Finding fundamental equation of the ideal VanWaals in Hemoltz

[Kidphysics]

Homework Statement

fundamental equation of the ideal Van de Waals fluid in the Helmholtz and
enthalpy representation.

Homework Equations

H(s,p)= U-TS (1)

dh=-Sdt-PdV (2)

KT=pv+a'/v-b'p-a'b'/v^2 (3)

The Attempt at a Solution

First I thought to use (2) dh=-Sdt-PdV and took the differential dt of (3) where t(p,v) and then I thought to take the total differential dv where v(p,t) and plug into dh and integrate to get H.. this has gotten very messy very quickly and I was wondering if there is a smarter/better way to move forward

 

[Chestermiller]

For the case of enthalpy, I provided a derivation of the relevant relationship in the link:

https://www.physicsforums.com/showthread.php?p=4269223#post4269223

See the very last response that I posted. What you do is establish a reference state in the ideal gas region (low pressure), at T₀ and P₀. You set the enthalpy equal to zero at that reference state. To get the enthalpy at another state, you first integrate with respect to temperature at constant pressure from T₀ to T. In this integral, C_p is a function only of temperature (recall, ideal gas). The next step is to integrate with respect to pressure at constant temperature T, from T, P₀ to T, P. This is how you can calculate the enthaply relative to the reference state if you know the heat capacity in the ideal gas region and the (non-ideal) PVT behavior of the material (from an equation of state).

 

[Kidphysics]

For the case of enthalpy, I provided a derivation of the relevant relationship in the link:

https://www.physicsforums.com/showthread.php?p=4269223#post4269223

See the very last response that I posted. What you do is establish a reference state in the ideal gas region (low pressure), at T₀ and P₀. You set the enthalpy equal to zero at that reference state. To get the enthalpy at another state, you first integrate with respect to temperature at constant pressure from T₀ to T. In this integral, C_p is a function only of temperature (recall, ideal gas). The next step is to integrate with respect to pressure at constant temperature T, from T, P₀ to T, P. This is how you can calculate the enthaply relative to the reference state if you know the heat capacity in the ideal gas region and the (non-ideal) PVT behavior of the material (from an equation of state).

Hmm I see you have written

dH=CP+(V−T(∂V∂T)P)dP

but to find Cp you must find ∂H∂T also.. and if I wish to find dp I will need to take a total differential and I will have dp's and dt's instead. Also any ideas with relation to my hemholtz approach?

 

[Chestermiller]

Hmm I see you have written

dH=CP+(V−T(∂V∂T)P)dP

but to find Cp you must find ∂H∂T also.. and if I wish to find dp I will need to take a total differential and I will have dp's and dt's instead. Also any ideas with relation to my hemholtz approach?

No. You can measure C_p for the ideal gas region in a separate experiment (once-and-for-all) and then use the results to make predictions of the enthalpy change for any temperature change in the ideal gas region. And, if you have already fit PVT data for a specific gas to the Van Der Waals Equation, you can use the relationship I gave to determine the change in enthalpy at constant temperature from a pressure in the ideal gas region to a pressure beyond the ideal gas region. The net result is that, knowing C_p vs T for a gas and the PVT equation for the gas, you can predict in advance the enthalpy for any arbitrary temperature and pressure, and you can predict the enthalpy change between any two equilibrium states, even outside the ideal gas region.

Integrate the pressure term in my equation (from zero pressure to arbitrary pressure) at constant T for a gas that satisfies the Van Der Waals equation.

Yes. I have ideas on how to derive the corresponding relation to the helmholtz free energy. But I already showed you how to do it for the enthalpy. Just do the same kind of thing for the helmholtz free energy. Of course, you will need to identify a different Maxwell relationship to use.

Chet

 

[Kidphysics]

No. You can measure C_p for the ideal gas region in a separate experiment (once-and-for-all) and then use the results to make predictions of the enthalpy change for any temperature change in the ideal gas region. And, if you have already fit PVT data for a specific gas to the Van Der Waals Equation, you can use the relationship I gave to determine the change in enthalpy at constant temperature from a pressure in the ideal gas region to a pressure beyond the ideal gas region. The net result is that, knowing C_p vs T for a gas and the PVT equation for the gas, you can predict in advance the enthalpy for any arbitrary temperature and pressure, and you can predict the enthalpy change between any two equilibrium states, even outside the ideal gas region.

