试由波义尔温度[tex=1.143x1.214]Iut+51sQvaH3g40uPN2VQg==[/tex]的定义式，证明范德华气体的[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]可表示为[tex=4.714x1.357]tbsqI15yh+sdigLyysTB/ujdvLryTr8fxigaC4sPNrM=[/tex]式中[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]为范德华常数。

2022-10-25

试由波义尔温度[tex=1.143x1.214]Iut+51sQvaH3g40uPN2VQg==[/tex]的定义式，证明范德华气体的[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]可表示为[tex=4.714x1.357]tbsqI15yh+sdigLyysTB/ujdvLryTr8fxigaC4sPNrM=[/tex]式中[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]为范德华常数。

答案：

解：波义尔温度[tex=1.143x1.214]Iut+51sQvaH3g40uPN2VQg==[/tex]的定义式为[tex=8.286x2.929]7kMByFfHltGFyhym0UePrXLHiIRyr3W7S1KlaUWC84cW1Y4dCRPHVxXdqz8dTS/+R53xElSxNYAs8k5ljRtOYTdUvSwDVOzWA2g1UYG3J08YJvePedZ6bVLf9hiZL0Ib4CYyMrzLN4S6rE4yyXofPg==[/tex]先将范德华方程改写成[tex=12.571x2.786]538zUbDze/mrFiC//AyGOryY4kFqjQ/x594o35ygBizliIcUG8rT5o19Wnw/IP0Oom2mnjHTl350GKvCc4M1I1NspjbX05G7bskV2wxywniPwgzPA05PvURE3JJ1JVAbZnUbDijXNorYM8173Pq+cA==[/tex]根据复合函数微分法则对上式求微分，得[tex=25.429x3.357]7kMByFfHltGFyhym0UePrXLHiIRyr3W7S1KlaUWC84cW1Y4dCRPHVxXdqz8dTS/+R53xElSxNYAs8k5ljRtOYTdUvSwDVOzWA2g1UYG3J08YJvePedZ6bVLf9hiZL0Ibi+31EwZyeDUb5lKNlgxJu9xo4O/RWA2EtAxxknrY27ufXzVqEjvuWtxUKEJgOTfPQJxli5JnBWfg6ta1pBXXvJmbpx0vL+CPusy4wYNJ94ywxJ/8vBhj/QXB5BIaQFxJOCLMRhpm8cX9DFQ4+sTaJLrA2g/oxUmrOEt+QoPiBVgpzhrJGm/mL8lRNSPAnj7VaeZbNEMoupJ6glFSK22jlzy8BCERHvRjDRAnf4tKi+QavNc0LuqVg2PwhIWRirvTfhaCrk1qj2luTzZBw7WD3/nd2wq7KXYlTOLulXLt/RsavnAl77il90uDafA10pBCrQ0rp5tyV1WlDYrHlHPU7g==[/tex]在波义尔温度下，气体在几百千帕的压力范围内可较好地符合理想气体状态方程，因此，[tex=6.571x1.5]+lgFg97s3u46iEtzGrVgVfJOMuo+c+7zRR6mbcYTIAtZ09h+Rt1duNtVsoq0JXrrCxHPrfo0rsG91/8xLO3pn0sRUobwUpV4axHxapOlidU=[/tex]，所以只有[tex=10.286x2.857]76raDbgnwuvkBJF/fusDV9H2UXuc8mW2LWFSKHDikDXrWVy38lhusYN4Re+oRA6Y2WAgknn8UmVrwd0/7WFGRGisyijE2gpOnfDntrICVqWqWPAdjadZby054XpPS7GNeZwtr+HhRR8qsA5DYGaw2AlNg549OJWlndVbhcv7fGw=[/tex]由上式解出[tex=8.643x2.929]AyGcp6JX3BQKBVOsO0NYfHR7OmF/6edDPqGhRS2+70M8WhcQVpBWaCO9sbTNATEhaSTCumJ8wp2sxGNCwQkfiRzSLy2DTm4/VWlgI+3zZ1c=[/tex]因为[tex=16.857x1.357]fxIt1j6p3SJpJKLwgGa6iUInl0ILQnu07pdvCTVJCcMQNKmrlNUHhkA6yOeWVqUfq2dDE2793KyNkYtnIL4sg1QsSu/T5xW8GtmEl02Nsq+963nLpLl7UMhS63cx52PyCPEHU2E64opEW0VHojP3/A==[/tex]所以[tex=3.5x2.143]AyGcp6JX3BQKBVOsO0NYfN8T8+1jGNGY02o7M1AslQcTbkQz3Ntf9sAnxQwTqpP4[/tex]

举一反三

内容

0
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]HtShgpkNmCs66SDQIt6cYg==[/tex],使[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]带[tex=1.286x1.143]NeILClIc8twxeO5vjYm8Dw==[/tex][img=203x220]17e1aa8f4234a38.png[/img]
1
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]NeILClIc8twxeO5vjYm8Dw==[/tex],将[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的外表面接地。[img=203x220]17e1aa8f4234a38.png[/img]
2
设[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex],[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]是不等于零的整数.且满足下列两个条件的正整数[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex]叫做[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的最小公倍数：(i)[tex=3.571x1.357]2r4ZpNKLF6HpDoP4ji6v2g==[/tex]；(ii)如果[tex=1.929x1.071]rFBE4MTOSfVgaTsLfRa5FA==[/tex]且[tex=3.0x1.357]huACl7vUaYZTtkivcspxUA==[/tex]，则[tex=2.357x1.357]53n+iIHx1XAyRRtWGAbzKQ==[/tex].证明：[tex=1.357x1.357]TWUgLpDrEXIKICMuiEQPjw==[/tex]任意两个不等于零的整数[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex],[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]都有唯一的最小公倍数；[tex=1.214x1.357]vzdGmXlbw83hTiK2SebvEA==[/tex]令[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex]是[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的最小公倍数而[tex=3.357x1.357]Xxt8bFgvMkQLJViypSrDYg==[/tex],则[tex=4.0x1.357]Qf/TY1YnpQWchPW96yN99w==[/tex]
3
试证明，在环[tex=0.786x1.0]as0RCzgUx1oS48cKHRAVVg==[/tex]中，如对某两元素[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]有[tex=2.714x1.0]e5Tkis6ZqJFFkh05Fa5eXw==[/tex]，那么（1）[tex=4.571x1.214]G8LgwyeIG5Rh8byMRV1rYg==[/tex]（假定[tex=1.429x1.214]drLkwZObby+MgPbD6lCaLg==[/tex]存在）；（2）[tex=5.857x1.357]03Otkmf/H6NruAvI9Uocrw==[/tex]。
4
计算下述[tex=1.143x1.0]oTcZ8bPOd5+p8E1UHN7wXA==[/tex]阶行列式（主对角线上元素都是[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]，反对角线上元素都是[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]，空缺处的元素为0）：[tex=14.357x7.214]BafYOyQLkfv749f2fydiSmuFaORJrmT8ZJIXGdw44f1aEpC52UG/9KaK/rVnUNciUh3QoBqaPPxfmlIg/phge+h4iq0ABGDReZk1AL0sZKzKnThLESNQm78N48nK4v5O+GmV/flx/lbKKFGOzBOQhYxNt+leiRpulVjqMeOFBfnI0RXZdSR7MVvsUvgTHcf+ugGSltGkhpnoxUoJeFlxsZdoHZzCrpQrZ4mwb1kz/X/Fqe167F6aEL+T1v5e+y8WvVKhcL8g9UtTTbtwM8lNug==[/tex]