{"id":34759,"date":"2021-07-03T13:00:01","date_gmt":"2021-07-03T13:00:01","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34759"},"modified":"2025-09-20T08:16:32","modified_gmt":"2025-09-20T08:16:32","slug":"lex-prima-thermodynamicae","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/lex-prima-thermodynamicae\/","title":{"rendered":"Lex Prima Thermodynamicae"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\ttext-align:center;\n\t}\n<\/style>\n<h1>Lex Prima Thermodynamicae<\/h1>\n<p style=\"text-align:center;\"><em>Lex Prima Thermodynamicae est fundamentum quod coniungit notiones fundamentales ut calorem, laborem et energiam internam, statuens energiam nec creari nec deleri, sed tantum transformari. Hoc documentum explanat quomodo haec lex ad systemata clausa applicetur, penitius perscrutans analysin laboris thermodynamici, capacitates calorificas atque proprietates statisticas gasorum. Per coniunctionem formulationum mathematicarum et rationum physicorum, invenies instrumenta essentialia ad intellegendos processus energeticos in systematibus complexis.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi:<\/strong><br \/>\nPost hanc lectionem discipulus poterit\n<\/p>\n<ol>\n<li><strong>Justificare<\/strong> Legem Primam Thermodynamicae pro systematibus clausis, explicans relationes inter calorem, laborem et energiam internam.<\/li>\n<li><strong>Analyzare<\/strong> notionem laboris thermodynamici in processibus compressionis et expansionis, utens formulis differentialibus.<\/li>\n<li><strong>Computare<\/strong> capacitatem calorificam in condicionibus voluminis et pressionis constantibus, applicando restrictiones thermodynamicas.<\/li>\n<li><strong>Explicare<\/strong> distributionem Maxwell-Boltzmann atque principium aequipartitionis energiae in systematibus molecularibus.<\/li>\n<li><strong>Demonstrari<\/strong> relationes specificas inter capacitates calorificas, indicem adiabaticum et alias proprietates thermodynamicas gasorum idealiorem.<\/li>\n<\/ol>\n<p style=\"text-align:center\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\">Formulatio Legis Primae Thermodynamicae<\/a><br \/>\n<a href=\"#2\">Labor thermodynamicus<\/a><br \/>\n<a href=\"#3\">Capacitas calorifica<\/a><br \/>\n<a href=\"#4\">Distributio Maxwell-Boltzmann et aequipartitio Energiae<\/a><br \/>\n<a href=\"#5\">Exercitia<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/T6K1Nizc5NE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Formulatio Legis Primae Thermodynamicae<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=140s\" target=\"_blank\" rel=\"noopener\"><strong>Lex prima thermodynamicae<\/strong><\/a> statuit quod:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #c0ffc0;\"><span style=\"color: #000080;\">LEX PRIMA THERMODYNAMICA<br \/>\n<strong>Energia nec creatur nec destruitur; praeterea, calor et labor sunt formae energiae (emissae, absorptae vel adhibitae a processu) <\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Energia interna <span class=\"katex-eq\" data-katex-display=\"false\">U<\/span> est functio status quia habet valorem bene definitum pro unoquoque statu aequilibrii systematis. Variari potest energia interna systematis applicando calorem <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> aut per laborem <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span>; attamen, labor et calor non sunt functiones status. Hoc fit quia utrumque pendet a processu quo energia additur vel extrahitur et, semel absoluto processu, impossibile est scire quantitatem caloris aut laboris quae effecta sit ad talem statum aequilibrii obtinendum.<\/p>\n<p>Mutatio energiae internae systematis scribi potest hoc modo:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta U = \\Delta Q + \\Delta W<\/span><\/span>,<\/p>\n<p>ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> est quantitas caloris suppeditati et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> quantitas laboris in systema effecti. Conventione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> est positivum cum calor systemati suppeditatur; si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> est negativum, tunc calor e systemate extrahitur; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> est positivum pro labore in systema effectu; si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> est negativum, systema laborem in ambitum exercet.<\/p>\n<p>Relatio inter laborem, calorem et energiam internam etiam exprimi potest forma differentiali per relationem<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU = \\delta Q + \\delta W<\/span><\/span>.<\/p>\n<p>Hic littera <span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span> adhibetur ad repraesentandos differentiales inexactos.<\/p>\n<p>Systema thermaliter isolatum definitur ut illud systema quod calorem cum ambitu suo commutare non potest. Cum hoc fit, tunc habetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d U = \\delta W<\/span><\/span>. Hoc est <strong>Lex Prima Thermodynamicae<\/strong> ad systema adiabaticum restricta.