{"id":17,"date":"2020-06-28T01:48:07","date_gmt":"2020-06-28T01:48:07","guid":{"rendered":"https:\/\/tbits.jp\/plus\/?p=17"},"modified":"2020-06-28T10:55:05","modified_gmt":"2020-06-28T10:55:05","slug":"entropy-calculation","status":"publish","type":"post","link":"https:\/\/tbits.jp\/plus\/entropy-calculation\/","title":{"rendered":"\u30e2\u30eb\u6570\uff08\u5206\u5b50\u6570\uff09\u3092\u5909\u6570\u306b\u3057\u305f\u7406\u60f3\u6c17\u4f53\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u306e\u6c42\u3081\u65b9"},"content":{"rendered":"<p>\u524d\u8a18\u4e8b\u300c<a href=\"https:\/\/tbits.jp\/plus\/calculation-example\/\" target=\"_blank\" rel=\"noopener noreferrer\">\u7406\u60f3\u6c17\u4f53\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u5909\u5316\u306e\u8a08\u7b97\u4f8b<\/a>\u300d\u306e\u4e2d\u3067\u3001\u30e2\u30eb\u6570$n$\u3092\u4e00\u5b9a\u3068\u3057\u305f\u3068\u304d\u306e\u7406\u60f3\u6c17\u4f53\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u5909\u5316\u3092\u8a08\u7b97\u3092\u3057\u307e\u3057\u305f\u3002<\/p>\n<p>\u4eca\u56de\u306f\u3001\u30e2\u30eb\u6570$n$\u3082\u5909\u5316\u3059\u308b\u5834\u5408\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u306e\u5f0f\u3092\u5c0e\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<p>\u306a\u304a\u3001\u30e2\u30eb\u6570\u3067\u306f\u306a\u304f\u5206\u5b50\u6570N\u3067\u8868\u73fe\u3057\u305f\u3044\u65b9\u306f\u3001\u6c17\u4f53\u5b9a\u6570R\u3092\u30dc\u30eb\u30c4\u30de\u30f3\u5b9a\u6570k\u306b\u5909\u3048\u3066\u4e0b\u3055\u3044\u3002<\/p>\n<p>\u500b\u4eba\u7684\u306b\u306f\u3001\u5206\u5b501\u3064\u5f53\u305f\u308a\u306e\u71b1\u5bb9\u91cf\u3068\u304b\u3001\u5206\u5b50\u3072\u3068\u3064\u5f53\u305f\u308a\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u3068\u3044\u3046\u306e\u304c\u30a4\u30e1\u30fc\u30b8\u3057\u306b\u304f\u3044\u3068\u601d\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u71b1\u529b\u5b66\u306e\u7bc4\u56f2\u3067\u306f\u30e2\u30eb\u6570\u3092\u4f7f\u3046\u3053\u3068\u306b\u3057\u307e\u3059\uff08\u7d71\u8a08\u529b\u5b66\u304c\u7d61\u3093\u3067\u304f\u308b\u3068\u305d\u3046\u3044\u3046\u8a33\u306b\u306f\u3044\u304b\u306a\u304f\u306a\u308a\u307e\u3059\u304c\uff09\u3002<\/p>\n<h2>\u3068\u308a\u3042\u3048\u305a\u8a08\u7b97\u3057\u3066\u307f\u308b<\/h2>\n<p>\u524d\u8a18\u4e8b\u3067\u306f\u3001$n$\u4e00\u5b9a\u3067\u3001\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc$S$\u3092\u6e29\u5ea6$T$\u3068\u4f53\u7a4d$V$\u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u3057\u3066\u3001\u4e0b\u306e\u5f0f\u3092\u7a4d\u5206\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u305f\u3002<\/p>\n<div class=\"horizontal-scroll\">$${\\small dS=\\frac{1}{T}\\frac{\\partial S}{\\partial T}dT+\\frac{1}{T}\\frac{\\partial S}{\\partial V}dV}$$<\/div>\n<p>\u3068\u308a\u3042\u3048\u305a\u540c\u3058\u65b9\u6cd5\u3067\u8a66\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u307e\u305a$n$\u306e\u9805\u3092\u52a0\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n<div