{"id":59576,"date":"2014-12-27T12:07:48","date_gmt":"2014-12-27T04:07:48","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=59576"},"modified":"2021-10-06T16:12:42","modified_gmt":"2021-10-06T08:12:42","slug":"%e5%88%86%e5%ad%90%e9%80%9f%e7%8e%87%e7%9a%84%e9%a6%ac%e5%85%8b%e5%a3%ab%e5%a8%81%e5%88%86%e5%b8%83-maxwells-distribution-of-molecular-speed","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%88%86%e5%ad%90%e9%80%9f%e7%8e%87%e7%9a%84%e9%a6%ac%e5%85%8b%e5%a3%ab%e5%a8%81%e5%88%86%e5%b8%83-maxwells-distribution-of-molecular-speed\/","title":{"rendered":"\u5206\u5b50\u901f\u7387\u7684\u99ac\u514b\u58eb\u5a01\u5206\u5e03"},"content":{"rendered":"

\u5206\u5b50\u901f\u7387\u7684\u99ac\u514b\u58eb\u5a01\u5206\u5e03 (Maxwell\u2019s Distribution of Molecular Speed)<\/strong><\/span>
\n\u570b\u7acb\u81fa\u7063\u5927\u5b78\u7269\u7406\u5b78\u7cfb98\u7d1a \u8521\u4ea6\u6db5<\/strong><\/span><\/p>\n

\u5f9e\u5c0f\u5230\u5927\uff0c\u73ed\u969b\u904b\u52d5\u6703\u4e00\u5b9a\u6709\u4e00\u500b\u9805\u76ee\u2500\u5927\u968a\u63a5\u529b\uff0c\u7b2c\u4e00\u68d2\u7387\u5148\u5728\u8dd1\u9053\u4e0a\u5954\u99b3\uff0c\u63a5\u8457\u4ea4\u7d66\u7b2c\u4e8c\u68d2\uff0c\u7e7c\u7e8c\u5728\u8dd1\u9053\u4e0a\u63ee\u7051\u9752\u6625\u7684\u6c57\u6c34\uff0c\u7b49\u5230\u6700\u5f8c\u4e00\u68d2\u8dd1\u56de\u7d42\u9ede\u7684\u90a3\u4e00\u524e\u90a3\uff0c\u88c1\u5224\u6309\u4e0b\u624b\u4e2d\u7684\u78bc\u9336\uff0c\u6642\u9593\u975c\u6b62\u90a3\u4e00\u523b\uff0c\u78bc\u9336\u4e0a\u9762\u7684\u6578\u5b57\u8a18\u9304\u8457\u6211\u5011\u4e00\u8d77\u5b8c\u6210\u7684\u6545\u4e8b\u3002\u9019\u6642\u5019\u5982\u679c\u6211\u5011\u628a\u7e3d\u5171\u8dd1\u7684\u8ddd\u96e2\u9664\u4ee5\u9019\u500b\u6642\u9593\uff0c\u6703\u5f97\u5230\u4e00\u500b\u300c\u901f\u7387\u300d\u7684\u6982\u5ff5\uff0c\u90a3\u9019\u500b\u901f\u7387\u4ee3\u8868\u4ec0\u9ebc\u5462\uff1f\u5176\u5be6\u9019\u500b\u901f\u7387\u662f\u300c\u5e73\u5747\u901f\u7387\u300d\uff0c\u4e5f\u4ee3\u8868\u8457\u5e73\u5747\u6bcf\u4f4d\u9078\u624b\u7684\u901f\u7387\u3002<\/p>\n

