{"id":218485,"date":"2025-05-23T20:48:39","date_gmt":"2025-05-23T20:48:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=218485"},"modified":"2025-05-23T20:48:42","modified_gmt":"2025-05-23T20:48:42","slug":"the-escape-velocity-from-mercury-is-much-smaller-than-from-earth-and-is-only-4-25-km-s","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/23\/the-escape-velocity-from-mercury-is-much-smaller-than-from-earth-and-is-only-4-25-km-s\/","title":{"rendered":"The escape velocity from Mercury is much smaller than from Earth and is only 4.25 km\/s."},"content":{"rendered":"\n<p>The escape velocity from Mercury is much smaller than from Earth and is only 4.25 km\/s. At what temperature (in K) would nitrogen molecules (molar mass is equal to 28.01 g\/mol) have an average velocity vrms equal to Mercury&#8217;s escape velocity?<\/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 temperature at which nitrogen molecules (N2\\text{N}_2) have an average root-mean-square (rms) speed equal to the escape velocity from Mercury (4.25 km\/s), we use the formula for vrmsv_{\\text{rms}}: vrms=3kTmv_{\\text{rms}} = \\sqrt{\\frac{3kT}{m}}<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>vrmsv_{\\text{rms}} is the root-mean-square speed,<\/li>\n\n\n\n<li>kk is Boltzmann&#8217;s constant =1.38\u00d710\u221223\u2009J\/K= 1.38 \\times 10^{-23} \\, \\text{J\/K},<\/li>\n\n\n\n<li>TT is the temperature in Kelvin,<\/li>\n\n\n\n<li>mm is the mass of a nitrogen molecule in kilograms.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Convert molar mass to mass per molecule<\/h3>\n\n\n\n<p>The molar mass of nitrogen is 28.01 g\/mol. To convert this to kg per molecule: m=28.01\u00d710\u221236.022\u00d71023=4.65\u00d710\u221226\u2009kgm = \\frac{28.01 \\times 10^{-3}}{6.022 \\times 10^{23}} = 4.65 \\times 10^{-26} \\, \\text{kg}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Solve for temperature<\/h3>\n\n\n\n<p>Given vrms=4.25\u00d7103\u2009m\/sv_{\\text{rms}} = 4.25 \\times 10^3 \\, \\text{m\/s}, we solve: T=mvrms23k=(4.65\u00d710\u221226)(4.25\u00d7103)23(1.38\u00d710\u221223)T = \\frac{m v_{\\text{rms}}^2}{3k} = \\frac{(4.65 \\times 10^{-26})(4.25 \\times 10^3)^2}{3(1.38 \\times 10^{-23})} T=(4.65\u00d710\u221226)(1.80625\u00d7107)4.14\u00d710\u221223=8.4\u00d710\u2212194.14\u00d710\u221223\u224820,290\u2009KT = \\frac{(4.65 \\times 10^{-26})(1.80625 \\times 10^7)}{4.14 \\times 10^{-23}} = \\frac{8.4 \\times 10^{-19}}{4.14 \\times 10^{-23}} \\approx 20,290 \\, \\text{K}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>T\u224820,\u2009\u2063290\u2009K\\boxed{T \\approx 20,\\!290 \\, \\text{K}}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><\/h3>\n\n\n\n<p>The escape velocity from a planet is the minimum speed an object must have to overcome the planet&#8217;s gravitational pull without further propulsion. For Mercury, this velocity is 4.25 km\/s (4250 m\/s), which is significantly lower than Earth\u2019s escape velocity due to Mercury\u2019s smaller mass and radius.<\/p>\n\n\n\n<p>Molecules in a gas move with a distribution of speeds, but the root-mean-square (rms) speed provides a useful average when considering thermal motion. The rms speed of a gas molecule is related to the temperature and molecular mass by the equation: vrms=3kTmv_{\\text{rms}} = \\sqrt{\\frac{3kT}{m}}<\/p>\n\n\n\n<p>To determine the temperature at which nitrogen molecules achieve this escape speed, we set vrmsv_{\\text{rms}} equal to 4.25 km\/s and solve for TT. First, the molar mass of nitrogen (28.01 g\/mol) is converted to the mass of a single molecule in kilograms. Boltzmann\u2019s constant relates the average kinetic energy of particles in a gas to temperature.<\/p>\n\n\n\n<p>Using the derived equation, we calculate the required temperature to be approximately 20,290 K. This is extremely hot and far above Mercury\u2019s actual surface temperatures, which typically range from about 100 K at night to 700 K during the day. Therefore, under natural conditions, nitrogen gas on Mercury would not reach escape velocity just from thermal motion. However, this calculation helps illustrate why lighter gases like hydrogen and helium are more likely to escape a planet&#8217;s atmosphere over time\u2014because their rms speeds are higher at any given temperature.<\/p>\n\n\n\n<p>This method is a useful way to explore atmospheric retention and planetary habitability across the solar system.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-257.png\" alt=\"\" class=\"wp-image-218486\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The escape velocity from Mercury is much smaller than from Earth and is only 4.25 km\/s. At what temperature (in K) would nitrogen molecules (molar mass is equal to 28.01 g\/mol) have an average velocity vrms equal to Mercury&#8217;s escape velocity? The Correct Answer and Explanation is: To find the temperature at which nitrogen molecules [&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-218485","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218485","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=218485"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218485\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=218485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=218485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=218485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}