{"id":168329,"date":"2024-11-17T09:00:49","date_gmt":"2024-11-17T09:00:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=168329"},"modified":"2024-11-17T09:00:53","modified_gmt":"2024-11-17T09:00:53","slug":"what-is-the-age-in-years-of-a-bone-in-which-the-14c-12c-ratio-is-measured-to-be-4-45x10-132","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/17\/what-is-the-age-in-years-of-a-bone-in-which-the-14c-12c-ratio-is-measured-to-be-4-45x10-132\/","title":{"rendered":"What is the age in years of a bone in which the 14C\/12C ratio is measured to be 4.45&#215;10-132"},"content":{"rendered":"\n<p>What is the age in years of a bone in which the 14C\/12C ratio is measured to be 4.45&#215;10-132 Express your answer as a number of years.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine the age of the bone, we use the concept of radiocarbon dating, which relies on the decay of carbon-14 ((^{14}C)) to carbon-12 ((^{12}C)) over time. The ratio of (^{14}C\/^{12}C) in a sample decreases over time as carbon-14 decays, and by measuring this ratio, we can calculate the age of the sample.<\/p>\n\n\n\n<p>The ratio of (^{14}C\/^{12}C) in a living organism is roughly constant, but as the organism dies, the (^{14}C) begins to decay, while the (^{12}C) remains constant. The half-life of carbon-14 is about <strong>5730 years<\/strong>, which means that after 5730 years, half of the original amount of (^{14}C) in the sample will have decayed.<\/p>\n\n\n\n<p>The general equation for the decay of carbon-14 is given by:<\/p>\n\n\n\n<p>[<br>N(t) = N_0 e^{-\\lambda t}<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(N(t)) is the amount of (^{14}C) remaining after time (t),<\/li>\n\n\n\n<li>(N_0) is the initial amount of (^{14}C),<\/li>\n\n\n\n<li>(\\lambda) is the decay constant, and<\/li>\n\n\n\n<li>(t) is the time that has passed.<\/li>\n<\/ul>\n\n\n\n<p>We can also express the decay of carbon-14 in terms of its half-life using the following relationship between (\\lambda) and the half-life (t_{1\/2}):<\/p>\n\n\n\n<p>[<br>\\lambda = \\frac{\\ln(2)}{t_{1\/2}}<br>]<\/p>\n\n\n\n<p>Substituting the half-life of carbon-14 ((5730 \\, \\text{years})) into this equation:<\/p>\n\n\n\n<p>[<br>\\lambda = \\frac{\\ln(2)}{5730} \\approx 1.21 \\times 10^{-4} \\, \\text{per year}<br>]<\/p>\n\n\n\n<p>The ratio of the (^{14}C\/^{12}C) in the sample is given as (4.45 \\times 10^{-13}), which is a fraction of the original ratio. To find the age of the sample, we rearrange the decay formula:<\/p>\n\n\n\n<p>[<br>\\frac{N(t)}{N_0} = e^{-\\lambda t}<br>]<\/p>\n\n\n\n<p>Taking the natural logarithm of both sides:<\/p>\n\n\n\n<p>[<br>\\ln \\left( \\frac{N(t)}{N_0} \\right) = -\\lambda t<br>]<\/p>\n\n\n\n<p>Substituting the values:<\/p>\n\n\n\n<p>[<br>\\ln \\left( 4.45 \\times 10^{-13} \\right) = -1.21 \\times 10^{-4} t<br>]<\/p>\n\n\n\n<p>[<br>-28.92 = -1.21 \\times 10^{-4} t<br>]<\/p>\n\n\n\n<p>Solving for (t):<\/p>\n\n\n\n<p>[<br>t = \\frac{-28.92}{-1.21 \\times 10^{-4}} \\approx 2.39 \\times 10^5 \\, \\text{years}<br>]<\/p>\n\n\n\n<p>Thus, the age of the bone is approximately <strong>239,000 years<\/strong>.<\/p>\n\n\n\n<p>This calculation demonstrates the application of radiocarbon dating to determine the age of ancient organic materials based on the measurement of remaining carbon-14.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is the age in years of a bone in which the 14C\/12C ratio is measured to be 4.45&#215;10-132 Express your answer as a number of years. The Correct Answer and Explanation is: To determine the age of the bone, we use the concept of radiocarbon dating, which relies on the decay of carbon-14 ((^{14}C)) [&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-168329","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168329","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=168329"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168329\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=168329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=168329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=168329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}