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热力学一般关系式

发布于 6/9/2026, 11:30:42 AM

工程热力学

基本状态参数和状态方程

在简单可压缩系统(与外界只有准静态体积变化功交换)中有如下基本状态参数:

  • 压力p(力学参数)
  • 比体积v(位移变量)
  • 温度T(热力学第零定律)
  • 热力学能u(热一)
  • 熵s(热二)

在这之中,由三个可测基本状态参数pvT组成的函数关系F(p,v,T)=0F(p,v,T)=0称为状态方程

一般有如下常用状态方程:

{简单可压缩系统：F(P,T,V)=0,热力学能函数：u=u(T,v),焓函数：h=h(T,P),熵函数：s=s(T,v)=s(T,p).\left\{ \begin{aligned} &\text{简单可压缩系统：}\quad F(P,T,V)=0,\\ &\text{热力学能函数：}\quad u=u(T,v),\\ &\text{焓函数：}\quad h=h(T,P),\\ &\text{熵函数：}\quad s=s(T,v)=s(T,p). \end{aligned} \right.

Furthermore,我们得到状态公理:

在平衡状态下,一个热力系的内部强度状态参数可由n+1个独立的内部强度状态参数来确定.

这是吉布斯相律（Gibbs phase rule）在热力学基本概念中的体现，用于描述系统状态所需的独立变量数。实际系统中，n 取决于具体功的形式 (在简单可压缩系中,n=1 仅pdV)

要判断dz是否为状态参数,需满足全微分条件,i.e.

dz=(∂z∂x)ydx+(∂z∂y)xdy=M dx+N dydz = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy = M\,dx + N\,dy (∂M∂y)x=(∂N∂x)y\left( \frac{\partial M}{\partial y} \right)_x = \left( \frac{\partial N}{\partial x} \right)_y

基本热力学关系推导

  • 闭系能量平衡方程式(热一律):

du=δq−δwdu = \delta q - \delta w

对于简单可压缩工质,做功量为δw=pdV\delta w=pdV

对于可逆过程,由热力学第二定律,吸热量δq=Tds\delta q = Tds

  • 对于简单可压缩工质,其微分形式能量平衡方程为:

du=Tds−pdvdu = Tds -pdv

这是导出其它一般关系式的热力学依据,称为基本热力学关系

  • 组合状态参数,得到焓的定义:

h=u+pvh = u + pv

根据全微分公式,可以推导得到:

dh=Tds+vdpdh = Tds + vdp

进一步的,可以用不同的热力学参数表达基本热力学关系:

Legendre变换Legendre变换

{(∂u∂s)v=(∂h∂s)p=T−(∂u∂v)s=−(∂f∂v)T=P(∂h∂p)s=(∂g∂p)T=v−(∂f∂T)v=−(∂g∂T)p=s\left\{ \begin{aligned} \left( \dfrac{\partial u}{\partial s} \right)_v &= \left( \dfrac{\partial h}{\partial s} \right)_p = T \\[8pt] -\left( \dfrac{\partial u}{\partial v} \right)_s &= -\left( \dfrac{\partial f}{\partial v} \right)_T = P \\[8pt] \left( \dfrac{\partial h}{\partial p} \right)_s &= \left( \dfrac{\partial g}{\partial p} \right)_T = v \\[8pt] -\left( \dfrac{\partial f}{\partial T} \right)_v &= -\left( \dfrac{\partial g}{\partial T} \right)_p = s \end{aligned} \right.

