在热力学中,等容热容
以及等压热容
是单位为能量除以温度的广度性质,它们可通过膨胀系数和压缩系数联系在一起。
可根据热力学定律导出以下关系:[1]
![{\displaystyle C_{P}-C_{V}=VT{\frac {\alpha ^{2}}{\beta _{T}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c390cf97087a366531090b0dcbaf13d8fda545a)
![{\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8724863929180c9d93246866bb07ba781c24f0e0)
其中
是膨胀系数:
![{\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ec1ce9fef1bbd72c1462a1df8c8ea6d343e502)
是等温压缩系数(体积模量的倒数):
![{\displaystyle \beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/041cffa5820b54fc1c15d2963c2fc3ec1c2b3de2)
是等熵压缩系数:
![{\displaystyle \beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bf827ea3ee437e040b1d32f09df0f26a09f3c2)
定容和定压下比热容(强度特性)关系的对应表达式为:
![{\displaystyle c_{p}-c_{v}={\frac {T\alpha ^{2}}{\rho \beta _{T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e534f78f86fbbc1fdf47ccf68fdf08eab76da1a)
其中
是物质的密度。
比热容比的相应表达式保持不变,因为与热力学系统尺寸相关的量,无论是基于质量还是摩尔,在相除的时候都会被消掉,因为比热容是强度性质。因此:
![{\displaystyle {\frac {c_{p}}{c_{v}}}={\frac {\beta _{T}}{\beta _{S}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6515dff4b67e8fdd3dee581f9303a3b392dbbf)
热容间差的关系可被用于计算难以直接测定的固体恒容热容。我们也可通过热容比来表达等熵压缩系数。
在等容下:
![{\displaystyle C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}=\left({\frac {\partial U}{\partial S}}\right)_{V}\left({\frac {\partial S}{\partial T}}\right)_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3e2092c3d5f26ee3cd1ba4391e9cbc13b9f307)
同理可得
,作差:
![{\displaystyle C_{P}-C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{P}-T\left({\frac {\partial S}{\partial T}}\right)_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18926a7e5403ff6819c3e8d27e85426707098278)
由于熵
是温度、体积的函数,即
,体积
是温度、压强的函数,即
,根据复合函数偏微分的链式法则:
![{\displaystyle \left({\frac {\partial S}{\partial T}}\right)_{P}=\left({\frac {\partial S}{\partial T}}\right)_{V}+\left({\frac {\partial S}{\partial V}}\right)_{T}\left({\frac {\partial V}{\partial T}}\right)_{P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fedc6e80d6c86ac44ef29883156779cabfb9903)
带入:
![{\displaystyle {\begin{aligned}C_{P}-C_{V}&=T\left({\frac {\partial S}{\partial V}}\right)_{T}\left({\frac {\partial V}{\partial T}}\right)_{P}\\&=VT\alpha \left({\frac {\partial S}{\partial V}}\right)_{T}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7d62aa4c40d03dc0c44608ea614c9a33275736)
由麦克斯韦关系式和三乘积法则:
![{\displaystyle {\begin{aligned}C_{P}-C_{V}&=\left({\frac {\partial P}{\partial T}}\right)_{V}TV\alpha \\&={\frac {\alpha ^{2}TV}{\beta _{T}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd655ca58464bb00c1e83898b500abb722f5f7b)
若把
和
相除:
![{\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\left({\frac {\partial S}{\partial T}}\right)_{P}}{\left({\frac {\partial S}{\partial T}}\right)_{V}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8c03ff972570c7451f737751df42c4722ed990)
对分子分母分别使用三乘积法则,并重新组合:
![{\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\left({\frac {\partial P}{\partial T}}\right)_{S}\left({\frac {\partial V}{\partial S}}\right)_{T}}{\left({\frac {\partial P}{\partial S}}\right)_{T}\left({\frac {\partial V}{\partial T}}\right)_{S}}}={\frac {\left({\frac {\partial V}{\partial S}}\right)_{T}\left({\frac {\partial S}{\partial P}}\right)_{T}}{\left({\frac {\partial V}{\partial T}}\right)_{S}\left({\frac {\partial T}{\partial P}}\right)_{S}}}={\frac {\left({\frac {\partial V}{\partial P}}\right)_{T}}{\left({\frac {\partial V}{\partial P}}\right)_{S}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efea184e66534f7da96d045e4ceb2b7ebda0fb5a)
根据定义:
![{\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ac164aaa7359280fa7c576e90f99d82a0833b4)
理想气体满足理想气体状态方程:
![{\displaystyle PV=nRT}](https://wikimedia.org/api/rest_v1/media/math/render/svg/934032db2ac1f12624f85a90eeba651dcf4af377)
可由此求出理想气体的膨胀系数:
![{\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)={\frac {nR}{PV}}={\frac {1}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa16aab1898e751a9dcc34f564e478f31aa709c0)
由此求出理想气体的膨胀系数:
![{\displaystyle \beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}=-{\frac {1}{V}}\left(-{\frac {V}{P}}\right)={\frac {1}{P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/977d676b6a73af9be20fcacdce7284091ffafa5b)
带入关系式:
![{\displaystyle C_{P}-C_{V}={\frac {VP}{T}}=nR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff480db85f33157a20535706d3a393aa258e0b1)
- ^ Gaskell, David R. Introduction to the thermodynamics of materials 5th ed. New York: Taylor & Francis. 2008. ISBN 978-1-59169-043-6. OCLC 191024055.