H=U+pVF=U−TSG=U−TS+pV\begin{aligned} H =& U + pV\\ F =& U - TS\\ G =& U - TS + pV \end{aligned} H=F=G=U+pVU−TSU−TS+pV
微分形式:
U=TdS−pdVU = TdS - pdV U=TdS−pdV
F=−SdT−pdVF = -SdT - pdV F=−SdT−pdV
H=TdS+VdpH = TdS + Vdp H=TdS+Vdp
G=−SdT+VdpG = -SdT + Vdp G=−SdT+Vdp
H=G−T∂G∂TH=G-T\frac{\partial G}{\partial T} H=G−T∂T∂G
U=F−T∂F∂T=G−T∂G∂T−p∂G∂pU=F-T\frac{\partial F}{\partial T} =G-T\frac{\partial G}{\partial T}-p\frac{\partial G}{\partial p} U=F−T∂T∂F=G−T∂T∂G−p∂p∂G
(∂T∂V)S=−(∂p∂S)V\Big(\frac{\partial T}{\partial V}\Big)_S = -\Big(\frac{\partial p}{\partial S}\Big)_V (∂V∂T)S=−(∂S∂p)V
(∂T∂p)S=(∂V∂S)p\Big(\frac{\partial T}{\partial p}\Big)_S = \Big(\frac{\partial V}{\partial S}\Big)_p (∂p∂T)S=(∂S∂V)p
(∂S∂V)T=(∂p∂T)V\Big(\frac{\partial S}{\partial V}\Big)_T = \Big(\frac{\partial p}{\partial T}\Big)_V (∂V∂S)T=(∂T∂p)V
(∂S∂p)T=−(∂V∂T)p\Big(\frac{\partial S}{\partial p}\Big)_T = -\Big(\frac{\partial V}{\partial T}\Big)_p (∂p∂S)T=−(∂T∂V)p
一些扩展:
Cv=(∂U∂T)V=T(∂S∂T)VC_v = \Big(\frac{\partial U}{\partial T}\Big)_V = T \Big(\frac{\partial S}{\partial T}\Big)_V Cv=(∂T∂U)V=T(∂T∂S)V
Cp=(∂U∂T)p+p(∂V∂T)p=(∂H∂T)p=T(∂S∂T)pC_p = \Big(\frac{\partial U}{\partial T}\Big)_p + p \Big(\frac{\partial V}{\partial T}\Big)_p = \Big(\frac{\partial H}{\partial T}\Big)_p = T \Big(\frac{\partial S}{\partial T}\Big)_p Cp=(∂T∂U)p+p(∂T∂V)p=(∂T∂H)p=T(∂T∂S)p
(∂U∂V)T=T(∂p∂T)V−p=Tpβ−p\Big(\frac{\partial U}{\partial V}\Big)_T = T\Big(\frac{\partial p}{\partial T}\Big)_V - p = \frac{T}{p}\beta - p (∂V∂U)T=T(∂T∂p)V−p=pTβ−p
(∂H∂p)T=V−T(∂V∂T)p=V−TVα(\frac{\partial H}{\partial p})_T = V - T(\frac{\partial V}{\partial T})_p = V - \frac{T}{V}\alpha (∂p∂H)T=V−T(∂T∂V)p=V−VTα
气体在节流过程中焓不变。
μ=(∂T∂p)H=VCp(Tα−1)=1Cp[T(∂V∂T)p−V]\mu=\Big(\cfrac{\partial T}{\partial p}\Big)_H = \cfrac{V}{C_p}(T\alpha - 1)=\cfrac{1}{C_p}\Big[T\Big(\cfrac{\partial V}{\partial T}\Big)_p -V\Big]μ=(∂p∂T)H=CpV(Tα−1)=Cp1[T(∂T∂V)p−V] 称为焦汤系数。
可以利用节流过程中 μ>0\mu > 0μ>0 一侧制冷区，利用节流过程使得液体降温而液化。
U=∫{CvdT+[T(∂p∂T)V−p]dV}+U0U = \int\{C_vdT+[T(\frac{\partial p}{\partial T})_V -p]dV\}+U_0 U=∫{CvdT+[T(∂T∂p)V−p]dV}+U0
S=∫[CvTdT+(∂p∂TdV)]+S0S = \int[\frac{C_v}{T}dT+(\frac{\partial p}{\partial T}dV)]+S_0 S=∫[TCvdT+(∂T∂pdV)]+S0
辐射压强 p=13up=\cfrac{1}{3}up=31u，而能态密度 u=aT4u=aT^4u=aT4。
S=43aT3VS=\cfrac{4}{3}aT^3VS=34aT3V (可逆绝热下有 T3VT^3VT3V 常数)
辐射通量密度 Ju=14CUJ_u=\cfrac{1}{4}CUJu=41CU
m=MVm=MVm=MV 是介质的总磁矩
所做的功为 dW=μ0HdmdW =\mu_0HdmdW=μ0Hdm
磁介质的内能满足 dU=TdS+μ0hdmdU=TdS+\mu_0hdmdU=TdS+μ0hdm
磁介质的吉布斯函数满足 dG=−SdT−μ0mdHdG=-SdT-\mu_0mdHdG=−SdT−μ0mdH
磁介质的热容 CH=T(∂S∂T)HC_H=T(\cfrac{\partial S}{\partial T})_HCH=T(∂T∂S)H 则 (∂T∂H)S=−μ0TCH(∂m∂T)M(\cfrac{\partial T}{\partial H})_S=-\cfrac{\mu_0T}{C_H}(\cfrac{\partial m}{\partial T})_M(∂H∂T)S=−CHμ0T(∂T∂m)M
居里定律: m=CVTHm=\cfrac{C_V}{T}Hm=TCVH 可以得出: (∂T∂H)S=CVCHTμ0H(\cfrac{\partial T}{\partial H})_S=\cfrac{C_V}{C_HT}\mu_0H(∂H∂T)S=CHTCVμ0H 和 TdS=CVdT+TακTdVTdS =C_VdT+T\cfrac{\alpha}{\kappa_T}dVTdS=CVdT+TκTαdV
证明 κsκT=CvCp\cfrac{\kappa_s}{\kappa_T} = \cfrac{C_v}{C_p}κTκs=CpCv
证明 Cp−CV=−T(∂p∂T)V2(∂p∂V)TC_p - C_V = -T\cfrac{(\cfrac{\partial p}{\partial T})_V^2}{(\cfrac{\partial p}{\partial V})_T}Cp−CV=−T(∂V∂p)T(∂T∂p)V2
