams.<\/li>\n<li><b>Recognize<\/b> the range of validity of the ideal gas model and select alternative models (van der Waals, quantum, relativistic) when appropriate.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b>TABLE OF CONTENTS<\/b><br \/>\n<a href=\"#1\">Fundamental empirical laws<\/a><br \/>\n<a href=\"#2\">Combination of laws into the ideal gas equation<\/a><br \/>\n<a href=\"#3\">Process-based deductions<\/a><br \/>\n<a href=\"#4\">Comments and microscopic background<\/a><br \/>\n<a href=\"#5\">Range of validity and limitations<\/a><br \/>\n<a href=\"#6\">Practical notes<\/a>\n<\/p>\n<p><center><br \/>\n <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/7WkrH_FS290?si=xWJQ-VAtbWgzm9bQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Fundamental Empirical Laws<\/h2>\n<p>Experiments with gases show a dependence between pressure <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span>, volume <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span>, and temperature <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>. Under controlled conditions, three fundamental empirical laws are observed:<\/p>\n<ol>\n<li><strong>Boyle\u2013Mariotte\u2019s Law (isothermal):<\/strong> In a process at constant temperature, the volume and pressure of a gas are inversely proportional; that is:\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P \\propto \\dfrac{1}{V}\\quad\\Leftrightarrow\\quad PV=\\text{cte.}<\/span><\/div>\n<div class=\"example\">\n  <strong>Example:<\/strong> A gas that, at an initial pressure of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 15\\ \\mathrm{MPa}<\/span><\/span>, expands isothermally from an initial volume <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 1{,}00\\ \\mathrm{L}<\/span><\/span> to a final volume <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2 = 2{,}00\\ \\mathrm{L}<\/span><\/span> will have its pressure reduced by half. Since the product <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=\\text{cte.}<\/span><\/span>, it follows that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 V_1 = P_2 V_2<\/span><\/span>, leading to:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  P_2 = \\dfrac{P_1 V_1}{V_2}\n\n      = 15\\ \\mathrm{MPa}\\left(\\dfrac{1{,}00\\ \\mathrm{L}}{2{,}00\\ \\mathrm{L}}\\right)\n\n      = 7{,}50\\ \\mathrm{MPa}\n\n  <\/span>\n<p>  <center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico.jpg\" alt=\"PV diagram for isothermal process\" width=\"480\" height=\"293\" class=\"aligncenter size-full wp-image-34975 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico.jpg\" alt=\"PV diagram for isothermal process\" width=\"480\" height=\"293\" class=\"aligncenter size-full wp-image-34975 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico.jpg 480w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico-300x183.jpg 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/noscript><\/center>\n<\/div>\n<\/li>\n<li><strong>Charles\u2019s Law (isobaric):<\/strong> In a process at constant pressure, the volume and temperature of a gas are directly proportional; that is:\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V \\propto T \\quad\\Leftrightarrow\\quad \\dfrac{V}{T}=\\text{cte.}<\/span><\/div>\n<div class=\"example\">\n  <strong>Example:<\/strong> A gas that, at an initial temperature of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, is heated isobarically up to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> starting from an initial volume <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 2{,}00\\ \\mathrm{L}<\/span><\/span> will see its volume increase by 50&nbsp;% (a factor of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{3}{2}<\/span><\/span>). Since <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{V}{T}=\\text{cte.}<\/span><\/span>, it follows that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{V_1}{T_1}=\\dfrac{V_2}{T_2}<\/span><\/span>, leading to:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  V_2 = V_1 \\cdot \\dfrac{T_2}{T_1}\n\n      = 2{,}00\\ \\mathrm{L}\\left(\\dfrac{450\\ \\mathrm{K}}{300\\ \\mathrm{K}}\\right)\n\n      = 3{,}00\\ \\mathrm{L}\n\n  <\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc.jpg\" alt=\"\" width=\"480\" height=\"298\" class=\"aligncenter size-full wp-image-34984 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc.jpg\" alt=\"\" width=\"480\" height=\"298\" class=\"aligncenter size-full wp-image-34984 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc.jpg 480w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc-300x186.jpg 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/noscript><\/center><\/p>\n<\/div>\n<\/li>\n<li><strong>Gay-Lussac\u2019s Law (isochoric):<\/strong> In a process at constant volume, the pressure and temperature of a gas are directly proportional; that is:\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P \\propto T \\quad\\Leftrightarrow\\quad \\dfrac{P}{T}=\\text{cte.