{"id":258683,"date":"2025-07-18T08:50:10","date_gmt":"2025-07-18T08:50:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=258683"},"modified":"2025-07-18T08:50:12","modified_gmt":"2025-07-18T08:50:12","slug":"a-radioactive-isotope-has-a-half-life-of-2-5-years","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/18\/a-radioactive-isotope-has-a-half-life-of-2-5-years\/","title":{"rendered":"A radioactive isotope has a half-life of 2.5 years."},"content":{"rendered":"\n<p>A radioactive isotope has a half-life of 2.5 years. Which fraction of the original mass remains unchanged after 10 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-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the fraction of the original mass of a radioactive isotope that remains unchanged after 10 years, we can use the concept of half-life and the following formula:N(t)N0=(12)tT1\/2\\frac{N(t)}{N_0} = \\left( \\frac{1}{2} \\right)^{\\frac{t}{T_{1\/2}}}N0\u200bN(t)\u200b=(21\u200b)T1\/2\u200bt\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>N(t)N(t)N(t) is the amount of the isotope remaining after time ttt.<\/li>\n\n\n\n<li>N0N_0N0\u200b is the initial amount of the isotope.<\/li>\n\n\n\n<li>T1\/2T_{1\/2}T1\/2\u200b is the half-life of the isotope.<\/li>\n\n\n\n<li>ttt is the time that has passed (in years).<\/li>\n<\/ul>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Half-life T1\/2=2.5T_{1\/2} = 2.5T1\/2\u200b=2.5 years.<\/li>\n\n\n\n<li>Time t=10t = 10t=10 years.<\/li>\n<\/ul>\n\n\n\n<p>Plugging these values into the equation:N(t)N0=(12)102.5=(12)4\\frac{N(t)}{N_0} = \\left( \\frac{1}{2} \\right)^{\\frac{10}{2.5}} = \\left( \\frac{1}{2} \\right)^4N0\u200bN(t)\u200b=(21\u200b)2.510\u200b=(21\u200b)4N(t)N0=116\\frac{N(t)}{N_0} = \\frac{1}{16}N0\u200bN(t)\u200b=161\u200b<\/p>\n\n\n\n<p>Thus, after 10 years, only 116\\frac{1}{16}161\u200b of the original amount of the isotope remains.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The radioactive decay process follows an exponential decay model, where the substance loses half of its remaining mass after each half-life. In this case, the half-life is 2.5 years, meaning that after each 2.5-year period, half of the isotope&#8217;s mass is gone.<\/p>\n\n\n\n<p>After 10 years, the number of half-lives that have passed is:102.5=4&nbsp;half-lives\\frac{10}{2.5} = 4 \\text{ half-lives}2.510\u200b=4&nbsp;half-lives<\/p>\n\n\n\n<p>Each half-life reduces the remaining amount by half, so after four half-lives, the fraction of the isotope remaining is (12)4=116\\left( \\frac{1}{2} \\right)^4 = \\frac{1}{16}(21\u200b)4=161\u200b. This means only 1\/16 of the original mass of the isotope is still present after 10 years.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner9-212.jpeg\" alt=\"\" class=\"wp-image-258684\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A radioactive isotope has a half-life of 2.5 years. Which fraction of the original mass remains unchanged after 10 years? The Correct Answer and Explanation is: To find the fraction of the original mass of a radioactive isotope that remains unchanged after 10 years, we can use the concept of half-life and the following formula:N(t)N0=(12)tT1\/2\\frac{N(t)}{N_0} [&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-258683","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258683","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=258683"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258683\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=258683"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=258683"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=258683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}