{"id":30646,"date":"2021-04-22T13:00:24","date_gmt":"2021-04-22T13:00:24","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30646"},"modified":"2024-12-31T18:53:08","modified_gmt":"2024-12-31T18:53:08","slug":"the-heat-capacity","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/the-heat-capacity\/","title":{"rendered":"The Heat Capacity"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2 {\n        text-align: center;\n    }\n<\/style>\n<h1>The Heat Capacity<\/h1>\n<p style=\"text-align:center;\"><em>Have you ever wondered what really happens when you heat an object? The <strong>heat capacity<\/strong> is the key to understanding this fundamental phenomenon that connects energy, temperature, and the physical states of matter. This fascinating concept not only explains why water takes longer to heat up than metal but is also crucial in areas such as thermodynamics, engineering, and material sciences.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives<\/strong><\/p>\n<ol>\n<li><strong>Understand<\/strong> the concept of heat capacity as a measure of the amount of energy required to change the temperature of a system.<\/li>\n<li><strong>Apply<\/strong> the definitions of specific and molar heat capacity in practical contexts.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE OF CONTENTS<\/u>:<\/strong><br \/>\n<a href=\"#1\">The Problem of Heat Capacity<\/a><br \/>\n<a href=\"#2\">Types of Heat Capacities<\/a><br \/>\n<a href=\"#3\">Exercises<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/JkDcmM8CMv0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Problem of Heat Capacity<\/h2>\n<p>Discussing heat capacity places us in a peculiar situation. It is not possible to \u00abstore heat\u00bb in a body as if we were storing water in a bucket with a defined capacity, for example, 5 liters. This is not the meaning behind the words heat or heat capacity. In fact, this misunderstanding comes from a historical inheritance in physics, as in ancient times, heat was thought to be a kind of substance, an idea that we now know is incorrect. Today, we understand that heat is the energy in transit that causes temperature variations, analogous to how work is the energy used to change the state of a system.<\/p>\n<p>Since its creation, the term has had enough time to settle in the common usage of physics. Therefore, even though terms like \u00abcapacitance\u00bb might seem more appropriate, replacing it now would only mean renaming millions of books without providing any significant benefit.<\/p>\n<p>Nevertheless, and despite everything, the concept of heat and capacity is not particularly complicated. To understand the idea it conveys, it is enough to ask the following question:<\/p>\n<p><strong>What relationship exists between heat and the temperature change <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta T<\/span> of an object?<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=JkDcmM8CMv0&amp;t=184s\" target=\"_blank\" rel=\"noopener\"><strong>The answer to this question<\/strong><\/a> comes in the form of the differential relationship <span class=\"katex-eq\" data-katex-display=\"false\">dQ = CdT,<\/span> where <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> is the heat capacity.<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">C:= \\dfrac{dQ}{dT}.<\/span>\n<p>Here, as long as we remember that it tells us how much heat is needed to raise the temperature of a given object (it has nothing to do with an object&#8217;s capacity to heat other things), we will not risk making a mistake. As can be inferred from the definition of heat capacity, it has units of <span class=\"katex-eq\" data-katex-display=\"false\">[J\/K].<\/span>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Types of Heat Capacities<\/h2>\n<h3>Specific Heat Capacities<\/h3>\n<p>\n    <a href=\"https:\/\/www.youtube.com\/watch?v=JkDcmM8CMv0&amp;t=250s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong>When we talk about heat capacity,<\/strong><br \/>\n    <\/a><br \/>\n    there are two different presentations: the usual one, which we have already introduced, and the specific one. The difference between them is that specific heat capacity refers to heat capacity per unit mass, defined as:\n<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">c := \\dfrac{C}{m}<\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span> represents the mass of the body.<\/p>\n<p>If we multiply the specific heat capacity by the molar mass, we obtain the specific molar heat capacity, defined as:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">c_{mol} := c \\cdot m_{mol}<\/span>\n<p>It is important to note that lowercase letters are always used for specific magnitudes.<\/p>\n<h4>Example<\/h4>\n<p>\n    <a href=\"https:\/\/www.youtube.com\/watch?v=JkDcmM8CMv0&amp;t=301s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong>The specific heat capacity<\/strong><br \/>\n    <\/a><br \/>\n    of water at room temperature (<span class=\"katex-eq\" data-katex-display=\"false\">26^\\circ C<\/span>) is:\n<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">c = 4.181 \\cdot 10^3 \\left[\\dfrac{J}{kg \\cdot K}\\right]<\/span>\n<p>Calculate:<\/p>\n<table>\n<tbody>\n<tr>\n<td>\n<p>a) The energy required to increase the temperature of <span class=\"katex-eq\" data-katex-display=\"false\">2 \\, [kg]<\/span> of water by <span class=\"katex-eq\" data-katex-display=\"false\">14^\\circ C<\/span>.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>b) The heat capacity of <span class=\"katex-eq\" data-katex-display=\"false\">3 \\, [L]<\/span> of water.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>c) The molar specific heat capacity of water.