第二章

Mr.Hope ... 2020-12-28 物理
  • 热统
大约 2 分钟

# 四个热力学函数

H=U+pVF=UTSG=UTS+pV\begin{aligned} H =& U + pV\\ F =& U - TS\\ G =& U - TS + pV \end{aligned}

微分形式:

U=TdSpdVU = TdS - pdV

F=SdTpdVF = -SdT - pdV

H=TdS+VdpH = TdS + Vdp

G=SdT+VdpG = -SdT + Vdp

H=GTGTH=G-T\frac{\partial G}{\partial T}

U=FTFT=GTGTpGpU=F-T\frac{\partial F}{\partial T} =G-T\frac{\partial G}{\partial T}-p\frac{\partial G}{\partial p}

# 麦克斯韦关系

(TV)S=(pS)V(\frac{\partial T}{\partial V})_S = -(\frac{\partial p}{\partial S})_V

(Tp)S=(VS)p(\frac{\partial T}{\partial p})_S = (\frac{\partial V}{\partial S})_p

(SV)T=(pT)V(\frac{\partial S}{\partial V})_T = (\frac{\partial p}{\partial T})_V

(Sp)T=(VT)p(\frac{\partial S}{\partial p})_T = -(\frac{\partial V}{\partial T})_p

一些扩展:

Cv=(UT)V=T(ST)VC_v = (\frac{\partial U}{\partial T})_V = T (\frac{\partial S}{\partial T})_V

Cp=(UT)p+p(VT)p=(HT)p=T(ST)pC_p = (\frac{\partial U}{\partial T})_p + p (\frac{\partial V}{\partial T})_p = (\frac{\partial H}{\partial T})_p = T (\frac{\partial S}{\partial T})_p

(UV)T=T(pT)Vp=Tpβp(\frac{\partial U}{\partial V})_T = T(\frac{\partial p}{\partial T})_V - p = \frac{T}{p}\beta - p

(Hp)T=VT(VT)p=VTVα(\frac{\partial H}{\partial p})_T = V - T(\frac{\partial V}{\partial T})_p = V - \frac{T}{V}\alpha

# 节流过程

气体在节流过程中焓不变。

μ=(Tp)H=VCp(Tα1)=1Cp[T(VT)pV]\mu=(\cfrac{\partial T}{\partial p})_H = \cfrac{V}{C_p}(T\alpha - 1)=\cfrac{1}{C_p}[T(\cfrac{\partial V}{\partial T})_p -V] 称为焦汤系数。

可以利用节流过程中 μ>0\mu > 0 一侧制冷区,利用节流过程使得液体降温而液化。

# 内能与焓的积分形式

U={CvdT+[T(pT)Vp]dV}+U0U = \int\{C_vdT+[T(\frac{\partial p}{\partial T})_V -p]dV\}+U_0

S=[CvTdT+(pTdV)]+S0S = \int[\frac{C_v}{T}dT+(\frac{\partial p}{\partial T}dV)]+S_0

# 热辐射

辐射压强 p=13up=\cfrac{1}{3}u,而能态密度 u=aT4u=aT^4

S=43aT3VS=\cfrac{4}{3}aT^3V (可逆绝热下有 T3VT^3V 常数)

辐射通量密度 Ju=14CUJ_u=\cfrac{1}{4}CU

# 磁介质

m=MVm=MV 是介质的总磁矩

所做的功为 dW=μ0HdmdW =\mu_0Hdm

磁介质的内能满足 dU=TdS+μ0hdmdU=TdS+\mu_0hdm

磁介质的吉布斯函数满足 dG=SdTμ0mdHdG=-SdT-\mu_0mdH

磁介质的热容 CH=T(ST)HC_H=T(\cfrac{\partial S}{\partial T})_H(TH)S=μ0TCH(mT)M(\cfrac{\partial T}{\partial H})_S=-\cfrac{\mu_0T}{C_H}(\cfrac{\partial m}{\partial T})_M

居里定律: m=CVTHm=\cfrac{C_V}{T}H 可以得出: (TH)S=CVCHTμ0H(\cfrac{\partial T}{\partial H})_S=\cfrac{C_V}{C_HT}\mu_0HTdS=CVdT+TακTdVTdS =C_VdT+T\cfrac{\alpha}{\kappa_T}dV

# 重要习题

  1. 证明 κsκT=CvCp\cfrac{\kappa_s}{\kappa_T} = \cfrac{C_v}{C_p}

  2. 证明 CpCV=T(pT)V2(pV)TC_p - C_V = -T\cfrac{(\cfrac{\partial p}{\partial T})_V^2}{(\cfrac{\partial p}{\partial V})_T}