Janus van der Waals equations for real molecules with two-sided phase transitions

We obtain families of generalised van der Waals equations characterised by an even number $n=2,4,6$ and a continuous free parameter which is tunable for a critical compressibility factor. Each equation features two adjacent critical points which have a common critical temperature $T_{c}$ and arbitrarily close two critical densities. The critical phase transitions are naturally two-sided: the critical exponents are $\alpha_{\scriptscriptstyle{P}}=\gamma_{\scriptscriptstyle{P}}=\frac{2}{3}$, $\beta_{\scriptscriptstyle{P}}=\delta^{-1}=\frac{1}{3}$ for $T>T_{c}$ and $\alpha_{\scriptscriptstyle{P}}=\gamma_{\scriptscriptstyle{P}}=\frac{n}{n+1}$, $\beta_{\scriptscriptstyle{P}}=\delta^{-1}=\frac{1}{n+1}$ for $T<T_{c}$. In contrast with the original van der Waals equation, our novel equations all reduce consistently to the classical ideal gas law in low density limit. We test our formulas against NIST data for eleven major molecules and show agreements better than the original van der Waals equation, not only near to the critical points but also in low density regions.