{"id":30860,"date":"2021-07-03T13:00:53","date_gmt":"2021-07-03T13:00:53","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30860"},"modified":"2025-01-02T04:38:48","modified_gmt":"2025-01-02T04:38:48","slug":"%e7%83%ad%e5%8a%9b%e5%ad%a6%e7%ac%ac%e4%b8%80%e5%ae%9a%e5%be%8b","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e7%83%ad%e5%8a%9b%e5%ad%a6%e7%ac%ac%e4%b8%80%e5%ae%9a%e5%be%8b\/","title":{"rendered":"\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\t\ttext-align: center;\n\t}\n<\/style>\n<h1>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b<\/h1>\n<p style=\"text-align:center;\">\n\t<em><br \/>\n\t\t\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b\u662f\u8fde\u63a5\u57fa\u672c\u6982\u5ff5\uff08\u5982\u70ed\u91cf\u3001\u529f\u548c\u5185\u80fd\uff09\u7684\u57fa\u7840\uff0c\u5b83\u89c4\u5b9a\u80fd\u91cf\u65e2\u4e0d\u4f1a\u88ab\u521b\u9020\u4e5f\u4e0d\u4f1a\u88ab\u9500\u6bc1\uff0c\u53ea\u4f1a\u53d1\u751f\u8f6c\u5316\u3002\u672c\u6587\u4ecb\u7ecd\u4e86\u8be5\u5b9a\u5f8b\u5728\u5c01\u95ed\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\uff0c\u6df1\u5165\u5206\u6790\u4e86\u70ed\u529b\u5b66\u529f\u3001\u70ed\u5bb9\u4ee5\u53ca\u6c14\u4f53\u7684\u7edf\u8ba1\u7279\u6027\u3002\u901a\u8fc7\u6570\u5b66\u516c\u5f0f\u548c\u7269\u7406\u63a8\u7406\u7684\u7ed3\u5408\uff0c\u60a8\u5c06\u638c\u63e1\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u4e2d\u80fd\u91cf\u8fc7\u7a0b\u7684\u57fa\u672c\u5de5\u5177\u3002<br \/>\n\t<\/em>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><\/p>\n<p>\u5728\u672c\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u8bc1\u660e<\/strong>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b\u5728\u5c01\u95ed\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\uff0c\u89e3\u91ca\u70ed\u91cf\u3001\u529f\u548c\u5185\u80fd\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/li>\n<li><strong>\u5206\u6790<\/strong>\u70ed\u529b\u5b66\u529f\u5728\u538b\u7f29\u548c\u81a8\u80c0\u8fc7\u7a0b\u4e2d\u7684\u6982\u5ff5\uff0c\u4f7f\u7528\u5fae\u5206\u516c\u5f0f\u3002<\/li>\n<li><strong>\u8ba1\u7b97<\/strong>\u5728\u6052\u5b9a\u4f53\u79ef\u548c\u6052\u5b9a\u538b\u529b\u6761\u4ef6\u4e0b\u7684\u70ed\u5bb9\uff0c\u5e94\u7528\u70ed\u529b\u5b66\u7ea6\u675f\u6761\u4ef6\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong>\u9ea6\u514b\u65af\u97e6-\u73bb\u5c14\u5179\u66fc\u5206\u5e03\u4ee5\u53ca\u80fd\u91cf\u5747\u5206\u539f\u7406\u5728\u5206\u5b50\u7cfb\u7edf\u4e2d\u7684\u4f5c\u7528\u3002<\/li>\n<li><strong>\u8bc1\u660e<\/strong>\u70ed\u5bb9\u3001\u7edd\u70ed\u6307\u6570\u53ca\u5176\u4ed6\u7406\u60f3\u6c14\u4f53\u70ed\u529b\u5b66\u7279\u6027\u4e4b\u95f4\u7684\u7279\u5b9a\u5173\u7cfb\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center\"><strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>\uff1a<\/strong><br \/>\n\t<a href=\"#1\">\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b\u7684\u516c\u5f0f\u5316<\/a><br \/>\n\t<a href=\"#2\">\u70ed\u529b\u5b66\u529f<\/a><br \/>\n\t<a href=\"#3\">\u70ed\u5bb9<\/a><br \/>\n\t<a href=\"#4\">\u9ea6\u514b\u65af\u97e6-\u73bb\u5c14\u5179\u66fc\u5206\u5e03\u4e0e\u80fd\u91cf\u5747\u5206\u539f\u7406<\/a><br \/>\n\t<a href=\"#5\">\u7ec3\u4e60<\/a>\n<\/p>\n<p><center><br \/>\n\t<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><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b\u7684\u516c\u5f0f\u5316<\/h2>\n<p>\n\t<a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=140s\" target=\"_blank\" rel=\"noopener\"><br \/>\n\t\t<strong>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b<\/strong><br \/>\n\t<\/a>\u89c4\u5b9a\uff1a\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #c0ffc0;\">\n\t<span style=\"color: #000080;\"><br \/>\n\t\t<strong>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b<\/strong><br \/>\n\t\t\u80fd\u91cf\u65e2\u4e0d\u4f1a\u88ab\u521b\u9020\u4e5f\u4e0d\u4f1a\u88ab\u9500\u6bc1\uff1b\u6b64\u5916\uff0c\u70ed\u91cf\u548c\u529f\u662f\u80fd\u91cf\u7684\u5f62\u5f0f\uff08\u7531\u8fc7\u7a0b\u53d1\u51fa\u3001\u5438\u6536\u6216\u4f7f\u7528\uff09\u3002<br \/>\n\t<\/span>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u5185\u80fd <span class=\"katex-eq\" data-katex-display=\"false\">U<\/span> \u662f\u4e00\u79cd\u72b6\u6001\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u5bf9\u7cfb\u7edf\u7684\u6bcf\u4e2a\u5e73\u8861\u72b6\u6001\u90fd\u6709\u660e\u786e\u7684\u503c\u3002\u901a\u8fc7\u65bd\u52a0\u70ed\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> \u6216\u8fdb\u884c\u529f <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span>\uff0c\u53ef\u4ee5\u6539\u53d8\u7cfb\u7edf\u7684\u5185\u80fd\uff1b\u7136\u800c\uff0c\u529f\u548c\u70ed\u91cf\u4e0d\u662f\u72b6\u6001\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u4eec\u53d6\u51b3\u4e8e\u6dfb\u52a0\u6216\u79fb\u9664\u80fd\u91cf\u7684\u8fc7\u7a0b\u3002\u4e00\u65e6\u8fc7\u7a0b\u7ed3\u675f\uff0c\u5c31\u65e0\u6cd5\u77e5\u9053\u4e3a\u4e86\u8fbe\u5230\u8be5\u5e73\u8861\u72b6\u6001\u8fdb\u884c\u4e86\u591a\u5c11\u70ed\u91cf\u6216\u529f\u3002<\/p>\n<p>\u7cfb\u7edf\u5185\u80fd\u7684\u53d8\u5316\u53ef\u4ee5\u5199\u4e3a\uff1a<\/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>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> \u662f\u63d0\u4f9b\u7684\u70ed\u91cf\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> \u662f\u5bf9\u7cfb\u7edf\u8fdb\u884c\u7684\u529f\u3002\u6839\u636e\u7ea6\u5b9a\uff0c\u5f53\u70ed\u91cf\u4f20\u9012\u5230\u7cfb\u7edf\u4e2d\u65f6\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> \u4e3a\u6b63\uff1b\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> \u4e3a\u8d1f\uff0c\u5219\u8868\u793a\u4ece\u7cfb\u7edf\u4e2d\u79fb\u9664\u70ed\u91cf\u3002\u540c\u6837\uff0c\u5f53\u5bf9\u7cfb\u7edf\u65bd\u52a0\u529f\u65f6\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> \u4e3a\u6b63\uff1b\u5f53\u7cfb\u7edf\u5bf9\u73af\u5883\u505a\u529f\u65f6\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> \u4e3a\u8d1f\u3002<\/p>\n<p>\u529f\u3001\u70ed\u91cf\u548c\u5185\u80fd\u4e4b\u95f4\u7684\u5173\u7cfb\u4e5f\u53ef\u4ee5\u901a\u8fc7\u5fae\u5206\u5f62\u5f0f\u8868\u8fbe\uff1a<\/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>\u8fd9\u91cc\uff0c\u5b57\u6bcd <span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span> \u7528\u4e8e\u8868\u793a\u4e0d\u5b8c\u5168\u5fae\u5206\u3002<\/p>\n<p>\u4e00\u4e2a\u70ed\u7edd\u7f18\u7cfb\u7edf\u88ab\u5b9a\u4e49\u4e3a\u4e0d\u80fd\u4e0e\u73af\u5883\u4ea4\u6362\u70ed\u91cf\u7684\u7cfb\u7edf\u3002\u5f53\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\u65f6\uff0c\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU = \\delta W<\/span><\/span>\u3002\u8fd9\u662f <strong>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b<\/strong> \u5728\u7edd\u70ed\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\u3002<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u70ed\u5bb9<\/h2>\n<p>\n\t<a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=677s\" target=\"_blank\" rel=\"noopener\"><br \/>\n\t\t<strong>\u73b0\u5728\u6211\u4eec\u5047\u8bbe<\/strong><br \/>\n\t<\/a> \u60f3\u66f4\u8be6\u7ec6\u5730\u4e86\u89e3\u5f53\u70ed\u91cf\u88ab\u6dfb\u52a0\u65f6\u7cfb\u7edf\u7684\u5185\u80fd\u5982\u4f55\u53d8\u5316\u3002\u4e00\u822c\u6765\u8bf4\uff0c\u5185\u80fd\u662f\u6e29\u5ea6\u548c\u4f53\u79ef\u7684\u51fd\u6570\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=U(T,V)<\/span><\/span>\u3002\u7531\u4e8e\u80fd\u91cf\u662f\u4e00\u4e2a\u5b8c\u5168\u5fae\u5206\uff0c\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u5173\u7cfb\u8868\u8fbe <span class=\"katex-eq\" data-katex-display=\"false\">U<\/span> \u5bf9 <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u7684\u53d8\u5316\uff1a\n<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<p>\u73b0\u5728\uff0c\u6839\u636e\u5173\u7cfb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU=\\delta Q + \\delta W<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W=-PdV<\/span><\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u63a8\u7406\u91cd\u65b0\u8868\u8ff0 <strong>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b<\/strong>\uff1a<\/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>\u8fd9\u662f\u4e00\u4e2a\u9002\u7528\u4e8e\u4efb\u4f55\u6e29\u5ea6\u548c\u4f53\u79ef\u53d8\u5316\u7684\u901a\u7528\u5173\u7cfb\u3002<\/p>\n<p>\u57fa\u4e8e\u6b64\u7ed3\u679c\uff0c\u6211\u4eec\u53ef\u4ee5\u786e\u5b9a\u5728\u7279\u5b9a\u9650\u5236\u4e0b\u4e3a\u4e86\u4ea7\u751f\u6e29\u5ea6\u53d8\u5316\u9700\u8981\u6dfb\u52a0\u7684\u70ed\u91cf\u3002<\/p>\n<h3>\u6052\u5b9a\u4f53\u79ef\u9650\u5236<\/h3>\n<p>\u4e3a\u4e86\u5206\u6790\u5728\u6052\u5b9a\u4f53\u79ef\u4e0b\u4f1a\u53d1\u751f\u4ec0\u4e48\uff0c\u8bf7\u8bb0\u4f4f\uff0c\u6052\u5b9a\u4f53\u79ef\u4e0b\u70ed\u5bb9\u7684\u5b9a\u4e49\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V=(\\partial Q\/ \\partial T)_V<\/span><\/span>\u3002\u5982\u679c\u6211\u4eec\u5728\u4e4b\u524d\u7684\u5206\u6790\u4e2d\u9650\u5236\u4f53\u79ef\u4fdd\u6301\u6052\u5b9a\uff0c\u90a3\u4e48\u5728\u8868\u8fbe\u5f0f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta Q\/dT<\/span><\/span> \u4e2d\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dV\/dT<\/span><\/span> \u9879\u5c06\u88ab\u6d88\u9664\u3002\u8fd9\u4f7f\u6211\u4eec\u80fd\u591f\u5199\u4f5c\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  C_V = \\left(\\frac{\\partial U}{\\partial T} \\right)_V<\/span><\/span>.