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# Newbie Thermo Question: How to derive Cp and Cv from equation of state

## Newbie Thermo Question: How to derive Cp and Cv from equation of state - Physics Forum

### Newbie Thermo Question: How to derive Cp and Cv from equation of state - Physics Forum. Discuss and ask physics questions, kinematics and other physics problems.

#1
07-26-2005, 07:20 PM
 Monty Hall Guest Posts: n/a
Newbie Thermo Question: How to derive Cp and Cv from equation of state

I was wondering if anybody would know of any text regarding deriving Cp and
Cv from an EOS.

Monty

#2
07-27-2005, 09:13 PM
Newbie Thermo Question: How to derive Cp and Cv from equation of state

"Monty Hall" <[Only registered users see links. ]> wrote in message
news:H1wFe.488\$[Only registered users see links. ].prodigy.com. ..
and

Any first year text in Physical Chemistry or Thermodynamics will do.

Try this:
[Only registered users see links. ]

You have not specified *which* Equation of State (EOS) you are using, so
"...like a sewer, what you get out of it depends on what you put into it."
(thanks to Tom Lehrer)

Cp and Cv are heat capacities (for constant Pressure P and constant Volume V
respectively), so you will need to be able to differentiate your EOS with
respect to temperature while holding either pressure or volume constant.
[How are you with partial derivatives - second year calculus?]

Thermodynamics teaches us a whole slew of identities which may help us get
from one expression to another, much like solving for trigonometric
identities.

Basically, let q be the heat absorbed by the system. Then Cv is the
derivative of q with respect to T at constant V, and Cp is the derivative of
q with respect to T at constant P.

Of course, if your EOS doesn't describe q explicitly, you will have a little
algebra to do so that you can isolate q.

Tom Davidson
Richmond, VA

#3
07-28-2005, 07:00 PM
 Monty Hall Guest Posts: n/a
Newbie Thermo Question: How to derive Cp and Cv from equation of state

Thanks,

My post came after reading about the thermodynamics of sonic and super sonic
fluid flow (was looking @ my undergrad text from Smith and VanNess) - which
incls. the use of heat capacity ratios for the adiabatic expansion process.

From here, I went on a tangent and was wondering if the heat capacity could
be calculated from an EOS either in terms of deviation from ideal, some
clever partial derivative manipulation, or something/anything else - hence
my post. Mayer's formula for Cp-Cv in terms of PVT differentials is in the
ballpark of what I was looking for. I just remembered as an undergrad -
doing homework sets - being told to use ideal gas behavior and then use some
polynomial curve fit model of heat capacity. I was just wondering if I had
a very accurate EOS would Cp and Cv be obtainable rather than running to
curve fits(if I could find them). Also, I'm under the presumption
explicit/implicitly differentiating an EOS should be straight foward that is
why I never spec'd one in my post. If I could obtain Cp and Cv from an EOS,
it would have been interesting to compare the curve fit to the calculated
model under the curve fit range and then at various temps & pressures. I
wanted to size a supersonic nozzle and was curious of the impact of various
models of heat capacity and EOS's on the sizing results.

After reading Mathworld's article on heat capacities, there are many flavors
of heat capacity modelling making it seem easier/quicker to look for/create
a curve fit (if it works well enough for an application) than find an
analytical model (estimate vibrational energies, line spectra, rotational
inertia etc.). I was under the assumption that while I would not be able to
determine an abolute energy the EOS could at least determine changes ie:Cp &
Cpv, in terms of PVT differentials or some kind of deviation from ideal.

Monty

news:[Only registered users see links. ]...

#4
08-03-2005, 07:44 PM
 W H G Guest Posts: n/a
Newbie Thermo Question: How to derive Cp and Cv from equation of state

"Monty Hall" <[Only registered users see links. ]> wrote in message
news:EX9Ge.1990\$[Only registered users see links. ].prodigy.com. ..
Joseph H.Keenan's Thermodynamics (I have the 1970 reprint by MIT
of the 1941 book) in Chapter 20 covers what you want. He derives
equations for entropy and enthalpy in terms of a p-v-T equation of state,
that is, partial derivatives are required (you supply the EOS and do
the math). These are eq's [217] and [218]. The paragraph at the end of
It is conceivable that a chart might reprosent he p-v-T relation with
fair precision and yet be utterly inadequate for evaluating the derivatives
in [217] and [218]. This is probably ture of most charts that are derived
from generalized p-v-T data.

It appears that most authoities still would curve fit cp(T, p=0) and derive
h and s from that. For example,
h(T,p1) = h(T, 0) + Integral(p=0 to p1)(@cp/@p dp)
where @ is partial derivative.

 Tags derive , equation , newbie , question , state , thermo

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