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Labor thermodynamicus<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=418s\" target=\"_blank\" rel=\"noopener\"><strong>Cum comprimimus filum elasticum <\/strong><\/a>in distantiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dx<\/span><\/span>, respondet vi elastica magnitudinis <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> atque ideo perficiemus laborem<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W = Fdx<\/span><\/span>.<\/p>\n<p>Cum comprimimus gas, fingere possumus hoc ex multis filis elasticis iuxta positis et certum spatium occupantibus componi. Ita habemus quod, si vim <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> super superficiem <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> applicamus, tunc pressionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P=F\/A<\/span><\/span> applicabimus et scribere possumus igitur laborem effectum fore<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W = PAdx = -PdV <\/span><\/span>.<\/p>\n<p>Signum negativum quod in ultima aequatione apparet debetur quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Adx=-dV<\/span><\/span>, et hoc fit quia, cum \u00abfili elastici\u00bb comprimuntur in distantiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dx<\/span><\/span>, volumen ab his formatum diminuitur. Cum labor in systema thermodynamicum exercetur, hoc respondet minuendo suum volumen.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Capacitas calorifica<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=677s\" target=\"_blank\" rel=\"noopener\"><strong>Supponamus nunc<\/strong><\/a> nos velle plenius intellegere quomodo mutetur energia interna systematis cum calor additur. Generatim, energia interna est functio temperiei et voluminis, unde scribere possumus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=U(T,V)<\/span><\/span>. Deinde, cum energia sit differentiale exactum, possibile est exprimere mutationem <span class=\"katex-eq\" data-katex-display=\"false\">U<\/span> respectu <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> per relationem<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle dU = \\left(\\frac{\\partial U}{\\partial T}\\right)_V dT + \\left(\\frac{\\partial U}{\\partial V}\\right)_T dV<\/span><\/span>.<\/p>\n<p>Nunc, ex relationibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU=\\delta Q + \\delta W<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W=-PdV<\/span><\/span> possumus reformulare <strong>Legem Primam Thermodynamicae<\/strong> per rationes sequentes:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\delta Q &amp;= dU + PdV\\\\ \\\\\n\n&amp; \\displaystyle  =\\left(\\frac{\\partial U}{\\partial T}\\right)_V dT + \\left(\\frac{\\partial U}{\\partial V}\\right)_T dV + PdV\\\\ \\\\\n\n&amp; \\displaystyle =\\left(\\frac{\\partial U}{\\partial T}\\right)_V dT + \\left[\\left(\\frac{\\partial U}{\\partial V}\\right)_T + P\\right]dV \\\\ \\\\\n\n\\displaystyle  \\frac{\\delta Q}{dT} &amp; \\displaystyle  =\\left(\\frac{\\partial U}{\\partial T}\\right)_V + \\left[\\left(\\frac{\\partial U}{\\partial V}\\right)_T + P\\right]\\frac{dV}{dT}.\n\n\\end{array}\n\n<\/span>\n<p>Hoc ultimum est relatio valida pro qualibet mutatione temperiei et voluminis.<\/p>\n<p>Ex hoc quod obtinuimus determinare possumus quantitatem caloris quam addere debemus ut mutationem temperiei sub certis restrictionibus efficiamus.<\/p>\n<h3>Restrictio ad volumen constans<\/h3>\n<p>Ut videamus quid fiat ad volumen constans, recordemur definitionem capacitatis calorificae ad volumen constans esse <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V=(\\partial Q\/ \\partial T)_V<\/span><\/span>. Deinde, si nos coartamus ad volumen constans in analysi quam ante fecimus, tunc annullabimus terminum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dV\/dT<\/span><\/span> in expressione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta Q\/dT<\/span><\/span>. Hoc nobis valet tamquam iustificatio ad scribendum:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  C_V = \\left(\\frac{\\partial U}{\\partial T} \\right)_V<\/span><\/span><\/p>\n<h3>Restrictio ad pressionem constantem<\/h3>\n<p>Si pressionem constantem servamus, tunc habebitur<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_p =\\left(\\frac{\\partial Q}{\\partial T}\\right)_P=\\left(\\frac{\\partial U}{\\partial T}\\right)_V + \\left[\\left(\\frac{\\partial U}{\\partial V}\\right)_T + P\\right]\\left(\\frac{\\partial V}{\\partial T}\\right)_p<\/span><\/span>.<\/p>\n<h3>Capacitas calorifica gasi monoatomici<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=974s\" target=\"_blank\" rel=\"noopener\"><strong>Cum consideremus<\/strong><\/a> gas monoatomicum, energia interna propter energiam kineticam particularum eius est formae <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n U=\\frac{3}{2}Nk_BT<\/span>. Hoc effectum iustificatur per principium aequipartitionis energiae quod ex prospectu statistico motus particularum investigari potest.