故du可以表示为:

du=(∂u∂s)vds+(∂u∂v)sdvdu = \textcolor{teal}{\left( \dfrac{\partial u}{\partial s} \right)_v} ds + \textcolor{teal}{\left( \dfrac{\partial u}{\partial v} \right)_s} dv

又二元函数的二阶混合微商与求导顺序无关,可以得到(下标恒温/容/压):

(∂s∂v)T=(∂p∂T)v\left( \dfrac{\partial s}{\partial v} \right)_T = \left( \dfrac{\partial p}{\partial T} \right)_v (∂s∂p)T=−(∂v∂T)p\left( \dfrac{\partial s}{\partial p} \right)_T = -\left( \dfrac{\partial v}{\partial T} \right)_p

此即麦克斯韦关系式,将不可测熵s的偏微商与可测的基本状态参数p,v 和T(即状态方程)的偏微商相关联

吉布斯方程式

{du=Tds−pdvdh=Tds+vdp\begin{cases} du = Tds -pdv \\ dh = Tds + vdp \end{cases}

吉布斯自由能:

f≡u−Tsf\equiv u-Ts

df=−sdT−pdvdf = -sdT -pdv

吉布斯自由焓:

g≡h−Tsg \equiv h -Ts

dg=−sdT+vdpdg = -sdT + vdp

比热容

定容比热:

cv≡(∂u∂T)vcv=(δqdT)vc_v \equiv \left( \dfrac{\partial u}{\partial T} \right)_v \qquad c_v = \left( \dfrac{\delta q}{dT} \right)_v

定压比热:

cp≡(∂h∂T)pδq=dh∣p  cp=(δqdT)pc_p \equiv \left( \dfrac{\partial h}{\partial T} \right)_p \qquad \delta q = dh \bigg|_p \;\qquad c_p = \left( \dfrac{\delta q}{dT} \right)_p

比热容之间存在如下关系:

T(∂s∂T)v=cv,比热比 γ=cpcvT(∂s∂T)p=cp−T(∂v∂T)p(∂P∂T)v,cp−cv=Tvα2/KTcp−cv=T(∂v∂T)p(∂P∂T)v\begin{aligned} & T \left( \frac{\partial s}{\partial T} \right)_v = c_v, && \text{比热比 } \quad \gamma = \frac{c_p}{c_v} \\ & T \left( \frac{\partial s}{\partial T} \right)_p = c_p - T \left( \frac{\partial v}{\partial T} \right)_p \left( \frac{\partial P}{\partial T} \right)_v, && c_p - c_v = T v \alpha^2 / K_T \\ & c_p - c_v = T \left( \frac{\partial v}{\partial T} \right)_p \left( \frac{\partial P}{\partial T} \right)_v \end{aligned}

定义以下特殊偏微商为热系数:

{αv=1ν(∂ν∂T)p体积膨胀系数KT=−1ν(∂ν∂p)T 定温压缩率 (定温压缩系数)β=1p(∂p∂T)v压力温度系数Ks=−1ν(∂ν∂p)s等熵压缩率 (绝热压缩系数)μJ=(∂T∂p)h 绝热节流系数\left\{ \begin{array}{l l} \alpha_v = \dfrac{1}{\nu} \left( \dfrac{\partial \nu}{\partial T} \right)_p & \text{体积膨胀系数} \\[10pt] K_T = -\dfrac{1}{\nu} \left( \dfrac{\partial \nu}{\partial p} \right)_T & \text{\color{teal} 定温压缩率 (定温压缩系数)} \\[10pt] \beta = \dfrac{1}{p} \left( \dfrac{\partial p}{\partial T} \right)_v & \text{压力温度系数} \\[10pt] K_s = -\dfrac{1}{\nu} \left( \dfrac{\partial \nu}{\partial p} \right)_s & \text{等熵压缩率 (绝热压缩系数)} \\[10pt] \mu_J = \left( \dfrac{\partial T}{\partial p} \right)_h & \text{\color{teal} 绝热节流系数} \end{array} \right.

