导航：题目详情 | 答案解析 | 相关问题 | 最新问题

由巨正则分布导出双原子分子理想气体的物态方程内能熵和化学势．

大学 2023-12-20 12:04:03 物态,内能,正则已帮助人

由巨正则分布导出双原子分子理想气体的物态方程、内能、熵和化学势．

热搜：单原子理想气体的配分函数

答案解析

难度★★☆ 搜索

按照巨正则系综理论，定义巨配分函数为
$latex.action?latex=XFhpID1cc3VtIF97Tj0wfV57XGluZnR5fVxzdW0gX3tzfWVeey1cYWxwaGEgTi1cYmV0YSBFX3tz%0D%0AfX0%3D$ ， (1)
式中，求和包含粒子数为N时系统对所有可能的微观状态s的求和，以及对所有可能的N的求和．将巨配分函数换一种写法为
$latex.action?latex=XFhpID1cc3VtIF97Tj0wfV57XGluZnR5fWVeey1cYWxwaGEgTn1cc3VtIF97c31lXnstXGJldGEg%0D%0ARV97c319PVxzdW0gX3tOPTB9XntcaW5mdHkgfWVeey1cYWxwaGEgTn1a$ ， (2)
式中，Z为具有N个粒子的正则配分函数，对于理想气体，有
$latex.action?latex=Wj1cZnJhY3sxfXtOIX16XntOfQ%3D%3D$ ， (3)
式中，z为单粒子的配分函数．对于双原子分子理想气体，有
$latex.action?latex=Wj16Xnt0fXpee3J9el57dn09VlxsZWZ0KCBcZnJhY3syXHBpIG19e1xiZXRhIGheezJ9fVxyaWdo%0D%0AdClee1xmcmFjezN9ezJ9fVxjZG90IFxmcmFjezhccGkgXnsyfUl9e1xiZXRhIGheezJ9fVxjZG90%0D%0AIFxmcmFjezJccGkgfXtcYmV0YSBoXG9tZWdhIH0%3D$
$latex.action?latex=PVxmcmFjezE2XHBpIF57M31JVn17XGJldGEgXnsyfWheezN9XG9tZWdhIH1cbGVmdCggXGZyYWN7%0D%0AMlxwaSBtfXtcYmV0YSBoXnsyfX1ccmlnaHQpXntcZnJhY3szfXsyfX0%3D$ ， (4)
所以
$latex.action?latex=XFhpID1cc3VtIF97Tj0wfV57XGluZnR5IH1cZnJhY3sxfXtOIX1cbGVmdChlXnstXGFscGhhfXpc%0D%0AcmlnaHQpXntOfT1leHBcbGVmdCggZV57LVxhbHBoYSB9elxyaWdodCk%3D$ . (5)
巨配分函数的对数为
$latex.action?latex=bG5cWGkgPWVeey1cYWxwaGEgfXo9ZV57LVxhbHBoYSB9XGZyYWN7MTZccGkgXnszfUlWfXtcYmV0%0D%0AYSBeezJ9aF57M31cb21lZ2EgfVxsZWZ0KCBcZnJhY3syXHBpIG19e1xiZXRhIGheezJ9fVxyaWdo%0D%0AdClee1xmcmFjezN9ezJ9fQ%3D%3D$ . (6)
气体的平均粒子数为
$latex.action?latex=XGJhcntOfT0tXGZyYWN7XHBhcnRpYWwgfXtccGFydGlhbCBcYWxwaGEgfWxuXFhpID1sblxYaSA9%0D%0AZV57LVxhbHBoYSB9XGZyYWN7MTZccGkgXnszfUlWfXtcYmV0YSBeezJ9aF57M31cb21lZ2EgfVxs%0D%0AZWZ0KCBcZnJhY3syXHBpIG19e1xiZXRhIGheezJ9fVxyaWdodClee1xmcmFjezN9ezJ9fQ%3D%3D$ . (7)
由此可得
$latex.action?latex=XGFscGhhID1sblxsZWZ0WyBcZnJhY3sxNlxwaSBeezN9SVZ9e1xiYXIge059XGJldGEgXnsyfWhe%0D%0AezN9XG9tZWdhIH1cbGVmdCggXGZyYWN7MlxwaSBtfXtcYmV0YSBoXnsyfX1ccmlnaHQpXntcZnJh%0D%0AY3szfXsyfX1ccmlnaHRd$ . (8)
化学势为
$latex.action?latex=XG11ID0ta1RcYWxwaGE9a1RsblxsZWZ0WyBcZnJhY3tcYmFyIHtOfVxiZXRhIF57Mn1oXnszfVxv%0D%0AbWVnYSB9ezE2XHBpIF57M31JVn1cbGVmdCggXGZyYWN7XGJldGEgaF57Mn19ezJccGkgbX1ccmln%0D%0AaHQpXntcZnJhY3szfXsyfX1ccmlnaHRd$ . (9)
气体的内能为
$latex.action?latex=RT0tXGZyYWN7XHBhcnRpYWwgfXtccGFydGlhbCBcYmV0YSB9bG5cWGkg$
$latex.action?latex=PWVeey1cYWxwaGEgfVxmcmFjezE2XHBpIF57M31JVn17XGJldGEgXnsyfWheezN9XG9tZWdhIH1c%0D%0AbGVmdCggXGZyYWN7XGJldGEgaF57Mn19ezJccGkgbX1ccmlnaHQpXntcZnJhY3szfXsyfX1cY2Rv%0D%0AdCBcZnJhY3s3fXsyXGJldGF9$
$latex.action?latex=PVxmcmFjezd9ezJ9XGJhciB7Tn1rVA%3D%3D$ . (10)
压强为
$latex.action?latex=cD1cZnJhY3sxfXtcYmV0YSB9XGZyYWN7XHBhcnRpYWwgfXtccGFydGlhbCBWfWxuXFhpID1cZnJh%0D%0AY3trVH17Vn1cYmFyIHtOfQ%3D%3D$ ， (11)
即
$latex.action?latex=cFY9XGJhciB7Tn1rVA%3D%3D$ . (12)
气体的熵为
$latex.action?latex=Uz1rXGxlZnQoIGxuXFhpIC1cYWxwaGEgXGZyYWN7XHBhcnRpYWwgfXtccGFydGlhbCBcYWxwaGEg%0D%0AfWxuXFhpIC1cYmV0YSBcZnJhY3tccGFydGlhbCB9e1xwYXJ0aWFsIFxiZXRhIH1sblxYaSBccmln%0D%0AaHQp$
$latex.action?latex=PWtcbGVmdCggbG5cWGkgK1xhbHBoYSBcYmFyIHtOfStcYmV0YSBFXHJpZ2h0KQ%3D%3D$ $latex.action?latex=PVxiYXIge059XGxlZnQoIDErXGFscGhhICtcZnJhY3s3fXsyfVxyaWdodCk%3D$ . (13)
将式(8)代入，得
$latex.action?latex=Uz1cYmFyIHtOfWtsblxsZWZ0WyBcZnJhY3sxNlxwaSBeezN9SVZ9e1xiYXIge059XGJldGEgXnsy%0D%0AfWheezN9XG9tZWdhIH1cbGVmdCggXGZyYWN7XGJldGEgaF57Mn19ezJccGkgbX1ccmlnaHQpXntc%0D%0AZnJhY3szfXsyfX1ccmlnaHRdK1xmcmFjezl9ezJ9XGJhciB7Tn1r$ . (14)