Solving an Equation of State Using Compressibility & Expansivity

[Mic :)]

Homework Statement

Obtaining an equation of state from compressibility and expansivity.

States of superheated steam are observed to have an isothermal compressibility k=(rNT) / (VP^2)
and a volume expansivity B=(N/V)((r/P)+(am / T^m+1)).

r,m and a are constants.

a) Find dv in terms of dP and dT
b) Deduce the equation of state for superheated steam up to an undetermined constant

Homework Equations

B= (1/V)(dV/dT) P const

k= (-1/V)(dV/dP) T const

The Attempt at a Solution

a) I have arrived at:

(rTN^2 / P^2) ((r/P)+(am / T^m+1)) = -(dV)^2 / dPdT

I can simplify and rearange that for dV = -sqrt((rTN^2 / P^2) ((r/P)+(am / T^m+1)) dP dTIf a) is correct, then I need help with b); how would I go about integrating the expression.

Do I start with (dV)^2 = ?

Thank you very much!
Any help would be sincerely appreciated (and needed).

 

[BiGyElLoWhAt]

I guess I'm not sure what exactly you're trying to do. Are you looking for 1 equation that relates all 3 differentials? Or are you looking for 2 equations i.e. partials? It appears as though you did some form of substitution to get the 2 equations together, but I'm not seeing what you did. Could you post your work?

 

[BiGyElLoWhAt]

Also, I don't think your notation is quite right:

$\frac{(dV)^2}{dPdT}$ is not the same as $\frac{\partial^2V}{\partial P\partial T}$

 

[Mic :)]

Also, I don't think your notation is quite right:

$\frac{(dV)^2}{dPdT}$ is not the same as $\frac{\partial^2V}{\partial P\partial T}$

Hi!

Are you referring to disparity between d and the other thing, or d^2 V and (dV)^2?

I came by it by multiplying two equations together (dV)(dV)

 

[Mic :)]

I guess I'm not sure what exactly you're trying to do. Are you looking for 1 equation that relates all 3 differentials? Or are you looking for 2 equations i.e. partials? It appears as though you did some form of substitution to get the 2 equations together, but I'm not seeing what you did. Could you post your work?

I did (1/V)(dV/dT) = (N/V)((r/P)+(am / T^m+1))

(-1/V)(dV/dP) =(rNT) / (VP^2)

then muliplied them together.

For a) I a trying to find dV=.

For b), I think that I am trying trying to find V= (or P= or T=)

Thanks!

 

[BiGyElLoWhAt]

Both actually. I am assuming that youre not looking for a partials solution.

Also, all of temperature, pressure, and volume are changing?

$\frac{1}{V}\frac{dV}{dT} = B$ is only true when P is constant, no? If they're changing then that relationship isn't true.
The same for the latter equation for constant T.

Is this for thermo or something?

 

[Mic :)]

Both actually. I am assuming that youre not looking for a partials solution.

Also, all of temperature, pressure, and volume are changing?

$\frac{1}{V}\frac{dV}{dT} = B$ is only true when P is constant, no? If they're changing then that relationship isn't true.
The same for the latter equation for constant T.

Is this for thermo or something?

Tis indeed for thremo

 

[BiGyElLoWhAt]

Also, what made you decide to multiply the equations, rather than adding or subtracting?

 

[Mic :)]

Also, what made you decide to multiply the equations, rather than adding or subtracting?

It looked like the easier way to go about combining the equations (at the time).
Would you suggest the latter?

 

[BiGyElLoWhAt]

Yea I would if that's what you're intending to do. Did you read my post about the legitimacy of the equations? Double check that what you're doing is correct before proceeding (don't want to do a lot of unnecessary work XD)

 

[Mic :)]

Ok. I've come to:

dV = [Nam / T^(m+1)] [1+ TdP / PdT]

Could someone help with the integration?

 

[BiGyElLoWhAt]

That definitely looks more reasonable. (I didn't work through it, so I'm taking what you did as correct)

As far as the integration goes, try distrubiting the first bracketed stuff through to the second
bracketed stuff, get your exponent for T in line, and then it should be fairly simple, as most of what you will have left is constants.

Just to clarify, is this what you have?
$dV = [\frac{N*a*m}{T^{m+1}}][1+\frac{T}{P}\frac{dP}{dT}]$

 

[Mic :)]

Just to clarify, is this what you have?
$dV = [\frac{N*a*m}{T^{m+1}}][1+\frac{T}{P}\frac{dP}{dT}]$[/QUOTE]

Ya

 

[BiGyElLoWhAt]

Actually maybe that won't be as easy to get into total differential form as I thought...