}<\/span><\/div>\n<div class=\"example\">\n  <strong>Example:<\/strong> A gas that, at an initial temperature of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, is heated isochorically up to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> starting from an initial pressure <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 1{,}00\\ \\mathrm{MPa}<\/span><\/span> will experience an increase in pressure in the same proportion. Since <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{P}{T}=\\text{cte.}<\/span><\/span>, it follows that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{P_1}{T_1}=\\dfrac{P_2}{T_2}<\/span><\/span>, leading to:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  P_2 = P_1 \\cdot \\dfrac{T_2}{T_1}\n\n      = 1{,}00\\ \\mathrm{MPa}\\left(\\dfrac{450\\ \\mathrm{K}}{300\\ \\mathrm{K}}\\right)\n\n      = 1{,}50\\ \\mathrm{MPa}\n\n  <\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc.jpg\" alt=\"\" width=\"480\" height=\"291\" class=\"aligncenter size-full wp-image-34987 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc.jpg\" alt=\"\" width=\"480\" height=\"291\" class=\"aligncenter size-full wp-image-34987 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc.jpg 480w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc-300x182.jpg 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/noscript><\/center>\n<\/div>\n<\/li>\n<\/ol>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Combination of the Laws into the Ideal Gas Equation<\/h2>\n<p>These three laws can be synthesized into a single proportional relationship:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV \\propto T<\/span>\n<p>Where, based on experimental and microscopic considerations, it is possible to infer the proportionality constant as the product of the number of particles <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> and the Boltzmann constant <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k_B = 1{,}380\\,649\\times10^{-23}\\ \\mathrm{J\\,K^{-1}}<\/span><\/span>, obtaining the microscopic relation:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{PV = N\\,k_B\\,T}<\/span>\n<p>Similarly, in molar terms, the proportionality constant is obtained as the product of the number of moles <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> and the universal gas constant <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=8{,}314\\,462\\,6\\ \\mathrm{J\\,mol^{-1}\\,K^{-1}}=0{,}082\\,057\\ \\mathrm{L\\,atm\\,mol^{-1}\\,K^{-1}}<\/span><\/span>, leading to the molar relation:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{PV = n\\,R\\,T}<\/span>\n<p>Regardless of the case, the fact remains that there is a direct proportionality between the product <span class=\"katex-eq\" data-katex-display=\"false\">PV<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>, which is equivalent to stating that if an ideal gas moves between two states, one with initial values <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha, V_\\alpha, T_\\alpha)<\/span><\/span> and another with final values <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega, V_\\omega, T_\\omega)<\/span><\/span>, then they satisfy the relation<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} =  \\dfrac{P_\\omega V_\\omega}{T_\\omega}<\/span>\n<p>and therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte.<\/span><\/span><\/p>\n<p>This relationship, which can serve as an experimental foundation to formulate both the microscopic and molar expressions, can be directly inferred from the experimental laws of Boyle-Mariotte, Charles, and Gay-Lussac. The reasoning is as follows:<\/p>\n<p>This can be demonstrated through three paths:<\/p>\n<ol>\n<li>A change in <b>volume<\/b> through an isothermal and an isobaric process<\/li>\n<li>A change in <b>pressure<\/b> through an isothermal and an isochoric process<\/li>\n<li>A change in <b>temperature<\/b> through an isobaric and an isochoric process<\/li>\n<\/ol>\n<p>For the development of these three cases, we will need an intermediate state with values <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Deductions by Processes<\/h2>\n<h3>Deduction by Change of Volume<\/h3>\n<p>If the initial state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> is connected to the intermediate state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> through an isothermal process, and then the intermediate state is connected to the final state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> through an isobaric process, then:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rclcl}\n\n &amp; P_\\alpha V_\\alpha= P_i V_i &amp; &amp; V_i\/T_i = V_\\omega\/T_\\omega   &amp; \\\\\n\n &amp;\\text{isothermal}&amp; &amp;\\text{isobaric} &amp; \\\\\n\nP_\\alpha &amp; \\longrightarrow &amp; P_i = \\dfrac{P_\\alpha V_\\alpha}{V_i}  &amp; \\longrightarrow &amp; P_\\omega = P_i \\\\ \\\\\n\nV_\\alpha &amp; \\longrightarrow &amp; V_i = \\dfrac{P_\\alpha V_\\alpha}{P_i} &amp; \\longrightarrow &amp; V_\\omega = \\dfrac{V_i T_\\omega}{T_i} \\\\ \\\\\n\nT_\\alpha &amp; \\longrightarrow &amp; T_i = T_\\alpha &amp; \\longrightarrow &amp; T_\\omega = \\dfrac{V_\\omega T_i}{V_i}\n\n\\end{array}\n\n<\/span>\n<p>From this, it follows that:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; V_\\omega = \\left(\\dfrac{T_\\omega}{T_i}\\right) V_i = \\left(\\dfrac{T_\\omega}{T_i}\\right) \\left(\\dfrac{P_\\alpha}{P_i} \\right) V_\\alpha = \\dfrac{T_\\omega P_\\alpha V_\\alpha}{T_\\alpha P_\\omega} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} = \\dfrac{P_\\omega V_\\omega}{T_\\omega}\n\n\\end{array}<\/span>\n<h3>Deduction by Change of Pressure<\/h3>\n<p>If the initial state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> is connected to the intermediate state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> through an isothermal process, and then the intermediate state is connected to the final state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> through an isochoric process, then:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rclcl}\n\n &amp; P_\\alpha V_\\alpha= P_i V_i &amp; &amp; P_i\/T_i = P_\\omega\/T_\\omega   &amp; \\\\\n\n &amp;\\text{isothermal}&amp; &amp;\\text{isochoric} &amp; \\\\\n\nP_\\alpha &amp; \\longrightarrow &amp; P_i = \\dfrac{P_\\alpha V_\\alpha}{V_i}  &amp; \\longrightarrow &amp; P_\\omega = \\dfrac{P_i T_\\omega}{T_i} \\\\ \\\\\n\nV_\\alpha &amp; \\longrightarrow &amp; V_i = \\dfrac{P_\\alpha V_\\alpha}{P_i} &amp; \\longrightarrow &amp; V_\\omega = V_i \\\\ \\\\\n\nT_\\alpha &amp; \\longrightarrow &amp; T_i = T_\\alpha &amp; \\longrightarrow &amp; T_\\omega = \\dfrac{V_\\omega T_i}{V_i}\n\n\\end{array}\n\n<\/span>\n<p>From this, it follows that:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n &amp; P_\\omega = \\left(\\dfrac{T_\\omega}{T_i}\\right) P_i = \\left(\\dfrac{T_\\omega}{T_i}\\right) \\left(\\dfrac{V_\\alpha}{V_i}\\right)P_\\alpha = \\dfrac{T_\\omega V_\\alpha P_\\alpha}{T_\\alpha V_\\omega} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} = \\dfrac{P_\\omega V_\\omega}{T_\\omega}\n\n\\end{array}<\/span>\n<h3>Deduction by Change of Temperature<\/h3>\n<p>If the initial state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> is connected to the intermediate state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> through an isobaric process, and then the intermediate state is connected to the final state <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> through an isochoric process, then:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rclcl}\n\n &amp; V_\\alpha\/ T_\\alpha= V_i \/ T_i &amp; &amp; P_i\/T_i = P_\\omega\/T_\\omega   &amp; \\\\\n\n &amp;\\text{isobaric}&amp; &amp;\\text{isochoric} &amp; \\\\\n\nP_\\alpha &amp; \\longrightarrow &amp; P_i = P_\\alpha  &amp; \\longrightarrow &amp; P_\\omega = \\dfrac{P_i T_\\omega}{T_i} \\\\ \\\\\n\nV_\\alpha &amp; \\longrightarrow &amp; V_i = \\dfrac{V_\\alpha T_i}{T_\\alpha} &amp; \\longrightarrow &amp; V_\\omega = V_i \\\\ \\\\\n\nT_\\alpha &amp; \\longrightarrow &amp; T_i = \\dfrac{V_i T_\\alpha}{V_\\alpha} &amp; \\longrightarrow &amp; T_\\omega = \\dfrac{P_\\omega T_i}{P_i}\n\n\\end{array}\n\n<\/span>\n<p>From this, it follows that:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n &amp; T_\\omega = \\left(\\dfrac{P_\\omega}{P_i}\\right) T_i = \\left(\\dfrac{P_\\omega}{P_i}\\right) \\left(\\dfrac{V_i}{V_\\alpha}\\right)T_\\alpha = \\dfrac{P_\\omega V_\\omega T_\\alpha}{P_\\alpha V_\\alpha}  \\\\ \\\\\n\n\\equiv &amp; \\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} = \\dfrac{P_\\omega V_\\omega}{T_\\omega}\n\n\\end{array}<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Comments and Microscopic Background<\/h2>\n<p>Although the previous formulation is empirical, it can be derived from first principles through the Kinetic Theory of Gases. In this model, the gas is a collection of particles that move and collide with each other and with the walls of the container. It is idealized with assumptions such as:<\/p>\n<ol>\n<li>Absence of long-range attractive or repulsive forces between particles.<\/li>\n<li>Point-like or negligibly small spherical particles.<\/li>\n<li>Perfectly elastic collisions between particles and with the walls.