\n<\/p>\n<h3>\u6052\u5b9a\u538b\u529b\u9650\u5236<\/h3>\n<p>\u5982\u679c\u4fdd\u6301\u538b\u529b\u6052\u5b9a\uff0c\u5219\u6709\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<h3>\u5355\u539f\u5b50\u6c14\u4f53\u7684\u70ed\u5bb9<\/h3>\n<p>\n\t<a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=974s\" target=\"_blank\" rel=\"noopener\"><br \/>\n\t\t<strong>\u5f53\u6211\u4eec\u8003\u8651<\/strong><br \/>\n\t<\/a> \u5355\u539f\u5b50\u6c14\u4f53\u65f6\uff0c\u5176\u5185\u80fd\u7531\u7c92\u5b50\u7684\u52a8\u80fd\u51b3\u5b9a\uff0c\u5f62\u5f0f\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\tU=\\frac{3}{2}Nk_BT<\/span>\u3002\u8fd9\u4e2a\u7ed3\u679c\u662f\u7531\u80fd\u91cf\u5747\u5206\u539f\u7406\u5f97\u51fa\u7684\uff0c\u8be5\u539f\u7406\u53ef\u4ee5\u4ece\u7c92\u5b50\u8fd0\u52a8\u7684\u7edf\u8ba1\u65b9\u6cd5\u7814\u7a76\u4e2d\u5f97\u5230\u89e3\u91ca\u3002\n<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u9ea6\u514b\u65af\u97e6-\u73bb\u5c14\u5179\u66fc\u5206\u5e03\u4e0e\u80fd\u91cf\u5747\u5206\u539f\u7406<\/h2>\n<p>\n\t<a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=1027s\" target=\"_blank\" rel=\"noopener\"><br \/>\n\t\t<strong>\u7531\u4e8e\u7cfb\u7edf\u7684\u80fd\u91cf<\/strong><br \/>\n\t<\/a> \u4e0e\u5176 <strong>\u73bb\u5c14\u5179\u66fc\u56e0\u5b50<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-E\/(k_BT)}<\/span><\/span> \u6210\u6b63\u6bd4\uff0c\u6839\u636e\u8fd9\u4e00\u70b9\u5e76\u8003\u8651\u5230\u7c92\u5b50\u7684\u52a8\u80fd\u5f62\u5f0f\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  E_{cin}=\\frac{1}{2}mv^2<\/span><\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u65ad\u7cfb\u7edf\u7c92\u5b50\u5728\u4e00\u4e2a\u5750\u6807\u8f74\u65b9\u5411\u4e0a\u7684\u8fd0\u52a8\u80fd\u91cf\u5206\u5e03\uff08\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span> \u8f74\u4e3a\u4f8b\uff09\u5bf9\u5e94\u4e8e\u901f\u5ea6\u5206\u5e03 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)<\/span><\/span>\uff0c\u5176\u5f62\u5f0f\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-mv_x^2\/(2k_BT)}<\/span><\/span>\u3002\u5373\uff1a\n<\/p>\n<p style=\"text-align: center;\">\n\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)= A e^{-mv_x^2\/(2k_BT)}<\/span><\/span>,\n<\/p>\n<p>\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u662f\u4e00\u4e2a\u5f85\u786e\u5b9a\u7684\u5e38\u6570\u3002\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)<\/span><\/span> \u662f\u4e00\u4e2a\u5206\u5e03\u51fd\u6570\uff0c\u5b83\u5fc5\u987b\u88ab\u5f52\u4e00\u5316\uff0c\u4f7f\u5f97\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\int_{-\\infty}^{+\\infty} g(v_x)dv_x= 1<\/span><\/span>.\n<\/p>\n<p>\u4e00\u4e2a\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u975e\u5e38\u6709\u7528\u7684\u7ed3\u679c\u662f\u9ad8\u65af\u79ef\u5206\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\int_{-\\infty}^{+\\infty} e^{-x^2}dx= \\sqrt{\\pi}<\/span><\/span>.