{"id":242603,"date":"2025-07-04T08:34:34","date_gmt":"2025-07-04T08:34:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=242603"},"modified":"2025-07-04T08:34:36","modified_gmt":"2025-07-04T08:34:36","slug":"quantum-numbers-arise-naturally-from-the-mathematics-used-to-describe-the-possible-states-of-an-electron-in-an-atom","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/quantum-numbers-arise-naturally-from-the-mathematics-used-to-describe-the-possible-states-of-an-electron-in-an-atom\/","title":{"rendered":"Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom."},"content":{"rendered":"\n

Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (ml<\/sub>), and the spin quantum number (ms<\/sub>) have strict rules which govern the possible values. Identify allowable combinations of quantum numbers for an electron. Select all that apply. n = 2, l = 1, ml<\/sub> = -1, ms<\/sub> = 0 n = 3, l = -2, ml<\/sub> = 1, ms<\/sub> = +1\/2 n = 3, l = 2, ml<\/sub> = 1, ms<\/sub> = +1\/2 n = 5, l = 5, ml<\/sub> = -1, ms<\/sub> = +1\/2 n = 5, l = 3, ml<\/sub> = 3, ms<\/sub> = +1\/2 n = 4, l = 3, ml<\/sub> = 4, ms<\/sub> = +1\/2<\/p>\n\n\n\n

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

Let’s analyze each combination of quantum numbers based on the rules for quantum numbers:<\/p>\n\n\n\n

    \n
  1. n = 2, l = 1, m<sub>l<\/sub> = -1, m<sub>s<\/sub> = 0<\/strong>
    • The principal quantum number nnn must be a positive integer (2 is fine).<\/li>
    • The angular momentum quantum number lll can range from 0 to n\u22121n-1n\u22121. Here, l=1l = 1l=1 is valid because 0\u2264l\u2264n\u221210 \\leq l \\leq n-10\u2264l\u2264n\u22121, where n=2n = 2n=2, so lll can be 0 or 1.<\/li>
    • The magnetic quantum number mlm_lml\u200b ranges from \u2212l-l\u2212l to +l+l+l, so for l=1l = 1l=1, mlm_lml\u200b can be -1, 0, or +1. Therefore, ml=\u22121m_l = -1ml\u200b=\u22121 is valid.<\/li>
    • The spin quantum number msm_sms\u200b can only be +1\/2 or -1\/2, so ms=0m_s = 0ms\u200b=0 is not valid<\/strong>.<\/li><\/ul>Conclusion<\/strong>: This combination is invalid<\/strong> because msm_sms\u200b cannot be 0.<\/li>\n\n\n\n
    • n = 3, l = -2, m<sub>l<\/sub> = 1, m<sub>s<\/sub> = +1\/2<\/strong>
      • The angular momentum quantum number lll must be between 0 and n\u22121n-1n\u22121, so lll can range from 0 to 2 when n=3n = 3n=3. Therefore, l=\u22122l = -2l=\u22122 is not valid<\/strong>.<\/li><\/ul>Conclusion<\/strong>: This combination is invalid<\/strong> because lll cannot be negative.<\/li>\n\n\n\n
      • n = 3, l = 2, m<sub>l<\/sub> = 1, m<sub>s<\/sub> = +1\/2<\/strong>
        • For n=3n = 3n=3, lll can be 0, 1, or 2, so l=2l = 2l=2 is valid.<\/li>
        • The magnetic quantum number mlm_lml\u200b ranges from \u2212l-l\u2212l to +l+l+l, so for l=2l = 2l=2, mlm_lml\u200b can be -2, -1, 0, +1, or +2. Therefore, ml=1m_l = 1ml\u200b=1 is valid.<\/li>
        • The spin quantum number ms=+1\/2m_s = +1\/2ms\u200b=+1\/2 is valid.<\/li><\/ul>Conclusion<\/strong>: This combination is valid<\/strong>.<\/li>\n\n\n\n
        • n = 5, l = 5, m<sub>l<\/sub> = -1, m<sub>s<\/sub> = +1\/2<\/strong>
          • The angular momentum quantum number lll must be between 0 and n\u22121n-1n\u22121. Here, n=5n = 5n=5, so lll can range from 0 to 4. Therefore, l=5l = 5l=5 is not valid<\/strong> because it exceeds n\u22121n-1n\u22121.<\/li><\/ul>Conclusion<\/strong>: This combination is invalid<\/strong> because lll must be less than nnn.<\/li>\n\n\n\n
          • n = 5, l = 3, m<sub>l<\/sub> = 3, m<sub>s<\/sub> = +1\/2<\/strong>
            • For n=5n = 5n=5, l=3l = 3l=3 is valid (since lll ranges from 0 to 4).<\/li>
            • The magnetic quantum number mlm_lml\u200b must be between \u2212l-l\u2212l and +l+l+l, so for l=3l = 3l=3, mlm_lml\u200b can range from -3 to +3. Therefore, ml=3m_l = 3ml\u200b=3 is valid.<\/li>
            • The spin quantum number ms=+1\/2m_s = +1\/2ms\u200b=+1\/2 is valid.<\/li><\/ul>Conclusion<\/strong>: This combination is valid<\/strong>.<\/li>\n\n\n\n
            • n = 4, l = 3, m<sub>l<\/sub> = 4, m<sub>s<\/sub> = +1\/2<\/strong>
              • For n=4n = 4n=4, l=3l = 3l=3 is valid.<\/li>
              • The magnetic quantum number mlm_lml\u200b can range from \u2212l-l\u2212l to +l+l+l, so for l=3l = 3l=3, mlm_lml\u200b can be -3, -2, -1, 0, 1, 2, or 3. Therefore, ml=4m_l = 4ml\u200b=4 is not valid<\/strong> because it exceeds lll.<\/li><\/ul>Conclusion<\/strong>: This combination is invalid<\/strong> because mlm_lml\u200b must be within the range of \u2212l-l\u2212l to +l+l+l.<\/li>\n<\/ol>\n\n\n\n

                Final Allowable Combinations:<\/h3>\n\n\n\n
                  \n
                • Valid combinations<\/strong>:\n
                    \n
                  • n = 3, l = 2, m<sub>l<\/sub> = 1, m<sub>s<\/sub> = +1\/2<\/li>\n\n\n\n
                  • n = 5, l = 3, m<sub>l<\/sub> = 3, m<sub>s<\/sub> = +1\/2<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
                    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

                    Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms) have strict rules which govern the possible values. Identify allowable […]<\/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-242603","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/242603","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=242603"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/242603\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=242603"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=242603"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=242603"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}