\u9019\u6642\u5019\u554f\u984c\u5c31\u4f86\u4e86\uff0c\u65e2\u7136\u9019\u500b\u53eb\u505a\u5e73\u5747\u901f\u7387\uff0c\u96e3\u9053\u5927\u5bb6\u90fd\u7528\u9019\u500b\u901f\u7387\u5728\u8dd1\u55ce\uff1f\u5f88\u660e\u986f\u7684\uff0c\u4e26\u4e0d\u662f\u6bcf\u4f4d\u9078\u624b\u90fd\u8dd1\u4e00\u6a23\u7684\u901f\u7387\uff0c\u5176\u5be6\u5927\u5bb6\u6709\u5feb\u6709\u6162\uff0c\u9019\u500b\u300c\u5e73\u5747\u300d\u4ee3\u8868\u7684\u662f\u6211\u5011\u9019\u4e00\u500b\u300c\u7fa4\u9ad4\u300d\uff0c\u300c\u500b\u5225\u300d\u662f\u6709\u5dee\u7570\u7684\u3002<\/p>\n

\u73fe\u5728\u8003\u616e\u4e00\u500b\u65b9\u76d2\u5b50\uff0c\u6211\u5011\u53ef\u4ee5\u6e2c\u5f97\u65b9\u76d2\u5b50\u5167\u7684\u6eab\u5ea6\uff0c\u6839\u64da\u6c23\u9ad4\u52d5\u529b\u8ad6(kinetic theory of gas)\uff0c\u6211\u5011\u77e5\u9053\u6eab\u5ea6\u8ddf\u5206\u5b50\u901f\u7387\u6709\u95dc\uff0c\u4f46\u6211\u5011\u6e2c\u5f97\u7684\u9019\u500b\u901f\u7387\u53ea\u80fd\u300c\u4ee3\u8868\u7fa4\u9ad4\u300d\uff0c\u5982\u540c\u4e0a\u8ff0\u5927\u968a\u63a5\u529b\u7684\u4f8b\u5b50\uff0c\u5206\u5b50\u904b\u52d5\u6709\u5feb\u6709\u6162\uff0c\u4f46\u5206\u5b50\u7684\u6578\u76ee\u5c31\u9019\u9ebc\u9f90\u5927\uff0c\u4e0d\u53ef\u80fd\u77e5\u9053\u6bcf\u500b\u5206\u5b50\u904b\u52d5\u7684\u901f\u7387\uff0c\u66f4\u5225\u8aaa\u904b\u52d5\u65b9\u5411\u4e86\uff0c\u9019\u500b\u7d50\u679c\u5be6\u5728\u4ee4\u4eba\u6cae\u55aa\u3002<\/p>\n

\u4f46\u4e0a\u5e1d\u95dc\u4e0a\u4e00\u6247\u9580\uff0c\u4e00\u5b9a\u6703\u70ba\u6211\u5011\u958b\u53e6\u4e00\u6247\u7a97\uff0c\u9019\u6642\u5019\u8070\u660e\u7684\u7269\u7406\u5b78\u5bb6\u2500\u99ac\u514b\u58eb\u5a01(Maxwell)\u548c\u6ce2\u8332\u66fc(Boltzmann)\uff0c\u63d0\u51fa\u4e86\u7d71\u8a08\u529b\u5b78(statistical mechanics)\u4e00\u500b\u5f88\u91cd\u8981\u7684\u7406\u8ad6\u2500\u99ac\u514b\u58eb\u5a01-\u6ce2\u8332\u66fc\u5206\u5e03(Maxwell-Boltzmann distribution)\uff1a<\/p>\n

\\(\\displaystyle\\frac{N_i}{N}=\\displaystyle\\frac{e^{\\displaystyle(-\\frac{E_i}{kT})}}{\\sum_j e^{\\displaystyle(-\\frac{E_j}{kT})}}\\)<\/p>\n

\u00a0\u5176\u4e2d \\(N\\) \u4ee3\u8868\u7e3d\u5206\u5b50\u6578\uff0c\\(N_i\\)\u00a0\u4ee3\u8868\u80fd\u91cf\u70ba \\(E_i\\)\u00a0\u7684\u5206\u5b50\u6578\u3002<\/p>\n

