First order equations Thermodynamic equations

- general system

- particular systems

just internal energy version of fundamental equation, chain rule can used on above equations find k+2 equations of state respect particular potential. if Φ thermodynamic potential, fundamental equation may expressed as:

d
Φ
=

∑

i





∂
Φ


∂

x

i





d

x

i




{\displaystyle d\phi =\sum _{i}{\frac {\partial \phi }{\partial x_{i}}}dx_{i}}

where




x

i




{\displaystyle x_{i}}

natural variables of potential. if




γ

i




{\displaystyle \gamma _{i}}

conjugate




x

i




{\displaystyle x_{i}}

have equations of state potential, 1 each set of conjugate variables.

γ

i


=



∂
Φ


∂

x

i







{\displaystyle \gamma _{i}={\frac {\partial \phi }{\partial x_{i}}}}

only 1 equation of state not sufficient reconstitute fundamental equation. equations of state needed characterize thermodynamic system. note commonly called equation of state mechanical equation of state involving helmholtz potential , volume:

(



∂
f


∂
v



)


t
,
{

n

i


}


=
−
p


{\displaystyle \left({\frac {\partial f}{\partial v}}\right)_{t,\{n_{i}\}}=-p}

for ideal gas, becomes familiar pv=nkbt.

euler integrals

because of natural variables of internal energy u extensive quantities, follows euler s homogeneous function theorem that

u
=
t
s
−
p
v
+

∑

i



μ

i



n

i





{\displaystyle u=ts-pv+\sum _{i}\mu _{i}n_{i}\,}

substituting expressions other main potentials have following expressions thermodynamic potentials:

f
=
−
p
v
+

∑

i



μ

i



n

i





{\displaystyle f=-pv+\sum _{i}\mu _{i}n_{i}\,}








h
=
t
s
+

∑

i



μ

i



n

i





{\displaystyle h=ts+\sum _{i}\mu _{i}n_{i}\,}








g
=

∑

i



μ

i



n

i





{\displaystyle g=\sum _{i}\mu _{i}n_{i}\,}

note euler integrals referred fundamental equations.

gibbs–duhem relationship

differentiating euler equation internal energy , combining fundamental equation internal energy, follows that:

0
=
s
d
t
−
v
d
p
+

∑

i



n

i


d

μ

i





{\displaystyle 0=sdt-vdp+\sum _{i}n_{i}d\mu _{i}\,}

which known gibbs-duhem relationship. gibbs-duhem relationship among intensive parameters of system. follows simple system r components, there r+1 independent parameters, or degrees of freedom. example, simple system single component have 2 degrees of freedom, , may specified 2 parameters, such pressure , volume example. law named after willard gibbs , pierre duhem.