不难发现:

cp−cv>0c_p-c_v > 0

在引入比热容后,可以简化各个状态参数的微分式

状态参数的微分式

热力学能的微分式

u(T,v)u(T, v)

du=(∂u∂T)vdT+(∂u∂v)Tdvdu = \left( \frac{\partial u}{\partial T} \right)_v dT + \left( \frac{\partial u}{\partial v} \right)_T dv

Tds=du+pdvTds = du + pdv

(∂u∂v)T=T(∂s∂v)T−p\left( \frac{\partial u}{\partial v} \right)_T = T \left( \frac{\partial s}{\partial v} \right)_T - p

(∂s∂v)T=(∂p∂T)v\left( \frac{\partial s}{\partial v} \right)_T = \left( \frac{\partial p}{\partial T} \right)_v

du=cvdT+[T(∂p∂T)v−p]dvdu = c_v dT + \left[ T \left( \frac{\partial p}{\partial T} \right)_v - p \right] dv

焓的微分式

h(T,p)h(T, p)

dh=(∂h∂T)pdT+(∂h∂p)Tdpdh = \left( \frac{\partial h}{\partial T} \right)_p dT + \left( \frac{\partial h}{\partial p} \right)_T dp

dh=Tds+vdpdh = T ds + v dp

(∂h∂p)T=−T(∂v∂T)p+v\left( \frac{\partial h}{\partial p} \right)_T = -T \left( \frac{\partial v}{\partial T} \right)_p + v

−(∂v∂T)p=(∂s∂p)T-\left( \frac{\partial v}{\partial T} \right)_p = \left( \frac{\partial s}{\partial p} \right)_T

dh=cpdT−[T(∂v∂T)p−v]dpdh = c_p dT - \left[ T \left( \frac{\partial v}{\partial T} \right)_p - v \right] dp

熵的微分式

ds=(∂s∂T)vdT+(∂s∂v)Tdvds = \left( \frac{\partial s}{\partial T} \right)_v dT + \left( \frac{\partial s}{\partial v} \right)_T dv

=(∂s∂T)pdT+(∂s∂P)TdP= \left( \frac{\partial s}{\partial T} \right)_p dT + \left( \frac{\partial s}{\partial P} \right)_T dP

(∂s∂T)v=(∂u∂T)v(∂u∂s)v=cvT\left( \frac{\partial s}{\partial T} \right)_v = \left( \frac{\partial u}{\partial T} \right)_v \left( \frac{\partial u}{\partial s} \right)_v = \frac{c_v}{T}

(∂s∂T)p=(∂h∂T)p(∂h∂s)p=cpT\left( \frac{\partial s}{\partial T} \right)_p = \left( \frac{\partial h}{\partial T} \right)_p \left( \frac{\partial h}{\partial s} \right)_p = \frac{c_p}{T}

(∂s∂v)T=(∂P∂T)v\left( \frac{\partial s}{\partial v} \right)_T = \left( \frac{\partial P}{\partial T} \right)_v

(∂s∂P)T=−(∂v∂T)P\left( \frac{\partial s}{\partial P} \right)_T = -\left( \frac{\partial v}{\partial T} \right)_P

ds=cvTdT+(∂p∂T)vdvds = \frac{c_v}{T} dT + \left( \frac{\partial p}{\partial T} \right)_v dv

ds=cpTdT−(∂v∂T)pdpds = \frac{c_p}{T} dT - \left( \frac{\partial v}{\partial T} \right)_p dp

(∂s∂p)v=cvT(∂T∂p)v\left( \frac{\partial s}{\partial p} \right)_v = \frac{c_v}{T} \left( \frac{\partial T}{\partial p} \right)_v

(∂s∂v)p=cpT(∂T∂v)p\left( \frac{\partial s}{\partial v} \right)_p = \frac{c_p}{T} \left( \frac{\partial T}{\partial v} \right)_p

s(p,v)s(p, v)

ds=cvT(∂T∂p)vdp+cpT(∂T∂v)pdvds = \frac{c_v}{T} \left( \frac{\partial T}{\partial p} \right)_v dp + \frac{c_p}{T} \left( \frac{\partial T}{\partial v} \right)_p dv