Inversion Curve for a gas obeying Dieterici's equation of state

[ncholland]

Homework Statement

For a gas obeying Dieterici's equation of state:

P(V-b) = RTexp(-a/RTV)

for one mole, prove that the equation of the inversion curve is

P = ((2a/b^2) - (RT/b)) * exp((1/2) - (a/(RTb)))

and hence find the maximum inversion temperature.

Homework Equations

N/A

The Attempt at a Solution

So, I know that for the inversion curve, the condition is (dV/dT) = V/T (where the derivative is evaluated at constant pressure). But this would need implicit differentiation to find dV/dt ... and it seems completely intractable - is there something I'm missing?

 

[haruspex]

Do the partial differentiation.
To arrive at the target equation, you need to eliminate ∂V/∂T (for which you have an equation) and V (for which you have the original equation).
If you're still stuck, please post your working.

 

[ncholland]

Hi, thanks for the help!

I've tried the partial differentiation, but when I try and eliminate V and ∂V/∂T between the equation I obtain and the equation of state I just get a horrible mess - I'm not sure if I'm doing the differentiation right...

I rearranged the equation to get:

$\frac{-a}{RTV}$ = ln(p(V-b)) - ln(RT)

So, differentiating wrt T:

$\frac{a}{RT^{2}V}$ + $\frac{a}{RTV^{2}}$*$\frac{∂V}{∂T}$ = $\frac{1}{V-b}$*$\frac{∂V}{∂T}$ - $\frac{1}{T}$

There's not really any point me posting any of the further work / manipulation I've done - I've tried a load of different things and nothing gets anywhere...

Is the initial differentiation correct?

Cheers!

 

[haruspex]

$\frac{a}{RT^{2}V}$ + $\frac{a}{RTV^{2}}$*$\frac{∂V}{∂T}$ = $\frac{1}{V-b}$*$\frac{∂V}{∂T}$ - $\frac{1}{T}$

Yes, that looks good. Substitute for ∂V/∂T and get it into the form V = ...

 

[ncholland]

Awesome, thanks! Got it now, much less nasty than I'd thought :-) (I was just messing up the cancellation of terms - which made me doubt I'd got the differentiation right in the first place because the whole thing looked such a mess!)

 

[ncholland]

And the maximum inversion temperature is just found by setting P = 0, right? So that gives T = 2a/bR ?

 

[haruspex]

And the maximum inversion temperature is just found by setting P = 0, right?

If you say so. I know nothing about this subject matter.