class=\"horizontal-scroll\">$${\\small dS=\\frac{1}{T}\\frac{\\partial S}{\\partial T}dT+\\frac{1}{T}\\frac{\\partial S}{\\partial V}dV+\\frac{1}{T}\\frac{\\partial S}{\\partial n}dn}$$<\/div>\n<p>\u5b9a\u7a4d\u30e2\u30eb\u6bd4\u71b1\u3092$c_v$\u3001\u5316\u5b66\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092$\\mu$\u3068\u3059\u308b\u3068\u3001<\/p>\n<p>&nbsp;<\/p>\n<div class=\"horizontal-scroll\">$${\\small dS=\\frac{nc_v}{T}dT-\\frac{p}{T}dV+\\frac{\\mu}{T}dn}$$<\/div>\n<div class=\"horizontal-scroll\">$${\\small=\\frac{nc_v}{T}dT-\\frac{nR}{V}dV+\\frac{\\mu}{T}dn}$$<\/div>\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3053\u308c\u3092\u7a4d\u5206\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3059\u304c\u3001\u5316\u5b66\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb$\\mu$\u3092\u3069\u3046\u3042\u3064\u304b\u3048\u3070\u3044\u3044\u306e\u304b\u56f0\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<p>\u53f3\u8fba\u306e\u7b2c\u4e00\u9805\u3001\u7b2c\u4e8c\u9805\u306b\u3082\u5909\u6570$n$\u304c\u5165\u3063\u3066\u3044\u3066\u5909\u6570\u5206\u96e2\u3067\u304d\u3066\u3044\u307e\u305b\u3093\u3002<\/p>\n<h2>\u89e3\u3051\u308b\u3088\u3046\u306b\u5de5\u592b\u3059\u308b<\/h2>\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\u3001\u89e3\u3051\u308b\u3088\u3046\u306b\u5de5\u592b\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u3084\u308b\u3053\u3068\u306f2\u3064\u3067\u3059\u3002<\/p>\n<div class=\"box_style box_style_gray\">\n<div class=\"box_inner\">\n<ul style=\"list-style-type: disc;\">\n<li><strong>\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc$S$\u3067\u306f\u306a\u304f\u30011\u30e2\u30eb\u5f53\u305f\u308a\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc$S\/n$\u3092\u6c42\u3081\u308b<\/strong><\/li>\n<li><strong>${\\small T\u3001V\u3001n}$\u3067\u306f\u306a\u304f\u3001${\\small T\u3001P\u3001n}$\u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u3059<\/strong><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>\u65e9\u901f\u3084\u3063\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>$S$\u3082$n$\u3082\u72b6\u614b\u91cf\u306a\u306e\u3067$S\/n$\u3082\u72b6\u614b\u91cf\u3067\u3059\u3002<\/p>\n<div class=\"horizontal-scroll\">$${\\small d \\left(\\frac{S}{n}\\right)=\\frac{1}{T}\\frac{\\partial (\\frac{S}{n})}{\\partial T}dT+\\frac{1}{T}\\frac{\\partial (\\frac{S}{n})}{\\partial P}dP+\\frac{1}{T}\\frac{\\partial (\\frac{S}{n})}{\\partial n}dn}$$<\/div>\n<p>\u3068\u8868\u305b\u307e\u3059\u3002<\/p>\n<p>\u3053\u3053\u3067\u3001<strong>\u6e29\u5ea6\u3068\u5727\u529b\u4e00\u5b9a\u306e\u5834\u5408\u306f\u3001\u30e2\u30eb\u6570\u3068\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u306f\u6bd4\u4f8b<\/strong>\u3059\u308b\u306e\u3067\u3001\u6700\u5f8c\u306e\u9805\u306e$S\/n$\u306f$n$\u306b\u3088\u3063\u3066\u5909\u5316\u3057\u306a\u3044\u5024\u3067\u3059\u3002<\/p>\n<p>\u3053\u306e\u504f\u5fae\u5206\u306f\u30bc\u30ed\u306b\u306a\u3063\u3066\u9762\u5012\u306a\u9805\u304c\u6d88\u3048\u307e\u3059\u3002<\/p>\n<p>\u6c42\u3081\u308b\u5024\u3092$S\/n$\u306b\u3057\u3066\u3001$V$\u3067\u306f\u306a\u304f$P$\u306e\u5909\u6570\u306b\u5909\u3048\u305f\u7406\u7531\u306f\u3053\u3053\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n<div