\u9019\u500b\u7406\u8ad6\u5f88\u91cd\u8981\u7684\u5047\u8a2d\u662f\uff1a\u4e0d\u8003\u616e\u76f8\u5c0d\u8ad6\u6548\u61c9\u3001\u6c92\u6709\u91cf\u5b50\u73fe\u8c61\u3001\u5206\u5b50\u6578\u91cf\u76f8\u7576\u9f90\u5927\u4e14\u5ffd\u7565\u9664\u4e86\u78b0\u649e\u4e4b\u5916\u7684\u4ea4\u4e92\u4f5c\u7528\u3002\u9019\u500b\u516c\u5f0f\u63cf\u8ff0\u4e86\u5728\u67d0\u56fa\u5b9a\u6eab\u5ea6 \\(T\\) \u7684\u60c5\u5f62\u4e0b\uff0c\u5404\u80fd\u91cf\u5206\u5b50\u6240\u4f54\u6709\u7684\u6bd4\u4f8b\uff0c\\(k\\) \u70ba\u6ce2\u8332\u66fc\u5e38\u6578\u3002\u73fe\u5728\u6211\u5011\u628a\u516c\u5f0f\u505a\u500b\u6539\u5beb\uff0c\u5206\u5b50\u3001\u5206\u6bcd\u540c\u4e58\u67d0\u500b\u5e38\u6578 \\(C\\)\uff0c\u4f7f\u5206\u6bcd\u7684\u6578\u503c\u8b8a\u6210 \\(1\\)\uff0c\u90a3\u9ebc\u516c\u5f0f\u8b8a\u6210\uff1a<\/p>\n

\\(\\displaystyle\\frac{N_i}{N}=Ce^{(\\displaystyle-\\frac{E_i}{kT})}\\equiv P(E_i)\\)<\/p>\n

\\(P(E_i)\\)\u00a0<\/span>\u4ee3\u8868\u80fd\u91cf\u70ba \\(E_i\\)\u00a0\u7684\u6a5f\u7387\u3002\u66f4\u7cbe\u78ba\u4f86\u8aaa\u61c9\u8a72\u662f\u6a5f\u7387\u5bc6\u5ea6(probability density)\uff0c\u5927\u5bb6\u5982\u679c\u7b2c\u4e00\u6b21\u807d\u5230\u9019\u500b\u540d\u8a5e\uff0c\u53ef\u80fd\u6703\u89ba\u5f97\u5f88\u7591\u60d1\uff0c\u4ee5\u4e0b\u5c31\u7528\u4e00\u4e9b\u5716\u4f86\u8aaa\u660e\u3002<\/p>\n

\"q3\"<\/a><\/p>\n

\u4e0a\u5716\u7e3d\u5171\u6709 \\(20\\) \u500b\u7c92\u5b50\uff0c\u5404\u80fd\u91cf\u5177\u6709\u7684\u7c92\u5b50\u6578\u4e5f\u90fd\u6a19\u793a\u51fa\u4f86\uff0c\u73fe\u5728\u4ee5 \\(E=2\\) \u70ba\u4f8b\uff0c\u6a5f\u7387\u660e\u986f\u770b\u51fa\u662f\uff1a<\/p>\n

\u00a0\\(\\displaystyle\\frac{N_2}{N}=\\frac{3}{20}\\)
\n<\/a><\/p>\n

\u800c\u9019\u500b\u7b97\u51fa\u4f86\u7684\u6578\u503c\u6070\u5de7\u7b49\u65bc \\(E=2\\) \u90a3\u689d\u9577\u689d\u300c\u9762\u7a4d\u300d\u6240\u4f54\u6574\u500b\u85cd\u8272\u90e8\u5206\u9762\u7a4d\u7684\u6bd4\u503c\uff0c\u65bc\u662f\u6211\u5011\u628a\u5716\u8f49\u63db\u6210\u9019\u6a23\uff1a<\/p>\n