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Distributio Maxwell-Boltzmann et aequipartitio Energiae<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=1027s\" target=\"_blank\" rel=\"noopener\"><strong>Quoniam energia systematis<\/strong><\/a> est proportionalis ad suum <strong>Factor Boltzmann<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-E\/(k_BT)}<\/span><\/span>. Ratiocinando ex hoc et considerantes energiam kineticam particularum formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  E_{cin}=\\frac{1}{2}mv^2<\/span><\/span>, possumus inferre energiam motui particularum systematis associatam, in unum ex tribus axibus coordinatis proiectam (Consideremus nunc axem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span>), fore distributionem velocitatum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)<\/span><\/span> proportionalem ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-mv_x^2\/(2k_BT)}<\/span><\/span>. Id est:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)= A e^{-mv_x^2\/(2k_BT)}<\/span><\/span>,<\/p>\n<p>ubi <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> est aliqua constans determinanda. Nunc, quoniam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)<\/span><\/span> est functio distributionis, haec normalizata esse debet ita ut:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\int_{-\\infty}^{+\\infty} g(v_x)dv_x= 1<\/span><\/span><\/p>\n<p>Effectus utilis ad hanc condicionem analysandam est integralis Gaussiana<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\int_{-\\infty}^{+\\infty} e^{-x^2}dx= \\sqrt{\\pi}<\/span><\/span><\/p>\n<p>Ex hoc infertur quod<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1= \\int_{-\\infty}^{+\\infty} Ae^{\\frac{-mv_x^2}{2k_BT}}dv_x= A\\sqrt{\\frac{\\pi}{m\/(2k_BT)}} = A\\sqrt{\\frac{2\\pi k_BT}{m}}<\/span><\/span><\/p>\n<p>Et ideo<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  g(v_x) = \\sqrt{\\frac{m}{2\\pi k_BT}}e^{-mv_x^2\/(2k_BT)}<\/span><\/span><\/p>\n<p>His habita, nunc possibile est computare velocitatem mediam in axem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span> proiectam, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; v_x^2\\right&gt;<\/span><\/span>. Cuius effectus est:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left&lt; v_x^2\\right \\gt = \\int_{-\\infty}^{+\\infty} v_x^2 g(v_x) dv_x = \\sqrt{\\frac{m}{2\\pi k_BT}} \\int_{-\\infty}^{+\\infty} v_x^2 e^{-mv_x^2\/(2k_BT)} = \\frac{k_BT}{m} <\/span><\/span><\/p>\n<p>Et quoniam velocitas quadratica media decomponi potest hoc modo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  \\left\\lt v^2\\right\\gt = \\left\\lt v_x^2\\right\\gt + \\left\\lt v_y^2\\right\\gt + \\left\\lt v_z^2\\right\\gt<\/span><\/span> atque singulae partes idem progressionem et effectum habent, possibile est scribere energiam kineticam mediam systematis particularum per<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  \\left\\lt E_{cin}\\right\\gt =\\frac{1}{2}m\\left\\lt v^2\\right\\gt  = \\frac{1}{2}m \\cdot 3\\frac{k_BT}{m}= \\frac{3}{2}k_BT<\/span><\/span>.<\/p>\n<p>Hoc est quod vocamus \u00abprincipium aequipartitionis energiae\u00bb. Ex hoc possumus dicere quod si systema constat ex <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> particularibus cum energia cinetica media <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\lt E_{cin}\\right\\gt<\/span><\/span> et energia totalis systematis est origine pure cinetica, tunc non solum habebimus energiam internam systematis fore <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n U=3Nk_BT\/2<\/span> (sicut praedictum est), sed etiam evidenter ostenditur energiam internam solum a temperie systematis pendere, unde:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{\\partial U}{\\partial V}\\right)_T = 0<\/span><\/span><\/p>\n<h3>Evolutio pro gase ideali<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=1027s\" target=\"_blank\" rel=\"noopener\"><strong>Nunc, recordantes aequationem gasorum idealium<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=Nk_BT =nRT<\/span><\/span>. Si volumen explicamus obtinebimus<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  V= \\frac{nRT}{P}<\/span><\/span><\/p>\n<p>Ergo habebitur<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  \\left(\\frac{\\partial V}{\\partial T} \\right)_P = \\frac{nR}{P}<\/span><\/span><\/p>\n<p>Et si ad expressiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span> accedamus, animadvertimus quod<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nC_P - C_V &amp; \\displaystyle = \\left[\\left(\\frac{\\partial U}{\\partial V} \\right)_T + P \\right]\\left(\\frac{\\partial V}{\\partial T} \\right)_P = P\\cdot \\frac{nR}{P} = nR\n\n\\end{array}<\/span>\n<p>Nunc, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  