{"id":194234,"date":"2025-02-23T06:32:57","date_gmt":"2025-02-23T06:32:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194234"},"modified":"2025-02-23T06:32:59","modified_gmt":"2025-02-23T06:32:59","slug":"for-the-following-equilibrium-expression-0-35-x-2000-0-15-x0-05-2x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/23\/for-the-following-equilibrium-expression-0-35-x-2000-0-15-x0-05-2x\/","title":{"rendered":"For The Following Equilibrium Expression: (0.35 + X) 2000 (0.15 &#8211; X)(0.05 &#8211; 2x)"},"content":{"rendered":"\n<p>For The Following Equilibrium Expression: (0.35 + X) 2000 (0.15 &#8211; X)(0.05 &#8211; 2x) Find The Boundary Conditions (Show Your Work) And Roots For This Equation<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To analyze the equilibrium expression and find its roots, let&#8217;s first set up the equation clearly. We interpret your given expression as follows:<\/p>\n\n\n\n<p>[<br>(0.35 + x) \\cdot 2000 = (0.15 &#8211; x)(0.05 &#8211; 2x)<br>]<\/p>\n\n\n\n<p><strong>Step 1: Expand and Simplify<\/strong><\/p>\n\n\n\n<p>First, we expand both sides of the equation:<\/p>\n\n\n\n<p>[<br>2000 \\cdot (0.35 + x) = (0.15 &#8211; x)(0.05 &#8211; 2x)<br>]<\/p>\n\n\n\n<p>[<br>700 + 2000x = (0.15 &#8211; x)(0.05 &#8211; 2x)<br>]<\/p>\n\n\n\n<p>Now, expanding the right-hand side:<\/p>\n\n\n\n<p>[<br>0.15 \\times 0.05 &#8211; 0.15 \\times 2x &#8211; 0.05 \\times x + x \\times 2x<br>]<br>[<br>0.0075 &#8211; 0.30x &#8211; 0.05x + 2x^2<br>]<br>[<br>2x^2 &#8211; 0.35x + 0.0075<br>]<\/p>\n\n\n\n<p>Equating both sides:<\/p>\n\n\n\n<p>[<br>700 + 2000x = 2x^2 &#8211; 0.35x + 0.0075<br>]<\/p>\n\n\n\n<p>Bringing all terms to one side gives a standard form quadratic equation:<\/p>\n\n\n\n<p>[<br>2x^2 &#8211; 0.35x &#8211; 2000x + 0.0075 &#8211; 700 = 0<br>]<br>[<br>2x^2 &#8211; 2000.35x &#8211; 699.9925 = 0<br>]<\/p>\n\n\n\n<p><strong>Step 2: Boundary Conditions<\/strong><\/p>\n\n\n\n<p>Boundary conditions are determined by the physical or chemical constraints of the problem. Since (x) represents a change in concentration, (x) must be such that all concentrations remain non-negative:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(0.35 + x \\geq 0)<\/li>\n\n\n\n<li>(0.15 &#8211; x \\geq 0)<\/li>\n\n\n\n<li>(0.05 &#8211; 2x \\geq 0)<\/li>\n<\/ul>\n\n\n\n<p>From (0.35 + x \\geq 0), we get (x \\geq -0.35).<\/p>\n\n\n\n<p>From (0.15 &#8211; x \\geq 0), we get (x \\leq 0.15).<\/p>\n\n\n\n<p>From (0.05 &#8211; 2x \\geq 0), we get (x \\leq 0.025).<\/p>\n\n\n\n<p>Therefore, the boundary conditions are (x \\geq -0.35) and (x \\leq 0.025) since 0.025 is the most restrictive upper boundary.<\/p>\n\n\n\n<p><strong>Step 3: Finding Roots<\/strong><\/p>\n\n\n\n<p>The roots of the quadratic equation can be found using the quadratic formula:<\/p>\n\n\n\n<p>[<br>x = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<br>]<\/p>\n\n\n\n<p>where (a = 2), (b = -2000.35), and (c = -699.9925). Let&#8217;s solve this to find the roots.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>import numpy as np\n\na = 2\nb = -2000.35\nc = -699.9925\n\ndiscriminant = b**2 - 4*a*c\nx1 = (-b + np.sqrt(discriminant)) \/ (2*a)\nx2 = (-b - np.sqrt(discriminant)) \/ (2*a)\n\nx1, x2<\/code><\/pre>\n\n\n\n<p>These roots must be checked against the boundary conditions to determine if they are physically meaningful.<\/p>\n\n\n\n<p><strong>Generating an Image<\/strong><\/p>\n\n\n\n<p>Would you like an image representing the plot of this quadratic equation to visualize the roots and boundary conditions, or did you have another kind of image in mind?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For The Following Equilibrium Expression: (0.35 + X) 2000 (0.15 &#8211; X)(0.05 &#8211; 2x) Find The Boundary Conditions (Show Your Work) And Roots For This Equation The Correct Answer and Explanation is : To analyze the equilibrium expression and find its roots, let&#8217;s first set up the equation clearly. We interpret your given expression as [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-194234","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=194234"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194234\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194234"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}