Thermodynamics: DT/DV at constant entropy? (last maxwell relation I haven't figured)


by tsuwal
Tags: maxwell, thermodynamics
tsuwal
tsuwal is offline
#1
Jan22-13, 10:31 AM
tsuwal's Avatar
P: 103
So, until now I know:
(DV/DS)p=(DT/Dp)s=a*T/cp*(rho) (enthalpy)
(Dp/DT)v=(DS/DV)t=-a/k (helmoltz)
(DS/Dp)t=-(DV/DT)p=-Va (gibbs)

a=expansion coefficient
k=isothermal compression coefficent
cp=heat capacity at constante pressure

I want to deduce DT/DV at constant entropy=(DT/DV)s. BUT HOW?
Let me try to write S(T,V), then,
dS=Cv/T*dT-a/k*dV
putting S=0, i get,
a/k*dV=Cv/T*dT <=> (DT/DV)s=a*T/Cv*k

am I right?
Phys.Org News Partner Physics news on Phys.org
Physicists design quantum switches which can be activated by single photons
'Dressed' laser aimed at clouds may be key to inducing rain, lightning
Higher-order nonlinear optical processes observed using the SACLA X-ray free-electron laser
Studiot
Studiot is offline
#2
Jan22-13, 12:35 PM
P: 5,462
Is this the one you want?

[tex]\begin{array}{l}
T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_V} \\
{\left( {\frac{{\partial T}}{{\partial V}}} \right)_S} = \left[ {\frac{\partial }{{\partial V}}{{\left( {\frac{{\partial U}}{{\partial S}}} \right)}_V}} \right] = \frac{{{\partial ^2}U}}{{\partial V\partial S}} \\
\end{array}[/tex]

and

[tex]\begin{array}{l}
P = - {\left( {\frac{{\partial U}}{{\partial V}}} \right)_S} \\
{\left( {\frac{{\partial P}}{{\partial S}}} \right)_V} = - \left[ {\frac{\partial }{{\partial S}}{{\left( {\frac{{\partial U}}{{\partial S}}} \right)}_V}} \right] = - \frac{{{\partial ^2}U}}{{\partial S\partial V}} \\
\end{array}[/tex]

Therefore

[tex]{\left( {\frac{{\partial T}}{{\partial V}}} \right)_S} = - {\left( {\frac{{\partial P}}{{\partial S}}} \right)_V}[/tex]
tsuwal
tsuwal is offline
#3
Jan22-13, 02:21 PM
tsuwal's Avatar
P: 103
Hey, thanks for worring so much, but until there I knew...
I want to evaluate that derivative further and write in terms of a,k,Cv,Cp,T,p,... as I did
(∂T/∂p)s=a*T/cp*(rho)


Register to reply

Related Discussions
Gibbs Maxwell Relation Advanced Physics Homework 4
About the Maxwell's relation Engineering, Comp Sci, & Technology Homework 0
Maxwell relation with 3 variables? Classical Physics 1
Thermodynamics: Maxwell relation and thermal expansion Advanced Physics Homework 2
+ instead of - for Maxwell Relation? Advanced Physics Homework 4