\"q5\"<\/a><\/p>\n

\u9019\u6a23\u5c31\u4e00\u76ee\u4e86\u7136\u4e86\uff0c\u6a5f\u7387\u672c\u8eab\u6709\u300c\u9762\u7a4d\u300d\u7684\u611f\u89ba\uff0c\u5b83\u662f\u7531\u300c\u6a5f\u7387\u5bc6\u5ea6\u300d\u4e58\u4ee5\u300c\u5e95\u300d\u5f97\u5230\u7684\uff0c\u56e0\u70ba\u73fe\u5728\u80fd\u91cf\u662f\u4e0d\u9023\u7e8c\u7684\uff0c\u6bcf\u4e00\u500b\u80fd\u91cf\u5dee\u90fd\u662f \\(1\\)\uff0c\u6240\u4ee5\u6a5f\u7387\u5bc6\u5ea6\u6070\u5de7\u7b49\u65bc\u6a5f\u7387\u3002<\/p>\n

\u00a0\u6b64\u6642\u6211\u5011\u628a\u80fd\u91cf\u7576\u6210\u9023\u7e8c\u5206\u5e03\uff0c\u90a3\u9ebc\u5728\u80fd\u91cf \\(E_i\\)\u00a0\u9644\u8fd1\u7684\u6a5f\u7387\u8b8a\u6210\uff1a<\/p>\n

\\(Ce^{(-\\displaystyle\\frac{E_i}{kT})}\\mathrm{d}E\\)<\/p>\n

\u73fe\u5728\u6211\u5011\u60f3\u8981\u77e5\u9053\u901f\u7387\u7684\u5206\u5e03\uff0c\u65bc\u662f\u6211\u5011\u628a \\(E_i\\)\u00a0\u7528 \\(\\frac{1}{2}m(v_x^2+v_y^2+v_z^2)\\)\u00a0\u4ee3\u5165\uff0c\u56e0\u70ba\u6709\u4e09\u500b\u65b9\u5411\uff0c\u6240\u4ee5\u6a5f\u7387\u5beb\u6210\uff1a<\/p>\n

\\(Ce^{(-\\displaystyle\\frac{\\frac{1}{2}m(v_x^2+v_y^2+v_z^2)}{kT})}\\mathrm{d}v_x\\mathrm{d}v_y\\mathrm{d}v_z\\)<\/p>\n

\u5206\u5225\u5c0d\u4e09\u500b\u65b9\u5411\u7a4d\u5206\uff0c\u7a4d\u5206\u7bc4\u570d\u7686\u70ba \\(-\\infty\\) \u5230\\(\\infty\\)\uff0c\u4e14\u6a5f\u7387\u7684\u7e3d\u548c\u5fc5\u9808\u70ba \\(1\\)\uff0c\u65bc\u662f\uff1a<\/p>\n

\\(\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}Ce^{(-\\displaystyle\\frac{\\frac{1}{2}m(v_x^2+v_y^2+v_z^2)}{kT})}\\mathrm{d}v_x\\mathrm{d}v_y\\mathrm{d}v_z=1\\)<\/p>\n

\u7d93\u904e\u6574\u7406\u6211\u5011\u5f97\u5230\u6a5f\u7387\u5bc6\u5ea6\uff1a<\/p>\n

\\(P(v_x,v_y,v_z)=\\sqrt{\\displaystyle\\frac{m}{2\\pi kT}}^3e^{(-\\displaystyle\\frac{m(v_x^2+v_y^2+v_z^2)}{2kT})}\\)<\/p>\n

\u9019\u500b\u65b9\u7a0b\u5f0f\u544a\u8a34\u6211\u5011\u5404\u500b\u5206\u91cf\u7684\u6a5f\u7387\u5bc6\u5ea6\uff0c\u4f46\u662f\u6211\u5011\u6bd4\u8f03\u611f\u8208\u8da3\u7684\u662f\u901f\u7387\uff0c\u5373\u70ba\u901f\u5ea6\u5927\u5c0f\uff0c\u4e0d\u7ba1\u4ed6\u7684\u65b9\u5411\uff0c\u65bc\u662f\u6211\u5011\u6709\u4ee5\u4e0b\u5f0f\u5b50\uff1a<\/p>\n

