热容间的关系

在热力学中，等容热容 C V {\displaystyle C_{V}} 以及等压热容 C P {\displaystyle C_{P}} 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： C P − C V = V T α 2 β T {\displaystyle C_{P}-C_{V}=VT{\frac {\alpha ^{2}}{\beta _{T}}}\,} C P C V = β T β S {\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}\,} 其中 α {\displaystyle \alpha } 是膨胀系数： α = 1 V ( ∂ V ∂ T ) P {\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\,} β T {\displaystyle \beta _{T}} 是等温压缩系数（体积模量的倒数）： β T = − 1 V ( ∂ V ∂ P ) T {\displaystyle \beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\,} β S {\displaystyle \beta _{S}} 是等熵压缩系数： β S = − 1 V ( ∂ V ∂ P ) S {\displaystyle \beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\,} 定容和定压下比热容（强度特性）关系的对应表达式为： c p − c v = T α 2 ρ β T {\displaystyle c_{p}-c_{v}={\frac {T\alpha ^{2}}{\rho \beta _{T}}}} 其中 ρ {\displaystyle \rho } 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： c p c v = β T β S {\displaystyle {\frac {c_{p}}{c_{v}}}={\frac {\beta _{T}}{\beta _{S}}}\,} 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： C V = ( ∂ U ∂ T ) V = ( ∂ U ∂ S ) V ( ∂ S ∂ T ) V = T ( ∂ S ∂ T ) V {\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}} 同理可得 C P = T ( ∂ S ∂ T ) P {\displaystyle C_{P}=T\left({\frac {\partial S}{\partial T}}\right)_{P}} ，作差： C P − C V = T ( ∂ S ∂ T ) P − T ( ∂ S ∂ T ) V {\displaystyle C_{P}-C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{P}-T\left({\frac {\partial S}{\partial T}}\right)_{V}} 由于熵 S {\displaystyle S} 是温度、体积的函数，即 S = S ( T , V ) {\displaystyle S=S(T,V)} ，体积 V {\displaystyle V} 是温度、压强的函数，即 V = V ( T , P ) {\displaystyle V=V(T,P)} ，根据复合函数偏微分的链式法则： ( ∂ S ∂ T ) P = ( ∂ S ∂ T ) V + ( ∂ S ∂ V ) T ( ∂ V ∂ T ) P {\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}} 带入： C P − C V = T ( ∂ S ∂ V ) T ( ∂ V ∂ T ) P = V T α ( ∂ S ∂ V ) T {\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}}} 由麦克斯韦关系式和三乘积法则： C P − C V = ( ∂ P ∂ T ) V T V α = α 2 T V β T {\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}}} 若把 C P {\displaystyle C_{P}} 和 C V {\displaystyle C_{V}} 相除： C P C V = ( ∂ S ∂ T ) P ( ∂ S ∂ T ) V {\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\left({\frac {\partial S}{\partial T}}\right)_{P}}{\left({\frac {\partial S}{\partial T}}\right)_{V}}}} 对分子分母分别使用三乘积法则，并重新组合： C P C V = ( ∂ P ∂ T ) S ( ∂ V ∂ S ) T ( ∂ P ∂ S ) T ( ∂ V ∂ T ) S = ( ∂ V ∂ S ) T ( ∂ S ∂ P ) T ( ∂ V ∂ T ) S ( ∂ T ∂ P ) S = ( ∂ V ∂ P ) T ( ∂ V ∂ P ) S {\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}}}} 根据定义： C P C V = β T β S {\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}} 理想气体满足理想气体状态方程： P V = n R T {\displaystyle PV=nRT} 可由此求出理想气体的膨胀系数： α = 1 V ( ∂ V ∂ T ) = n R P V = 1 T {\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)={\frac {nR}{PV}}={\frac {1}{T}}} 由此求出理想气体的膨胀系数： β T = − 1 V ( ∂ V ∂ P ) T = − 1 V ( − V P ) = 1 P {\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}}} 带入关系式： C P − C V = V P T = n R {\displaystyle C_{P}-C_{V}={\frac {VP}{T}}=nR} Gaskell, David R. Introduction to the thermodynamics of materials 5th ed. New York: Taylor & Francis. 2008. ISBN 978-1-59169-043-6. OCLC 191024055.  引文格式1维护：冗余文本 (link)