<\/li>\n<\/ol>\n<p>These idealizations simplify the analysis and, although no real gas perfectly satisfies them, they describe many gases well across a wide range of conditions and provide a foundation for <strong>Classical Thermodynamics<\/strong>, with applications ranging from heat engines to atmospheric and astrophysical physics.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Scope of Validity and Limitations<\/h2>\n<p>The ideal gas law is not universal. It deviates when the above assumptions cease to be reasonable or when effects beyond classical physics emerge.<\/p>\n<ul>\n<li><strong>High pressures and low temperatures:<\/strong> interactions between molecules are no longer negligible and the finite size of the particles becomes significant. A common correction is the van der Waals equation:\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(P + a\\left(\\dfrac{n}{V}\\right)^2\\right)\\,(V - nb)=nRT<\/span><\/div>\n<p>    with parameters <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> characteristic of each gas.\n  <\/li>\n<li><strong>Quantum regime:<\/strong> at very low temperatures or high densities, Bose\u2013Einstein or Fermi\u2013Dirac statistics appear, requiring <em>quantum gas<\/em> models.<\/li>\n<li><strong>Relativistic regime:<\/strong> if particles move at speeds close to the speed of light, relativistic corrections are required.<\/li>\n<\/ul>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Practical Notes<\/h2>\n<ul>\n<li>Always use temperature in <strong>Kelvin<\/strong> in the formulas: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T(\\mathrm{K}) = T(^{\\circ}\\mathrm{C}) + 273{,}15<\/span><\/span>.<\/li>\n<li>Ensure unit consistency: if working with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{atm}<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{L}<\/span><\/span>, use <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=0{,}082\\,057\\ \\mathrm{L\\,atm\\,mol^{-1}\\,K^{-1}}<\/span><\/span>; if using <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{Pa}<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{m^3}<\/span><\/span>, use <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=8{,}314\\,462\\,6\\ \\mathrm{J\\,mol^{-1}\\,K^{-1}}<\/span><\/span>.<\/li>\n<li>Remember that each empirical law was obtained by keeping one variable constant. Combining results requires understanding which thermodynamic process is being performed at each stage.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Empirical Formulation of the Ideal Gas Have you ever wondered why a balloon expands when heated or why a tire\u2019s pressure changes with altitude? In this lesson, we will review the laws governing these behaviors and how they lead to the ideal gas equation, along with its considerations and key points. Learning Objectives At the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35092,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":17,"footnotes":""},"categories":[635,919],"tags":[],"class_list":["post-30065","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physics","category-thermodynamics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Ideal Gas Equation - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\ud83c\udf21\ufe0f Descubre la fascinante ecuaci\u00f3n de los gases ideales: fundamentos, aplicaciones en termodin\u00e1mica, y sus l\u00edmites \ud83d\ude80\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/en\/the-ideal-gas-equation\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Ideal Gas Equation\" \/>\n<meta property=\"og:description\" content=\"\ud83c\udf21\ufe0f Discover the fascinating Ideal Gas Equation: principles, thermodynamics applications, and its limitations \ud83d\ude80\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/the-ideal-gas-equation\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-10T13:00:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-20T07:24:29+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-1.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1536\" \/>\n\t<meta property=\"og:image:height\" content=\"1024\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The Ideal Gas Equation\" \/>\n<meta name=\"twitter:description\" content=\"\ud83c\udf21\ufe0f Discover the fascinating Ideal Gas Equation: principles, thermodynamics applications, and its limitations \ud83d\ude80\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-1.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Ideal Gas Equation\",\"datePublished\":\"2021-03-10T13:00:00+00:00\",\"dateModified\":\"2025-10-20T07:24:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/\"},\"wordCount\":1849,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/min3-1.jpg\",\"articleSection\":[\"Physics\",\"Thermodynamics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-ideal-gas-equation\\\/\",\"name\":\"The Ideal Gas Equation - 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