class=\"horizontal-scroll\">$${\\small d\\left(\\frac{S}{n}\\right)=\\frac{c_p}{T}dT-\\frac{V}{nT}dP=\\frac{c_p}{T}dT-\\frac{R}{P}dP}$$<\/div>\n<p>$c_P$\u304c\u5b9a\u6570\u306e\u5834\u5408\u3001\u5909\u6570\u5206\u96e2\u3082\u3067\u304d\u307e\u3057\u305f\uff08\u5727\u529b\u4e00\u5b9a\u3067\u306e\u504f\u5fae\u5206\u306a\u306e\u3067$c_v$\u3067\u306f\u306a\u304f\u5b9a\u5727\u6bd4\u71b1$c_p$\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\uff09\u3002<\/p>\n<p>${\\small S_0(n_0,T_0,P_0)}$\u304b\u3089\u3001${\\small S(n,T,P)}$\u307e\u3067\u7a4d\u5206\u3059\u308b\u3068<\/p>\n<div class=\"horizontal-scroll\">$${\\small \\frac{S}{n}-\\frac{S_0}{n_0}=c_p\\log\\frac{T}{T_0}dT-R\\log\\frac{P}{P_0}}$$<\/div>\n<div class=\"horizontal-scroll\">$${\\small S=nc_p\\log\\frac{T}{T_0}-nR\\log\\frac{P}{P_0}+n\\frac{S_0}{n_0}}$$<\/div>\n<p>\u306f\u3044\u3001\u7406\u60f3t\u6c17\u4f53\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u306e\u5f0f\u304c\u5c0e\u3051\u307e\u3057\u305f\u3002<\/p>\n<h2>\u4f53\u7a4dV\u306e\u95a2\u6570\u306b\u3059\u308b<\/h2>\n<p>$P$\u3067\u306f\u306a\u304f$V$\u306e\u95a2\u6570\u306b\u3057\u305f\u65b9\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u591a\u3044\u306e\u3067\u3001$V$\u306e\u95a2\u6570\u306b\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>${\\small PV=nRT}$\u3092\u4f7f\u3063\u3066<\/p>\n<div class=\"horizontal-scroll\">$${\\small S=nc_p\\log\\frac{T}{T_0}-nR\\log\\frac{nRTV_0}{n_0RT_0V}+n\\frac{S_0}{n_0}}$$<\/div>\n<div class=\"horizontal-scroll\">$${\\small S=nc_p\\log\\frac{T}{T_0}+nR\\log\\frac{n_0V}{nV_0}-nR\\log\\frac{T}{T_0}+n\\frac{S_0}{n_0}}$$<\/div>\n<p>${\\small c_p-c_v=R}$\u3088\u308a<\/p>\n<div class=\"horizontal-scroll\">$${\\small S=nc_v\\log\\frac{T}{T_0}+nR\\log\\frac{n_0V}{nV_0}+n\\frac{S_0}{n_0}}$$<\/div>\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<h2>S\/n\u306e\u4e2d\u8eab\u3092\u8003\u3048\u3066\u307f\u308b<\/h2>\n<p>\u3053\u3053\u3067\u4f7f\u3063\u305f$S\/n$\u306f\u30011\u30e2\u30eb\u5f53\u305f\u308a\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u5909\u5316\u3067\u3059\u3002<\/p>\n<p>\u6700\u7d42\u7684\u306b\u3067\u304d\u305f\u5f0f\u306b\u3082$S_0\/n_0$\u3068\u3044\u3046\u9805\u304c\u3042\u308a\u307e\u3059\u306e\u3067\u3001\u5c11\u3057\u4e2d\u8eab\u3092\u307f\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>$${\\small S=\\frac{1}{T}(U+PV-G)}$$<\/p>\n<p>$${\\small \\frac{S}{n}=\\frac{U}{nT}+\\frac{PV}{nT}-\\frac{G}{nT}}$$<\/p>\n<p>\u5185\u90e8\u30a8\u30cd\u30eb\u30ae\u30fc$U$\u304c\u7d76\u5bfe\u96f6\u5ea6\u30670\u3001$c_v$\u304c\u4e00\u5b9a\u3068\u3059\u308c\u3070<\/p>\n<div class=\"horizontal-scroll\">$${\\small \\frac{S}{n}=c_v+R-\\frac{\\mu}{T}=c_p-\\frac{\\mu}{T}}$$<\/div>\n<p>\u3068\u306a\u3063\u3066\u3001\u5316\u5b66\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3068\u306e\u9593\u306b\u7c21\u5358\u306a\u95a2\u4fc2\u304c\u306a\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<p>\u3053\u306e\u3053\u3068\u3092\u3064\u304b\u3063\u3066\u3001<\/p>\n<div class=\"horizontal-scroll\">$${\\small S=nc_v\\log\\frac{T}{T_0}+nR\\log\\frac{n_0V}{nV_0}+n\\frac{S_0}{n_0}}$$<\/div>\n<p>\u306e\u6700\u5f8c\u306e\u9805\u3092\u5909\u5f62\u3057\u3066<\/p>\n<div class=\"horizontal-scroll\">$${\\small S=nc_v\\log\\frac{T}{T_0}+nR\\log\\frac{n_0V}{nV_0}+nc_p-n\\frac{ \\mu _0}{T_0}}$$<\/div>\n<p>\u3068\u8868\u3059\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n<h2>\u4efb\u610f\u306e\u5024\u3092\u6e1b\u3089\u3059<\/h2>\n<p>\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u3092\u8a08\u7b97\u3057\u305f\u3068\u304d\u306b\u4f7f\u3063\u305f\u3001${\\small n_0\u3001V_0\u3001T_\uff10}$\u306f\u4efb\u610f\u306e\u5024\u3067\u3059\u3002<\/p>\n<p>\u305d\u306e\u3068\u304d\u306e\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc${\\small S_0}$\u3082\u3001${\\small n_0\u3001V_0\u3001T_\uff10}$\u3092\u4e0e\u3048\u305f\u3060\u3051\u3067\u8a08\u7b97\u3067\u304d\u308b\u3082\u306e\u3067\u306f\u306a\u304f\u3001\u4efb\u610f\u6027\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>${\\small S_0\u3001n_0\u3001V_0\u3001T_\uff10}$\u3068\u30014\u3064\u3082\u4efb\u610f\u306e\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u610f\u5473\u306f\u306a\u3044\u306e\u3067\u3001\u6e1b\u3089\u3057\u3066\u3084\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u6b21\u306e\u3088\u3046\u306a\u8868\u3057\u65b9\u3092\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3067\u3059\u3002<\/p>\n<p>$${\\small S=nR \\log{\\alpha \\frac{T^cV}{n}}}$$<\/p>\n<p>${\\small c=c_v\/R}$\u3092\u4f7f\u3063\u3066\u5168\u3066\u3092\u5bfe\u6570\u306e\u4e2d\u306b\u5165\u308c\u305f\u5f62\u3067\u3059\u3002<\/p>\n<p>$${\\small \\alpha=\\frac {n_0}{T_0^{C}V}e^{S_0\/Rn_0}}$$<\/p>\n<p>\u3067\u3059\u306d\u3002<\/p>\n<p>\u305d\u308c\u4ee5\u5916\u306b\u3001\u3053\u3093\u306a\u8868\u3057\u65b9\u3092\u3057\u3066\u3044\u308b\u5834\u5408\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>$${\\small S=nc_v \\log{T}+nR\\log\\frac{V}{n}+n\\beta}$$<\/p>\n<p>$${\\small\\beta=\\frac{S_0}{n_0}-c_v\\log{T_0}-R\\log{\\frac{V_0}{n_0}}}$$<\/p>\n<p>\u5168\u3066\u3092\u5bfe\u6570\u306e\u5916\u306b\u51fa\u3057\u3059\u65b9\u6cd5\u3067\u3059\u3002<\/p>\n<p>\u5f0f\u306f\u5358\u7d14\u3067\u3059\u304c\u3001\u5bfe\u6570\u306e\u4e2d\u304c\u7121\u6b21\u5143\u306b\u306a\u3063\u3066\u3044\u306a\u3044\u306e\u304c\u6c17\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u5316\u5b66\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3068\u306e\u95a2\u4fc2\u3092\u4f7f\u3063\u3066<\/p>\n<p>$${\\small S=n(R\\log{\\frac{T^cV}{n}}+c_p+i)}$$<\/p>\n<p>\u3068\u304b<\/p>\n<p>$${\\small S=n(R 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