Integrate the pressure term in my equation (from zero pressure to arbitrary pressure) at constant T for a gas that satisfies the Van Der Waals equation.

Yes. I have ideas on how to derive the corresponding relation to the helmholtz free energy. But I already showed you how to do it for the enthalpy. Just do the same kind of thing for the helmholtz free energy. Of course, you will need to identify a different Maxwell relationship to use.

Chet

Thank you for the response. I'm not completely sure if we are on the same page, as I am not trying to model a current experiment- I don't have any means to measure Cp physically and I do not have fit PVT data. I do understand how knowing the thermodynamic potential H is useful for further conversions to A.

 

[Kidphysics]

The thing is, this should be a really simple problem but my math isn't all there..

 

[Chestermiller]

Thank you for the response. I'm not completely sure if we are on the same page, as I am not trying to model a current experiment- I don't have any means to measure Cp physically and I do not have fit PVT data. I do understand how knowing the thermodynamic potential H is useful for further conversions to A.

Where do you think the tables of the thermodynamic potential H as a function of temperature and pressure for a particular gas come from? To generate such a table, they experimentally determine how C_p varies as a function of temperature at low pressures, and they do PVT experiments to fit the Van der Waals (or other EOS) equation parameters. They then use the equation I gave to fill in the entire table. All your teacher is asking is for you to derive the final equation that I gave and then integrate the first term with respect to temperature at constant pressure in the ideal gas region, and the second term with respect to pressure at constant temperature between very low pressure and arbitrary pressure (for the specific case of a gas that is approximated by the Van der Waals equation).

 

[Kidphysics]

Where do you think the tables of the thermodynamic potential H as a function of temperature and pressure for a particular gas come from? To generate such a table, they experimentally determine how C_p varies as a function of temperature at low pressures, and they do PVT experiments to fit the Van der Waals (or other EOS) equation parameters. They then use the equation I gave to fill in the entire table. All your teacher is asking is for you to derive the final equation that I gave and then integrate the first term with respect to temperature at constant pressure in the ideal gas region, and the second term with respect to pressure at constant temperature between very low pressure and arbitrary pressure (for the specific case of a gas that is approximated by the Van der Waals equation).

I see. Your write up was very good and I plan on using it I just need to convert my EOS which is a function of T,p, and V:

KT=pv+a'/v-b'p-a'b'/v^2

into something with just two independent variables (t,p) possibly (?). I don't think I can solve for V here, and taking some derivative wrt V won't rid me of it either. I know I must brake it down into two variables and try to find derivative relationships as you did

 

[Kidphysics]

I need to get a thermodynamic potential out of it I believe

 

[Chestermiller]

I see. Your write up was very good and I plan on using it I just need to convert my EOS which is a function of T,p, and V:

KT=pv+a'/v-b'p-a'b'/v^2

into something with just two independent variables (t,p) possibly (?). I don't think I can solve for V here, and taking some derivative wrt V won't rid me of it either. I know I must brake it down into two variables and try to find derivative relationships as you did

For the integrand of the second integral, you need to first evaluate the partial derivative of v with respect to T at constant p. This is the reciprocal of the partial derivative of T with respect to v at constant p. I think you can easily use the above equation to determine that.

You next multiply the partial derivative of v with respect to T at constant p by the temperature T, and subtract this term from v. This gives you the integrand in terms of p, v, and T. What does this integrand look like (i.e., what is it)? You will have to do some clever mathematics to manipulate this integral into an analytically integrable form.

 

[Kidphysics]

For the integrand of the second integral, you need to first evaluate the partial derivative of v with respect to T at constant p. This is the reciprocal of the partial derivative of T with respect to v at constant p. I think you can easily use the above equation to determine that.

You next multiply the partial derivative of v with respect to T at constant p by the temperature T, and subtract this term from v. This gives you the integrand in terms of p, v, and T. What does this integrand look like (i.e., what is it)? You will have to do some clever mathematics to manipulate this integral into an analytically integrable form.

I see. I guess I was just supposed to use your result this entire time, not use it like a template. I have to start reading Callen my teacher did me a diservice. Thanks for the help CM.