How Does the Dieterici Equation of State Compare to Reality?

[Aeon]

[SOLVED]Dieterici equation of state

Homework Statement

Establish the relationship between the critical constants and the parameters of Dieterici equation of state.
Show that $$Z_c$$$$=2e^{-2}$$ and deduce the reduced Dieterici equation of state.
Compare the predicted $$Z_c$$ from both Dieterici's and Van der Waals' equations. Which best approximates reality?

Homework Equations

http://img299.imageshack.us/i/dieterici.png/

The Attempt at a Solution

I have substituted p with $$p_c$$ and then tried to work my way around proving that the first and second partial derivatives relative to p evaluate to 0.

I have failed.

I know that at the critical point, any pair of first and second derivatives will evaluate to 0, since the critical point is an inflexion point. But other than that... I'm pretty much powerless.

Therefore, my question is: would someone please direct me so that I can establish the relationship between the critical constants ($$p_c, V_c, T_c$$) and the parameters (a,b) of the Dieterici equation of state please?

Thank you.

 

[Aeon]

I got it.

The Dieterici equation of state can be differentiated as it is, but it's more convenient to change its form before differentiating it. To help with differentiation, I used the natural logarithm of both sides.

I differentiated the log() form of the equation, twice. At the critical point (point of inflexion), $$\frac{\partial ln(p)}{\partial V_m} = 0$$ and $$\frac{\partial^2 ln(p)}{\partial V_m^2} = 0$$. You can then divide the first partial derivative by the second and everything clears up. Simple transformations then enable you to isolate $$V_c$$ (not $$V_m$$ since you're evaluating at the critical point).