\\(\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\sqrt{\\frac{m}{2\\pi kT}}^3e^{(-\\frac{m(v_x^2+v_y^2+v_z^2)}{2kT})} \\mathrm{d}v_x\\mathrm{d}v_y\\mathrm{d}v_z=\\int_{0}^{\\infty}\\sqrt{\\frac{m}{2\\pi kT}}^3e^{(-\\frac{mv^2}{2kT})}4\\pi v^2 \\mathrm{d}v\\)<\/p>\n

\u9019\u662f\u5c07\u76f4\u89d2\u5750\u6a19\u7cfb\u7684\u7a4d\u5206\u8f49\u63db\u6210\u7403\u5ea7\u6a19\u7cfb\u7684\u7a4d\u5206\uff0c\u7d93\u904e\u6574\u7406\uff1a<\/p>\n

\\(\\int_{0}^{\\infty}\\displaystyle\\sqrt{\\frac{m}{2\\pi kT}}^3e^{\\displaystyle(-\\frac{mv^2}{2kT})}4\\pi v^2 \\mathrm{d}v=\\int_0^{\\infty}P(v)\\mathrm{d}v\\)<\/p>\n

\u6211\u5011\u53ef\u4ee5\u5f97\u5230\u8ddf\u5206\u5b50\u901f\u7387\u6709\u95dc\u7684\u6a5f\u7387\u5206\u5e03\uff1a<\/p>\n

\\(P(v)=\\displaystyle\\sqrt{\\frac{2}{\\pi}(\\frac{m}{kT})^3}v^2e^{(\\frac{-m}{2kT})v^2}\\)<\/p>\n

\u9019\u5c31\u662f\u5206\u5b50\u901f\u7387\u7684\u99ac\u514b\u58eb\u5a01\u5206\u5e03(Maxwell\u2019s Distribution of Molecular Speed)\u3002<\/p>\n

\u9019\u500b\u6a5f\u7387\u5bc6\u5ea6\u516c\u5f0f\u544a\u8a34\u6211\u5011\uff0c\u5728\u67d0\u500b\u6eab\u5ea6 \\(T\\) \u7684\u60c5\u5f62\u4e0b\uff0c\u5206\u5b50\u901f\u7387\u70ba \\(v\\) \u6240\u4f54\u6709\u7684\u6a5f\u7387\u5bc6\u5ea6\u3002\u6211\u5011\u53ef\u4ee5\u5229\u7528\u9019\u500b\u516c\u5f0f\u9032\u4e00\u6b65\u7b97\u51fa\u54ea\u500b\u901f\u7387\u7684\u6a5f\u7387\u6700\u9ad8\u3001\u5e73\u5747\u901f\u7387\u3001\u65b9\u5747\u6839\u901f\u7387\u7b49\u7b49\uff0c\u5982\u4e0b\u6240\u793a\uff0c\u5176\u4e2d\u65b9\u5747\u6839\u901f\u7387\u4e0d\u504f\u4e0d\u501a\u7b49\u65bc\u6c23\u9ad4\u52d5\u529b\u8ad6\u88e1\u9762\u63a8\u5c0e\u904e\u7684\u65b9\u5747\u6839\u901f\u7387\u3002<\/p>\n

\u6a5f\u7387\u6700\u5927\u7684\u901f\u7387 \\(V_p=\\displaystyle\\sqrt{\\frac{2kT}{m}}\\)<\/p>\n

\u5e73\u5747\u901f\u7387\\(V_m=\\displaystyle\\sqrt{\\frac{8kT}{m}}\\)<\/p>\n

\u65b9\u5747\u6839\u901f\u7387\u00a0\\(V_{rms}=\\displaystyle\\sqrt{\\frac{3kT}{m}}\\)<\/p>\n