\n<\/p>\n<p>\u636e\u6b64\uff0c\u6211\u4eec\u63a8\u65ad\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<p>\u901a\u8fc7\u8fd9\u4e00\u7ed3\u679c\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u6cbf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span> \u8f74\u65b9\u5411\u7684\u5e73\u65b9\u5e73\u5747\u901f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left\\lt v_x^2\\right\\gt<\/span><\/span>\uff0c\u5176\u7ed3\u679c\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<p>\u7531\u4e8e\u5747\u65b9\u6839\u901f\u5ea6\u53ef\u4ee5\u5206\u89e3\u4e3a <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>\uff0c\u5e76\u4e14\u6bcf\u4e2a\u5206\u91cf\u7684\u63a8\u5bfc\u8fc7\u7a0b\u76f8\u540c\uff0c\u56e0\u6b64\u7cfb\u7edf\u7c92\u5b50\u7684\u5e73\u5747\u52a8\u80fd\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<p>\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u201c\u80fd\u91cf\u5747\u5206\u539f\u7406\u201d\u3002\u57fa\u4e8e\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u7ed3\u8bba\uff0c\u5982\u679c\u7cfb\u7edf\u7531 <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> \u4e2a\u7c92\u5b50\u7ec4\u6210\uff0c\u5176\u5e73\u5747\u52a8\u80fd\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\lt E_{cin}\\right\\gt<\/span><\/span>\uff0c\u4e14\u7cfb\u7edf\u603b\u80fd\u91cf\u7eaf\u7cb9\u662f\u52a8\u80fd\uff0c\u90a3\u4e48\u7cfb\u7edf\u7684\u5185\u80fd\u5c06\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\tU=3Nk_BT\/2<\/span>\uff08\u5982\u4e4b\u524d\u63a8\u5bfc\uff09\uff0c\u5e76\u4e14\u5185\u80fd\u4ec5\u53d6\u51b3\u4e8e\u7cfb\u7edf\u7684\u6e29\u5ea6\uff0c\u5373\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{\\partial U}{\\partial V}\\right)_T = 0<\/span><\/span>.\n<\/p>\n<h3>\u7406\u60f3\u6c14\u4f53\u7684\u63a8\u5bfc<\/h3>\n<p>\n\t<a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=1027s\" target=\"_blank\" rel=\"noopener\"><br \/>\n\t\t<strong>\u73b0\u5728\uff0c\u56de\u987e\u7406\u60f3\u6c14\u4f53\u65b9\u7a0b<\/strong><br \/>\n\t<\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=Nk_BT =nRT<\/span><\/span>\u3002\u901a\u8fc7\u89e3\u4f53\u79ef\u5f97\u5230\uff1a\n<\/p>\n<p style=\"text-align: center;\">\n\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  V= \\frac{nRT}{P}<\/span><\/span>.\n<\/p>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t<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>.\n<\/p>\n<p>\u7ed3\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span> \u7684\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5f97\u51fa\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\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}\n\n<\/span>\n<p>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  C_V=(\\partial U \/ \\partial T)_V<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=3Nk_BT\/2=3nRT\/2<\/span><\/span>\uff0c\u56e0\u6b64\uff1a<\/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>\u6240\u4ee5\uff1a<\/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>\u7edd\u70ed\u6307\u6570<\/h3>\n<p>\u5e38\u7528\u7684\u4e00\u4e2a\u91cf\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span> \u7684\u6bd4\u503c\uff0c\u79f0\u4e3a<strong>\u7edd\u70ed\u6307\u6570<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span>\u3002\u5b9a\u4e49\u5982\u4e0b\uff1a<\/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>\u5bf9\u4e8e\u7406\u60f3\u6c14\u4f53\uff0c\u7edd\u70ed\u6307\u6570\u7684\u503c\u4e3a\uff1