 

[Mic :)]

Actually maybe that won't be as easy to get into total differential form as I thought...

Still doable?

 

[BiGyElLoWhAt]

Haha, that's what I'm working on. It's not jumping out at me like it normally does, but I'm not necessarily a math wizard so that doesn't mean much. With this type of equation I think it should be setup as such:
http://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx
Once you get it into a form like that you can simply integrate each term, but I'm kind of struggling. I'm looking for a clever substitution for that pesky dT right now to at least get it out of the denominator in a useful fashion.

Edit
This function might not be nice enough for that though. Anyone else have any pointers? Otherwise I'm inclined to believe that you made some arithmetic error somewhere :(

 

[Mic :)]

Edit
This function might not be nice enough for that though. Anyone else have any pointers? Otherwise I'm inclined to believe that you made some arithmetic error somewhere :([/QUOTE]

Oh dear, I may have stuffed it up (sorry!)

I'm going through it again.

 

[Mic :)]

There should be a dT in the brackets with the Nam.

Does that help anything?

 

[Mic :)]

Haha, that's what I'm working on. It's not jumping out at me like it normally does, but I'm not necessarily a math wizard so that doesn't mean much. With this type of equation I think it should be setup as such:
http://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx
Once you get it into a form like that you can simply integrate each term, but I'm kind of struggling. I'm looking for a clever substitution for that pesky dT right now to at least get it out of the denominator in a useful fashion.

Edit
This function might not be nice enough for that though. Anyone else have any pointers? Otherwise I'm inclined to believe that you made some arithmetic error somewhere :(

If you ever get around to it, could you most what you get so that I can verify what I have?

Thanks so much for taking the time to help!

 

[BiGyElLoWhAt]

Yea no problem man, I'll try to get to it here soon... I'm trying to fix some stupid computer problems right now...

 

[Chestermiller]

The starting point for the solution of this problem should be as follows: If V = V(P,T), then

$$dV=\left(\frac{\partial V}{\partial T}\right)_PdT+\left(\frac{\partial V}{\partial P}\right)_TdP$$

Chet

 

[Mic :)]

The starting point for the solution of this problem should be as follows: If V = V(P,T), then

$$dV=\left(\frac{\partial V}{\partial T}\right)_PdT+\left(\frac{\partial V}{\partial P}\right)_TdP$$

Chet

Hi!

Are you referring to the integration required in part b ?

Thanks!

 

[Chestermiller]

Hi!

Are you referring to the integration required in part b ?

Thanks!

No, I'm referring to the first part. The second part seems pretty easy to do also. First you integrate one of the relations (either at constant T or at constant P, whichever is appropriate). Then you substitute into the second relation, cancel whatever terms cancel, and integrate the result. I would start out by integrating the k equation at constant T.

Chet

 

[Mic :)]

I've worked my way to this:

ΔV = N*a*[1/T_initial^(m) - 1/T_final^(m) + m*T*ln(p_final/p_initial]

 

[Chestermiller]

I've worked my way to this:

ΔV = N*a*[1/T_initial^(m) - 1/T_final^(m) + m*T*ln(p_final/p_initial]

That's not what I get. Please start by showing me your integration with respect to P (holding T constant) of:
$$\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T=-\frac{NRT}{VP^2}$$

Chet

 

[Mic :)]

That's not what I get. Please start by showing me your integration with respect to P (holding T constant) of:
$$\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T=-\frac{NRT}{VP^2}$$

Chet

NRT / 2P ?

 

[Chestermiller]

From this integration, I get:
$$V=\frac{NRT}{P}+f(T)$$
where f(T) is a function only of T, and represents the "constant of integration" for the partial integration with respect to P. See if this satisfies the equation for the partial of V with respect to P. You now need to substitute this into the other equation for the partial of V with respect to T in order to determine what the function f(T) is. What do you get for f(T)?

 

[BiGyElLoWhAt]

There should be a dT in the brackets with the Nam.

Does that help anything?

actually yes haha, I didn't see this message for some reason...
Distribute that whole expression with that dT in there and see what you get. Compare it to the equations in pauls notes and solve. You should be good assuming you algebra'd/calculus'd correctly.=)