{"id":268001,"date":"2025-07-23T18:37:56","date_gmt":"2025-07-23T18:37:56","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=268001"},"modified":"2025-07-23T18:37:59","modified_gmt":"2025-07-23T18:37:59","slug":"lithium-6-941-g-mol-packs-into-a-body-centered-cubic-bcc-unit-cell","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/lithium-6-941-g-mol-packs-into-a-body-centered-cubic-bcc-unit-cell\/","title":{"rendered":"Lithium (6.941 g\/mol) packs into a body-centered cubic (BCC) unit cell."},"content":{"rendered":"\n

Lithium (6.941 g\/mol) packs into a body-centered cubic (BCC) unit cell. If the unit cell has a known volume of 0.04350 nm^3, what is the atomic radius of Li in units of nanometers? Show all your work.<\/p>\n\n\n\n

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

To find the atomic radius of lithium (Li) in a body-centered cubic (BCC) unit cell, we need to use the information provided: the molar mass of lithium (6.941 g\/mol), the volume of the unit cell (0.04350 nm\u00b3), and the fact that lithium crystallizes in a BCC structure.<\/p>\n\n\n\n

Step-by-step solution:<\/h3>\n\n\n\n
    \n
  1. Understand the relationship between the BCC structure and the unit cell:<\/strong>
    In a BCC unit cell, there are two atoms per unit cell (one at the center and one-eighth of an atom at each of the eight corners). The relationship between the atomic radius RRR and the unit cell edge length aaa in a BCC structure is: Diagonal\u00a0of\u00a0the\u00a0unit\u00a0cell=4R\\text{Diagonal of the unit cell} = 4RDiagonal\u00a0of\u00a0the\u00a0unit\u00a0cell=4R The diagonal in terms of the edge length aaa is given by the Pythagorean theorem: Diagonal=3a\\text{Diagonal} = \\sqrt{3}aDiagonal=3\u200ba So, equating the two expressions for the diagonal: 3a=4R\\sqrt{3}a = 4R3\u200ba=4R Rearranging to solve for aaa: a=4R3a = \\frac{4R}{\\sqrt{3}}a=3\u200b4R\u200b<\/li>\n\n\n\n
  2. Use the volume of the unit cell to find the edge length aaa:<\/strong>
    The volume VVV of the unit cell is related to the edge length aaa by the equation: V=a3V = a^3V=a3 Given that the volume VVV is 0.04350 nm\u00b3, we can solve for aaa: a=V3=0.04350\u2009nm33\u22480.353\u2009nma = \\sqrt[3]{V} = \\sqrt[3]{0.04350 \\, \\text{nm}^3} \\approx 0.353 \\, \\text{nm}a=3V\u200b=30.04350nm3\u200b\u22480.353nm<\/li>\n\n\n\n
  3. Solve for the atomic radius RRR:<\/strong>
    Now that we have aaa, we can substitute it into the equation for aaa in terms of RRR from the BCC geometry: a=4R3a = \\frac{4R}{\\sqrt{3}}a=3\u200b4R\u200b Substituting a=0.353\u2009nma = 0.353 \\, \\text{nm}a=0.353nm: 0.353=4R30.353 = \\frac{4R}{\\sqrt{3}}0.353=3\u200b4R\u200b Solving for RRR: R=0.353\u00d734\u22480.153\u2009nmR = \\frac{0.353 \\times \\sqrt{3}}{4} \\approx 0.153 \\, \\text{nm}R=40.353\u00d73\u200b\u200b\u22480.153nm<\/li>\n<\/ol>\n\n\n\n

    Final Answer:<\/h3>\n\n\n\n

    The atomic radius of lithium (Li) in a BCC unit cell is approximately 0.153 nm<\/strong>.<\/p>\n\n\n\n

    Explanation:<\/h3>\n\n\n\n

    We used the relationship between the unit cell geometry in a BCC structure and the volume of the unit cell to find the edge length aaa. Using this, we derived the atomic radius RRR by applying the known relationship between aaa and RRR in the BCC structure. This method leverages both the geometric and volumetric properties of the unit cell.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Lithium (6.941 g\/mol) packs into a body-centered cubic (BCC) unit cell. If the unit cell has a known volume of 0.04350 nm^3, what is the atomic radius of Li in units of nanometers? Show all your work. The Correct Answer and Explanation is: To find the atomic radius of lithium (Li) in a body-centered cubic […]<\/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-268001","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268001","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=268001"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268001\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=268001"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=268001"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=268001"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}