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Heat Capacity at Constant Pressure and Volume<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=JkDcmM8CMv0&amp;t=706s\" target=\"_blank\" rel=\"noopener\"><strong>When we think of gases,<\/strong><\/a> we encounter an additional complication. In this case, we will be trying to find out how much heat we need to apply to a system to raise its temperature by <span class=\"katex-eq\" data-katex-display=\"false\">1[K].<\/span> But this can be done in two different ways:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 50px;\">(1)<\/td>\n<td>\n<p>Placing the gas in a sealed box and adding heat. As the temperature rises, the gas will be prevented from expanding so that its volume remains constant, but consequently, its pressure will increase. This method is known as \u00abat constant volume\u00bb.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50px;\">(2)<\/td>\n<td>\n<p>Placing the gas in a chamber with a movable piston. As the temperature rises, the gas will be allowed to push the piston, keeping its internal pressure constant, but consequently, its volume will increase. This method is known as \u00abat constant pressure\u00bb.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In both cases, we are applying <strong>constraints<\/strong> to the system. Under these circumstances, we need to modify our definition of heat capacity according to each case. Thus, we have heat capacities at constant volume and pressure: <span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span>, respectively. These magnitudes are expressed in terms of partial derivatives as follows:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V = \\left(\\dfrac{\\partial Q}{\\partial T}\\right)_V<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P = \\left(\\dfrac{\\partial Q}{\\partial T}\\right)_P<\/span>\n<h4>Example<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=JkDcmM8CMv0&amp;t=887s\" target=\"_blank\" rel=\"noopener\"><strong>The heat capacity of helium<\/strong><\/a> measured at constant volume is <span class=\"katex-eq\" data-katex-display=\"false\">3.12\\left[\\dfrac{kJ}{kg \\cdot K}\\right],<\/span> and when measured at constant pressure, it is <span class=\"katex-eq\" data-katex-display=\"false\">5.19\\left[\\dfrac{kJ}{kg \\cdot K}\\right].<\/span> Calculate the molar heat capacities at constant pressure and volume.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Exercises:<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/-LyA5Alrb8A\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<ol>\n<li>The world&#8217;s oceans contain approximately <span class=\"katex-eq\" data-katex-display=\"false\">10^{21}[kg]<\/span> of water. Calculate the heat capacity of the world&#8217;s oceans. <a href=\"https:\/\/www.youtube.com\/watch?v=-LyA5Alrb8A&amp;t=42s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>The world&#8217;s energy consumption is around <span class=\"katex-eq\" data-katex-display=\"false\">13[TW]<\/span> (and rising) (<span class=\"katex-eq\" data-katex-display=\"false\">1TW=10^{12}[W]<\/span>). Burning a ton of crude oil (approximately 7 barrels) produces around <span class=\"katex-eq\" data-katex-display=\"false\">42[GJ]<\/span> (<span class=\"katex-eq\" data-katex-display=\"false\">1[GJ]=10^9[J]<\/span>). If the world&#8217;s energy consumption relies solely on oil, how many barrels must be burned per second? <a href=\"https:\/\/www.youtube.com\/watch?v=-LyA5Alrb8A&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>The molar heat capacity of gold is <span class=\"katex-eq\" data-katex-display=\"false\">25.4\\left[\\dfrac{J}{mol \\cdot K}\\right].<\/span> Its density is <span class=\"katex-eq\" data-katex-display=\"false\">19.3\\cdot 10^3 \\left[\\frac{kg}{m^3}\\right].<\/span> Calculate the specific heat capacity of gold and its corresponding value per unit volume. <a href=\"https:\/\/www.youtube.com\/watch?v=-LyA5Alrb8A&amp;t=375s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>Two bodies with heat capacities <span class=\"katex-eq\" data-katex-display=\"false\">C_1<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">C_2<\/span> (assumed temperature-independent) and initial temperatures <span class=\"katex-eq\" data-katex-display=\"false\">T_1<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">T_2<\/span>, respectively, are put in contact. Show that the final temperature of the body <span class=\"katex-eq\" data-katex-display=\"false\">T_f<\/span> is given by:\n<span class=\"katex-eq\" data-katex-display=\"false\">T_f = \\dfrac{C_1 T_1 + C_2 T_2}{C_1 + C_2}<\/span>\n<p>And if <span class=\"katex-eq\" data-katex-display=\"false\">C_1<\/span> is much greater than <span class=\"katex-eq\" data-katex-display=\"false\">C_2,<\/span> then we have<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">T_f \\approx T_1 + \\dfrac{C_2}{C_1}(T_2 - T_1)<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=-LyA5Alrb8A&amp;t=726s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>The Heat Capacity Have you ever wondered what really happens when you heat an object? The heat capacity is the key to understanding this fundamental phenomenon that connects energy, temperature, and the physical states of matter. This fascinating concept not only explains why water takes longer to heat up than metal but is also crucial [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30645,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":7,"footnotes":""},"categories":[635,919],"tags":[],"class_list":["post-30646","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 v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Heat Capacity - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"What is heat capacity, and how does heat affect objects? 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