C_V=(\\partial U \/ \\partial T)_V<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=3Nk_BT\/2=3nRT\/2<\/span><\/span>, habetur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle  C_V = \\frac{3}{2}nR\n\n<\/span>\n<p>et, proinde<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nC_P = C_V + nR = \\displaystyle  \\frac{3}{2}nR + nR = \\frac{5}{2}nR\n\n<\/span>\n<h3>Index adiabaticus<\/h3>\n<p>Magnitudo usus frequens est ratio inter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span>, atque ob hanc causam nomen speciale accipit. Definiri solet <strong>index adiabaticus<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> per<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\gamma = \\displaystyle  \\frac{C_P}{C_V}\n\n<\/span>\n<p>In casu gasorum idealium, index adiabaticus valorem exactum habet:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\gamma = \\displaystyle \\frac{5}{3}\n\n<\/span>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercitia<\/h2>\n<ol>\n<li>Semprene verum erit quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU=C_VdT<\/span><\/span>? Compara casum generalem cum casu gasorum idealium et fundamenta responsionem tuam.<\/li>\n<li>Assumendo pro gasi ideali valere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=C_VT<\/span><\/span>, computa: (i) Energiam internam per unitatem massae et (ii) energiam internam per unitatem voluminis.<\/li>\n<li>Unus moles gasi idealis monoatomici in cylindro pistonis includitur et ad temperaturam constantem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_0<\/span><\/span> per contactum cum reservo thermico conservatur. Gas lente expanditur ab volumine <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1<\/span><\/span> ad aliud volumen <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2<\/span><\/span> manente temperatura constanti omni tempore. (i) Mutaturne energia interna gasi? (ii) Computa laborem a gaso effectum et fluxum caloris in gasum.<\/li>\n<li>Ostende quod, pro gasi ideali, relationes sequentes valent:\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{R}{c_V} = \\gamma-1<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{R}{c_P} = \\frac{\\gamma -1}{\\gamma}<\/span><\/span><\/p>\n<p>Ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c_V<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c_P<\/span><\/span> sunt capacitates calorificae molares.<\/p>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Lex Prima Thermodynamicae Lex Prima Thermodynamicae est fundamentum quod coniungit notiones fundamentales ut calorem, laborem et energiam internam, statuens energiam nec creari nec deleri, sed tantum transformari. Hoc documentum explanat quomodo haec lex ad systemata clausa applicetur, penitius perscrutans analysin laboris thermodynamici, capacitates calorificas atque proprietates statisticas gasorum. Per coniunctionem formulationum mathematicarum et rationum physicorum, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30854,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1250,1292],"tags":[],"class_list":["post-34759","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physica","category-thermodynamica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Lex Prima Thermodynamicae - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quomodo Lex Prima Thermodynamicae conectat calorem, laborem et energiam internam eiusque applicationem in systematibus clausis et gasibus idealibus.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/la\/lex-prima-thermodynamicae\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Lex Prima Thermodynamicae\" \/>\n<meta property=\"og:description\" content=\"Disce quomodo Lex Prima Thermodynamicae conectat calorem, laborem et energiam internam eiusque applicationem in systematibus clausis et gasibus idealibus.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/lex-prima-thermodynamicae\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-07-03T13:00:01+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-20T08:16:32+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/primeraley-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Lex Prima Thermodynamicae\" \/>\n<meta name=\"twitter:description\" content=\"Disce quomodo Lex Prima Thermodynamicae conectat calorem, laborem et energiam internam eiusque applicationem in systematibus clausis et gasibus idealibus.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/primeraley.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Lex Prima Thermodynamicae\",\"datePublished\":\"2021-07-03T13:00:01+00:00\",\"dateModified\":\"2025-09-20T08:16:32+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/\"},\"wordCount\":1659,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/primeraley.jpg\",\"articleSection\":[\"Physica\",\"Thermodynamica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lex-prima-thermodynamicae\\\/\",\"name\":\"Lex Prima Thermodynamicae - 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