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How/(Can?)Derive Cp or Cv from Equation of State 1
- Thread starter MontyHall
- Start date
Montemayor
Chemical
Monty:
Sorry. It is not easy to calculate the heat capacity of a molecule.
Heat capacity is the derivative of energy with respect to temperature:
(dE/dT)v = Cv
so that, by expressing the total energy as an algebraic sum, it is then possible to calculate heat capacities. The total energy which is temperature-dependent is divided into translational, rotational (external and internal), vibrational, electronic, and nuclear.
Quantum and statistical mechanics allow a mathematical formulation of the problem, but detailed knowledge of the structure and spectra is necessary to obtain numerical values.
For engineering work, it is usually impossible to obtain readily all the desired information such as the bond vibration frequencies from spectra, the molecular moments of inertia, and the steric blocking effects to internal rotation. Consequently, averaging techniques have been adopted and empirical structural correlations are employed.
Sorry. It is not easy to calculate the heat capacity of a molecule.
Heat capacity is the derivative of energy with respect to temperature:
(dE/dT)v = Cv
so that, by expressing the total energy as an algebraic sum, it is then possible to calculate heat capacities. The total energy which is temperature-dependent is divided into translational, rotational (external and internal), vibrational, electronic, and nuclear.
Quantum and statistical mechanics allow a mathematical formulation of the problem, but detailed knowledge of the structure and spectra is necessary to obtain numerical values.
For engineering work, it is usually impossible to obtain readily all the desired information such as the bond vibration frequencies from spectra, the molecular moments of inertia, and the steric blocking effects to internal rotation. Consequently, averaging techniques have been adopted and empirical structural correlations are employed.
This is a Montemayor also indicate fundamental thermodynamics (when I say fundamental i DONT mean easy).
I would recommend that you read a good textbook on the subject. The book i used while at univercity was: "Introduction to chemical engineering thermodynamics" by J. M. Smith and H.C. Van Ness. I covers this subject.
I would then like to point out section 4.1 (in the 4. ed) where the following formula for Cp (ideal gas) is given:
Cp,ig/R=A+BT+CT^2+DT^(-2)
The book then lists A,B,C,D for a number of substances.
To get from Cp to Cv use the relationship that
Cv,ig/R=Cp,ig/R-1
As you can see this is actually only valid for "ideal gasses". The error compared to "non ideal gasses" is however usually acceptable at pressure that is not "too high".
Best regards
Morten
I would recommend that you read a good textbook on the subject. The book i used while at univercity was: "Introduction to chemical engineering thermodynamics" by J. M. Smith and H.C. Van Ness. I covers this subject.
I would then like to point out section 4.1 (in the 4. ed) where the following formula for Cp (ideal gas) is given:
Cp,ig/R=A+BT+CT^2+DT^(-2)
The book then lists A,B,C,D for a number of substances.
To get from Cp to Cv use the relationship that
Cv,ig/R=Cp,ig/R-1
As you can see this is actually only valid for "ideal gasses". The error compared to "non ideal gasses" is however usually acceptable at pressure that is not "too high".
Best regards
Morten
DickRussell
Chemical
Tell us what you envision using the calculated Cp and Cv to do, and we can give more advice. Is it for doing compressor calculations? Control valves? Keep in mind that for many gases the Cp/Cv ratio changes considerably over a range of temperatures and pressures.
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1
- #5
Then there is the Mayer formula for the difference:
[α] being the volume expansivity = (1/V) ([∂] V/[∂] T)p
[κ], the isothermal compressibility = -(1/V) ([∂] V/[∂] P)T
The partial derivatives can be obtained from "applicable" EOS equations. The most common being the ideal gas, van der Waals, Redlich-Kwong, and, of course, virial equations.
When using the ideal gas EOS, PV=RT, one gets Cp-Cv = R.
Cp-Cv = [α]2T.V/[κ]
[α] being the volume expansivity = (1/V) ([∂] V/[∂] T)p
[κ], the isothermal compressibility = -(1/V) ([∂] V/[∂] P)T
The partial derivatives can be obtained from "applicable" EOS equations. The most common being the ideal gas, van der Waals, Redlich-Kwong, and, of course, virial equations.
When using the ideal gas EOS, PV=RT, one gets Cp-Cv = R.
- Thread starter
- #6
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. "25362 (Chemical)" 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). 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 Montemayor's post and looking @ Mathworld's article on heat capacities, there are many flavors of heat capacity modelling making it seem easier to look for/create a curve fit (if it works well enough for an application) than find an analytical model.
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. "25362 (Chemical)" 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). 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 Montemayor's post and looking @ Mathworld's article on heat capacities, there are many flavors of heat capacity modelling making it seem easier to look for/create a curve fit (if it works well enough for an application) than find an analytical model.
Montemayor
Chemical
Monty:
You Got IT!
Good rationalizing + common sense.
You Got IT!
Good rationalizing + common sense.
sailoday28
Mechanical
25362 answered the question and deserves the star.
Some time when I can understand how to input equations the below will not be frustrating to read.
However, a basic approach to deriving Cp and Cv from an EOS is:
Given an equation of state with
tau= 1/absolute temp=1/T
Cpo spec heat at const pressure and low pressure where pv=RT
Cvo specif heat at const vol at low pressure, where pv=RT
Change in enthalpy =integral Cpo dT {limits of ref temp to T
+integral partial of (v*tau) WRT tau at constant pressure *dP integral evaluated with temp, T
and limits of pressure from 0 to P
NEXT take derivative of enthalpy WRT to T at const pressure to obtain Cp
Similar apprach us used to obtain change in internal energy and Cv
Some time when I can understand how to input equations the below will not be frustrating to read.
However, a basic approach to deriving Cp and Cv from an EOS is:
Given an equation of state with
tau= 1/absolute temp=1/T
Cpo spec heat at const pressure and low pressure where pv=RT
Cvo specif heat at const vol at low pressure, where pv=RT
Change in enthalpy =integral Cpo dT {limits of ref temp to T
+integral partial of (v*tau) WRT tau at constant pressure *dP integral evaluated with temp, T
and limits of pressure from 0 to P
NEXT take derivative of enthalpy WRT to T at const pressure to obtain Cp
Similar apprach us used to obtain change in internal energy and Cv
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