{"id":231647,"date":"2025-06-11T09:29:04","date_gmt":"2025-06-11T09:29:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=231647"},"modified":"2025-06-11T09:29:06","modified_gmt":"2025-06-11T09:29:06","slug":"calculate-the-packing-efficiency-of-the-body-centered-cubic-unit-cell","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/calculate-the-packing-efficiency-of-the-body-centered-cubic-unit-cell\/","title":{"rendered":"Calculate the packing efficiency of the body-centered cubic unit cell"},"content":{"rendered":"\n


Calculate the packing efficiency of the body-centered cubic unit cell. Show your work. Drag the appropriate labels to their respective targets. 74<\/p>\n\n\n\n

52
68<\/p>\n\n\n\n

1 atom<\/p>\n\n\n\n

2 atoms<\/p>\n\n\n\n

4 atoms<\/p>\n\n\n\n

Submit Request Answer Group 1 Group 4 Group 2<\/p>\n\n\n\n

Group 2 Group 2 Group 2 %<\/p>\n\n\n\n

Group 2
Reset Help<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Step-by-Step Explanation (Like a Textbook)<\/strong><\/h3>\n\n\n\n

\ud83d\udd39 1. Understand the Structure<\/strong><\/h4>\n\n\n\n

A BCC unit cell has:<\/p>\n\n\n\n

    \n
  • 2 atoms per unit cell<\/strong> (1 from the center, 1 from the corners combined)<\/li>\n\n\n\n
  • Atoms touch along the body diagonal<\/strong> of the cube.<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    \ud83d\udd39 2. Volume of Atoms in the Unit Cell (V_atoms)<\/strong><\/h4>\n\n\n\n

    Each atom has volume: Vatom=43\u03c0r3V_{\\text{atom}} = \\frac{4}{3}\\pi r^3Vatom\u200b=34\u200b\u03c0r3<\/p>\n\n\n\n

    Since BCC has 2 atoms<\/strong> per unit cell: Vatoms=2\u00d743\u03c0r3=83\u03c0r3V_{\\text{atoms}} = 2 \\times \\frac{4}{3}\\pi r^3 = \\frac{8}{3}\\pi r^3Vatoms\u200b=2\u00d734\u200b\u03c0r3=38\u200b\u03c0r3<\/p>\n\n\n\n

      \n
    • \u2705 Group 1<\/strong>: 2 atoms<\/strong><\/li>\n\n\n\n
    • \u2705 Group 4<\/strong>: 43\u03c0r3\\frac{4}{3}\\pi r^334\u200b\u03c0r3<\/li>\n\n\n\n
    • \u2705 Group 2<\/strong>: 83\u03c0r3\\frac{8}{3}\\pi r^338\u200b\u03c0r3<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      \ud83d\udd39 3. Volume of the Unit Cell (V_unit cell)<\/strong><\/h4>\n\n\n\n

      Atoms in BCC touch along the body diagonal: body diagonal=4r=3l\u21d2l=4r3\\text{body diagonal} = 4r = \\sqrt{3}l \\Rightarrow l = \\frac{4r}{\\sqrt{3}}body diagonal=4r=3\u200bl\u21d2l=3\u200b4r\u200b<\/p>\n\n\n\n

      So, Vunit cell=l3=(4r3)3V_{\\text{unit cell}} = l^3 = \\left( \\frac{4r}{\\sqrt{3}} \\right)^3Vunit cell\u200b=l3=(3\u200b4r\u200b)3<\/p>\n\n\n\n

        \n
      • \u2705 Group 2<\/strong>: (4r3)3\\left(\\frac{4r}{\\sqrt{3}}\\right)^3(3\u200b4r\u200b)3<\/li>\n<\/ul>\n\n\n\n
        \n\n\n\n

        \ud83d\udd39 4. Packing Efficiency (%V_filled)<\/strong><\/h4>\n\n\n\n

        %Vfilled=VatomsVunit cell\u00d7100%=83\u03c0r3(4r3)3\u00d7100%\\%\\text{V}_{\\text{filled}} = \\frac{V_{\\text{atoms}}}{V_{\\text{unit cell}}} \\times 100\\% = \\frac{\\frac{8}{3}\\pi r^3}{\\left( \\frac{4r}{\\sqrt{3}} \\right)^3} \\times 100\\%%Vfilled\u200b=Vunit cell\u200bVatoms\u200b\u200b\u00d7100%=(3\u200b4r\u200b)338\u200b\u03c0r3\u200b\u00d7100%<\/p>\n\n\n\n

        Simplifying: =83\u03c0r364r333\u00d7100%=8\u03c064\/3\u00d7100%=\u03c038\u00d7100%\u224868%= \\frac{\\frac{8}{3}\\pi r^3}{\\frac{64r^3}{3\\sqrt{3}}} \\times 100\\% = \\frac{8\\pi}{64\/\\sqrt{3}} \\times 100\\% = \\frac{\\pi \\sqrt{3}}{8} \\times 100\\% \\approx 68\\%=33\u200b64r3\u200b38\u200b\u03c0r3\u200b\u00d7100%=64\/3\u200b8\u03c0\u200b\u00d7100%=8\u03c03\u200b\u200b\u00d7100%\u224868%<\/p>\n\n\n\n

          \n
        • \u2705 Group 3<\/strong>: \u03c038\\frac{\\pi \\sqrt{3}}{8}8\u03c03\u200b\u200b<\/li>\n\n\n\n
        • \u2705 Group 5<\/strong>: 68<\/strong><\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          \u2705 Final Label Placement<\/strong><\/h3>\n\n\n\n
          Label<\/th>Goes To<\/th><\/tr><\/thead>
          Group 1<\/strong><\/td>2 atoms<\/td><\/tr>
          Group 4<\/strong><\/td>43\u03c0r3\\frac{4}{3} \\pi r^334\u200b\u03c0r3<\/td><\/tr>
          Group 2<\/strong><\/td>83\u03c0r3\\frac{8}{3} \\pi r^338\u200b\u03c0r3, (4r3)3\\left(\\frac{4r}{\\sqrt{3}}\\right)^3(3\u200b4r\u200b)3<\/td><\/tr>
          Group 3<\/strong><\/td>\u03c038\\frac{\\pi \\sqrt{3}}{8}8\u03c03\u200b\u200b<\/td><\/tr>
          Group 5<\/strong><\/td>68<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
          \n\n\n\n

          \ud83e\udde0 Concept Summary<\/strong><\/h3>\n\n\n\n
            \n
          • BCC<\/strong> has 2 atoms per unit cell<\/strong>.<\/li>\n\n\n\n
          • Packing efficiency is the fraction of unit cell volume occupied by atoms<\/strong>.<\/li>\n\n\n\n
          • The derivation involves using geometry (atoms touch along the diagonal) and volume formulas.<\/li>\n\n\n\n
          • Final packing efficiency of BCC is approximately 68%<\/strong>, meaning 68% of the space is filled with atoms while the rest is empty space.<\/li>\n<\/ul>\n\n\n\n
            \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

            Calculate the packing efficiency of the body-centered cubic unit cell. Show your work. Drag the appropriate labels to their respective targets. 74 5268 1 atom 2 atoms 4 atoms Submit Request Answer Group 1 Group 4 Group 2 Group 2 Group 2 Group 2 % Group 2Reset Help The Correct Answer and Explanation is: Step-by-Step […]<\/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-231647","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231647","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=231647"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231647\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231647"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231647"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}