a<\/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>\u7ec3\u4e60<\/h2>\n<ol>\n<li>\n\t\t\u662f\u5426\u603b\u662f\u6b63\u786e\u7684\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU=C_VdT<\/span><\/span>? \u8bf7\u6bd4\u8f83\u4e00\u822c\u60c5\u51b5\u4e0e\u7406\u60f3\u6c14\u4f53\u7684\u60c5\u51b5\uff0c\u5e76\u8bf4\u660e\u7406\u7531\u3002\n\t<\/li>\n<li>\n\t\t\u5047\u8bbe\u5bf9\u4e8e\u7406\u60f3\u6c14\u4f53\uff0c\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=C_VT<\/span><\/span>\uff0c\u8ba1\u7b97\uff1a<\/p>\n<ol>\n<li>\u5355\u4f4d\u8d28\u91cf\u7684\u5185\u80fd\u3002<\/li>\n<li>\u5355\u4f4d\u4f53\u79ef\u7684\u5185\u80fd\u3002<\/li>\n<\/ol>\n<\/li>\n<li>\n\t\t\u4e00\u6469\u5c14\u5355\u539f\u5b50\u7406\u60f3\u6c14\u4f53\u88ab\u4e00\u4e2a\u6d3b\u585e\u9650\u5236\u5728\u4e00\u4e2a\u5706\u67f1\u4e2d\uff0c\u5e76\u901a\u8fc7\u4e0e\u70ed\u50a8\u76f8\u63a5\u89e6\u4fdd\u6301\u5728\u6052\u5b9a\u6e29\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_0<\/span><\/span>\u3002\u6c14\u4f53\u4ece\u4f53\u79ef <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1<\/span><\/span> \u7f13\u6162\u81a8\u80c0\u5230\u4f53\u79ef <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2<\/span><\/span>\uff0c\u5728\u6574\u4e2a\u8fc7\u7a0b\u4e2d\u6e29\u5ea6\u4fdd\u6301\u6052\u5b9a\u3002<\/p>\n<ol>\n<li>\u6c14\u4f53\u7684\u5185\u80fd\u662f\u5426\u53d1\u751f\u53d8\u5316\uff1f<\/li>\n<li>\u8ba1\u7b97\u6c14\u4f53\u6240\u505a\u7684\u529f\u4ee5\u53ca\u6c14\u4f53\u5438\u6536\u7684\u70ed\u91cf\u3002<\/li>\n<\/ol>\n<\/li>\n<li>\n\t\t\u8bc1\u660e\uff0c\u5bf9\u4e8e\u7406\u60f3\u6c14\u4f53\uff0c\u4ee5\u4e0b\u5173\u7cfb\u6210\u7acb\uff1a<\/p>\n<p style=\"text-align: center;\">\n\t\t\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{R}{C_V} = \\gamma - 1<\/span><\/span>\n\t\t<\/p>\n<p style=\"text-align: center;\">\n\t\t\t<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{R}{C_P} = \\frac{\\gamma - 1}{\\gamma}<\/span><\/span>\n\t\t<\/p>\n<p>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span> \u5206\u522b\u662f\u6469\u5c14\u5b9a\u5bb9\u70ed\u5bb9\u548c\u6469\u5c14\u5b9a\u538b\u70ed\u5bb9\u3002<\/p>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b \u70ed\u529b\u5b66\u7b2c\u4e00\u5b9a\u5f8b\u662f\u8fde\u63a5\u57fa\u672c\u6982\u5ff5\uff08\u5982\u70ed\u91cf\u3001\u529f\u548c\u5185\u80fd\uff09\u7684\u57fa\u7840\uff0c\u5b83\u89c4\u5b9a\u80fd\u91cf\u65e2\u4e0d\u4f1a\u88ab\u521b\u9020\u4e5f\u4e0d\u4f1a\u88ab\u9500\u6bc1\uff0c\u53ea\u4f1a\u53d1\u751f\u8f6c\u5316\u3002\u672c\u6587\u4ecb\u7ecd\u4e86\u8be5\u5b9a\u5f8b\u5728\u5c01\u95ed\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\uff0c\u6df1\u5165\u5206\u6790\u4e86\u70ed\u529b\u5b66\u529f\u3001\u70ed\u5bb9\u4ee5\u53ca\u6c14\u4f53\u7684\u7edf\u8ba1\u7279\u6027\u3002\u901a\u8fc7\u6570\u5b66\u516c\u5f0f\u548c\u7269\u7406\u63a8\u7406\u7684\u7ed3\u5408\uff0c\u60a8\u5c06\u638c\u63e1\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u4e2d\u80fd\u91cf\u8fc7\u7a0b\u7684\u57fa\u672c\u5de5\u5177\u3002 \u5b66\u4e60\u76ee\u6807\uff1a \u5728\u672c\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a 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