This is a follow-on from the very awesome video by Randall Munroe on making a wall made of 1L samples (10x10x10cm) of every element in the periodic table.<\/p>\n\n\n\n
The question is ‘what exactly is the power output of astatine?’<\/em><\/p>\n\n\n\n A solid $7\\text{ kg}$ sample of Astatine-210 (the most stable isotope) would generate a thermal output of approximately $233\\text{ MW}$ $(233,000,000\\text{ watts})$ at the moment of its creation. If the sample consisted of Astatine-211\u2014the isotope commonly created in particle accelerators\u2014the thermal output would climb even higher, reaching roughly $495\\text{ MW}$ $^{[1, 2]}$.<\/p>\n\n To put this scale into perspective, a 1-litre block of astatine would produce as much heat as a small commercial nuclear reactor, entirely concentrated within a handful of matter.<\/p>\n\n But don’t take my word for it; LET’S USE FIRST PRINCIPLES!!!.<\/p>\n\n To find out how much heat is released, we first determine how many atoms are present in a $7\\text{ kg}$ ($7,000\\text{ g}$) sample using Avogadro’s number:<\/p>\n\n $$N = \\frac{\\text{Mass}}{\\text{Molar Mass}} \\times N_A$$<\/p>\n\n Using Astatine-210 ($210\\text{ g\/mol}$) $^{[2]}$:<\/p>\n $$N = \\frac{7,000\\text{ g}}{210\\text{ g\/mol}} \\times 6.022 \\times 10^{23}\\text{ atoms\/mol}$$<\/p>\n\n $$\\approx 2.0073 \\times 10^{25}\\text{ atoms}$$<\/p>\n\n Next, we calculate the decay constant ($\\lambda$) using the half-life of Astatine-210, which stands at $8.1\\text{ hours}$ ($29,160\\text{ seconds}$) $^{[3]}$:<\/p>\n\n $$\\lambda = \\frac{\\ln(2)}{t_{1\/2}} = \\frac{0.69315}{29,160\\text{ s}}$$<\/p>\n\n $$\\approx 2.377 \\times 10^{-5}\\text{ s}^{-1}$$<\/p>\n\n The total radioactivity, or activity ($A$), measured in Becquerels ($\\text{Bq}$, decays per second) is the product of the number of atoms and the decay constant:<\/p>\n\n $$A = \\lambda \\times N$$<\/p>\n $$A = (2.377 \\times 10^{-5}\\text{ s}^{-1}) \\times (2.0073 \\times 10^{25})$$<\/p>\n\n $$\\approx 4.771 \\times 10^{20}\\text{ Bq}$$<\/p>\n\n Every individual decay of Astatine-210 releases an average combined radiation energy\u2014comprising alphas, electrons, and photons\u2014of approximately $3.0515\\text{ MeV}$. We convert this particle energy into Joules ($1\\text{ MeV} = 1.6022 \\times 10^{-13}\\text{ J}$) $^{[3, 4]}$:<\/p>\n\n $$\\text{Energy per decay} = 3.0515\\text{ MeV} \\times 1.6022 \\times 10^{-13}\\text{ J\/MeV}$$<\/p>\n\n $$\\approx 4.889 \\times 10^{-13}\\text{ J}$$<\/p>\n\n Multiplying the total decays per second by the energy per decay yields the absolute power output:<\/p>\n\n $$P = A \\times \\text{Energy per decay}$$<\/p>\n $$P = (4.771 \\times 10^{20}\\text{ Bq}) \\times (4.889 \\times 10^{-13}\\text{ J})$$<\/p>\n $$\\approx 2.333 \\times 10^8\\text{ W}$$<\/p>\n\n The hypothetical initial thermal output of a solid $7\\text{ kg}$ sample of Astatine-210 sits at $2.33 \\times 10^8\\text{ watts}$ ($233\\text{ MW}$).<\/p>\n\n For those who prefer programmatic verification, here is the Python script used to model these calculations based on specific activity and mean energy.<\/p><\/div>\n\n\n References<\/p>\n\n\n\n The Thermal Fury of Astatine: Calculating the Heat Output of an element you’ve never heard of.<\/p>\n This is a follow-on from the very awesome video by Randall Munroe on making a wall made of 1L samples (10x10x10cm) of every element in the periodic table.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[35],"class_list":["post-350","post","type-post","status-publish","format-standard","hentry","category-doing-a-thing","tag-zero-practical-application"],"_links":{"self":[{"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/posts\/350","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/comments?post=350"}],"version-history":[{"count":28,"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/posts\/350\/revisions"}],"predecessor-version":[{"id":390,"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/posts\/350\/revisions\/390"}],"wp:attachment":[{"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/media?parent=350"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/categories?post=350"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yoyodyne.com.au\/index.php\/wp-json\/wp\/v2\/tags?post=350"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n1. Count the Atoms<\/h3>\n
2. Determine the Decay Constant<\/h3>\n\n
3. Calculate Total Radioactivity<\/h3>\n
4. Convert Energy to Watts<\/h3>\n\n
5. Final Thermal Output Result<\/h3>\n
# Calculate the thermal power of Astatine-210 based on # specific activity and mean energy\n#\n# Specific activity # = 6.817e16 Bq\/g\n# Mass = 7000 g\n# Mean energy per decay = 0.0097 + 0.07962 + 2.96215 = # 3.05147 MeV\n\nspec_activity = 6.817e16 # Bq\/g\nmass = 7000 # g\ntotal_activity = spec_activity * mass # Bq\n\nmean_energy_mev = 3.05147\nenergy_j = mean_energy_mev * 1.60218e-13\n\npower_w = total_activity * energy_j\nprint(f\"At-210 Power: {power_w:.4e} Watts\")<\/code><\/pre>\n\n\n\n\n