How Does the Dieterici Equation of State Compare to Reality?

AI Thread Summary
The discussion focuses on establishing the relationship between the critical constants and the parameters of the Dieterici equation of state. It highlights the need to differentiate the equation effectively, particularly at the critical point where first and second partial derivatives equal zero. The solution involves using the natural logarithm to facilitate differentiation and isolate the critical volume. A comparison is made between the critical compressibility factor Z_c derived from both the Dieterici and Van der Waals equations to determine which better approximates real behavior. Ultimately, the discussion emphasizes the importance of transforming the equation for clearer analysis at critical conditions.
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[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.
 
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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).
 
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