\u6709\u8da3\u7684\u662f\uff0c\u5c31\u7b97\u6211\u5011\u6709\u80fd\u529b\u5728\u4e00\u958b\u59cb\u6642\u6545\u610f\u5b89\u6392\u6c23\u9ad4\u5206\u5b50\uff0c\u8b93\u5b83\u5011\u4e0d\u6eff\u8db3\u99ac\u514b\u58eb\u5a01(\u4e5f\u7b49\u540c\u65bc\u99ac\u514b\u58eb\u5a01-\u6ce2\u8332\u66fc)\u5206\u5e03\uff0c\u4f46\u6c23\u9ad4\u5206\u5b50\u9084\u662f\u6703\u900f\u904e\u5927\u91cf\u7684\u78b0\u649e\u53bb\u4ea4\u63db\u80fd\u91cf\u8207\u52d5\u91cf\uff0c\u65bc\u662f\u5728\u7d93\u904e\u4e00\u5c0f\u6bb5\u6642\u9593\u5f8c\uff0c\u7cfb\u7d71\u5c31\u6eff\u8db3\u99ac\u514b\u58eb\u5a01\u5206\u5e03\u4e86\uff01\u9019\u53ef\u4ee5\u5f9e\u57163\u7684\u96fb\u8166\u6a21\u64ec\u4e2d\u770b\u51fa\u4f86(\u57163\u662f\u500b\u5f88\u5927\u7684\u6a94\u6848\uff0c\u7d0425MB\uff0c\u6545\u5efa\u8b70\u8b80\u8005\u4e0b\u8f09\u56de\u53bb\u81ea\u884c\u89c0\u770b\u52d5\u756b)\u3002<\/p>\n

\"Fig<\/a>

\u57163 (\u9673\u7fa9\u88d5\u7e6a) \u6545\u610f\u5b89\u6392\u6c23\u9ad4\u5206\u5b50\u5728\u4e00\u500b\u65b9\u5f62\u7684\u7bb1\u5b50\u5167(\u4e0a\u5716)\uff0c\u8b93\u5b83\u5011\u4e0d\u6eff\u8db3\u99ac\u514b\u58eb\u5a01\u5206\u5e03\uff0c\u4f46\u5206\u5b50\u9084\u662f\u6703\u900f\u904e\u5927\u91cf\u7684\u78b0\u649e\u53bb\u4ea4\u63db\u80fd\u91cf\u8207\u52d5\u91cf\uff0c\u65bc\u662f\u5728\u7d93\u904e\u4e00\u5c0f\u6bb5\u6642\u9593\u5f8c\uff0c\u7cfb\u7d71\u5c31\u6eff\u8db3\u99ac\u514b\u58eb\u5a01\u5206\u5e03(\u6dfa\u85cd\u8272\u7684\u66f2\u7dda\u70ba\u7406\u8ad6\u9810\u6e2c)\u3002<\/p><\/div>\n

\n

\u53c3\u8003\u6587\u737b<\/b><\/p>\n

\u7dad\u57fa\u767e\u79d1. Maxwell-Boltzmann distribution. http:\/\/en.wikipedia.org\/wiki\/Maxwell\\(E2\\)80\\(93Boltzmann_distribution<\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u5206\u5b50\u901f\u7387\u7684\u99ac\u514b\u58eb\u5a01\u5206\u5e03 (Maxwell\u2019s Distribution of Molecular Speed)…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,16,107],"tags":[7579,7578,2205,2158,7580],"class_list":["post-59576","post","type-post","status-publish","format-standard","hentry","category-physics03-02","category-physics03","category-physics00","tag-7579","tag-7578","tag-2205","tag-2158","tag-7580","loop-entry","cat-18","cat-16","cat-107","no-thumbnail"],"views":20738,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/59576","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=59576"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/59576\/revisions"}],"predecessor-version":[{"id":86693,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/59576\/revisions\/86693"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=59576"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=59576"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=59576"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}