I have a function, an equation of state named the Birch-Murnaghan's equation of state, in which the Energy ( $E$ ) is a function of the Volume ($V$), where $E_{0}$, $V_{0}$, $B_{0}$ and $B_{0}^{'}$ are parameters that can be obtained from the fit:
\begin{equation} \label{BM-EOS-Energy-Volume} E\left ( V \right )=E_{0}+\frac{9V_{0}B_{0}}{16}\left \{ \left [ \left ( \frac{V_{0}}{V} \right )^\frac{2}{3} -1 \right ]^3 B_{0}^{'}+\left [ \left ( \frac{V_{0}}{V} \right )^{\frac{2}{3}}-1 \right ]^{2}\left [ 6-4\left (\frac{V_0}{V} \right )^{\frac{2}{3}} \right ]\right \}, \tag{1} \end{equation}
The pressure P may be written as a function of the volume V as:
\begin{equation} \label{BM-EOS-Energy-Volume-derivative} P\left ( V \right )=-\left ( \frac{\partial E}{\partial V} \right )_{S}, \tag{2} \end{equation}
so that we can get the expression of $P\left ( V \right )$ : \begin{equation} \label{BM-EOS-Pressure-Volume} P\left ( V \right )=\frac{3B_{0}}{2}\left [ \left ( \frac{V_{0}}{V} \right )^\frac{7}{3} - \left ( \frac{V_{0}}{V} \right )^\frac{5}{3}\right ]\left \{ 1+\frac{3}{4}\left ( B_{0}^{'}-4 \right )\left [ \left ( \frac{V_{0}}{V} \right )^\frac{2}{3}-1 \right ] \right \} \tag{3} \end{equation}
Now, I would like to get an expression of $E(P)$, but I do not find the way to accomplish that goal starting from the expression of $E(V)$ (Equation 1) and $P(V)$ (Equation 3).
I would appreciate if you could help me.
EDIT:
I have data of $E$ vs $V$. After plotting, I have fitted this data using Eq. (1).
I have transformed each $V$ data to $P$ using Eq. (3).
I have now plotted $P$ vs $V$, and fitted using Eq. (3).
Now I have plotted $E$ vs $P$ and I need to fit this data using a function. And this function would have to be the expression of $E(P)$, which I don't know a possible way of working it out.
