Inverting the Birch-Murnaghan Equation of State

[Polyamorph]

TL;DR

Efficient method to compute V at given P for a non-invertible equation of state

Hello,

https://www.physicsforums.com/lib/Eqn026.pngI have the pressure ($P$)-volume ($V$) Birch-Murnaghan equation of state coeffcients $(V_{0},K_{0}, K^{'}_{0}, K^{''}_{0})$ for a number of different compositions. I'm interested in the volume at very specific pressures only and ideally I would like to compute $V$ for each composition at a given $P$. My problem is the Birch-Murnaghan equation of state cannot be inverted and yields pressure for a given volume - the opposite of what I need:
$$P=3K_{0}f_{E}(1+2f_{E})^{5/2}\left(1+\frac{3}{2}(K^{'}_{0}-4)f_{E}+\frac{3}{2}\left(K_{0}K^{''}_{0}+(K^{'}_{0}-4)(K^{'}_{0}-3)+\frac{35}{9}\right)f^{2}_{E}\right)$$
where $f_{E}=\left[(V_{0}/V)^{2/3}-1\right]/2.$

So my question is, what would be the best approach to solve $V$ as a function of $P$?

I was going to calculate $P$ versus $V$, then interpolate the result over a specific $P$ range. But I have a large number of compositions (~100 or so) all with different volumes and equation of state coefficients so this seems quite inefficient, particularly as the range of $V$ to compute $P$ will vary depending on composition

What would be an efficient approach to find $V$ for a given $P$? Am I forgetting something quite trivial?

Thanks for your help!

 

[Charles Link]

I'm puzzled by the first term that contains $f_E$. ($3K_o f_E$). If $V \approx V_o$, then $f_E \approx 0$. Does this mean that $P \approx 0$? $\\$

 

[Polyamorph]

Does this mean that $P \approx 0$? $\\$

Thanks for your reply. Yes at V=V0 the pressure is zero (ambient pressure).

 

[Charles Link]

What that means is the equation really reads $P=P_o+$ "what you have as $P$". $\\$ You may try doing a first order solution of $f_E$ in terms of $P-P_o$, neglecting the $f_E^2$ and higher terms. It wouldn't be the perfect solution, but it would be a start. $\\$ Alternatively, try something like $f_E=A(P-P_o)+B(P-P_o)^2+...$, and see if you can determine $A$ and $B$. ($A$ and $B$ would come from Taylor series type calculations).

 

[David Lambert]

You are "interested in the volume at very specific pressures only". So use the Newton–Raphson method.

 

[Charles Link]

A follow-up to post 4: If $x=At+Bt^2$, if my algebra is correct, $t=\frac{x}{A}-\frac{Bx^2}{A^3}$, neglecting 3rd order and higher terms.

 

[Polyamorph]

You are "interested in the volume at very specific pressures only". So use the Newton–Raphson method.

I've done it this way, it works! Thanks to both of you for your help.