Coexistence curves at liquid-liquid critical points: Ising exponents and extended scaling

New measurements have been made of coexisting mass densities of isobutyric acid + water with a precision of 20 ppm and within 3.5°C of the critical temperature. The measurements were made using a single sample of composition very close to critical. It is found that the coexistence curve is more symmetric in terms of the difference in volume fraction ($\Delta \phi$) of coexisting phases than in terms of the difference in mass density. The difference $\Delta \phi$ is well fitted for $\epsilon =\frac{(T_c-T)}{T_c}<0.006\mathrm{}$ by the expression $\Delta \phi =B{\epsilon }^{\beta }$, where $B=1.071\mathrm{}\pm 0.023$ and $\beta =0.328\mathrm{}\pm 0.004$. (Uncertainties are given as 3 times the standard deviation.) A new analysis has been made of the recent data of Gopal et al. on the difference in volume fraction of coexisting densities of carbon disulfide + nitromethane. For this system, $\Delta \phi$ can be fitted for $\epsilon <0.2\mathrm{}$ by an extended scaling expression suggested by Wegner's work, $\Delta \phi =B{\epsilon }^{\beta }+B_1{\epsilon }^{\beta +{\Delta }_2}+B_2{\epsilon }^{\beta +2\mathrm{}{\Delta }_2}$. The exponent ${\Delta }_2$ is fixed at 0.5; the fit gives $\beta =0.316\mathrm{}\pm 0.008$, $B=1.63\mathrm{}\pm 0.09$, $B_1=0.77\mathrm{}\pm 0.31$, and $B_2=-2.43\mathrm{}\pm 0.40$. This work suggests that liquid-liquid critical phenomena are consistent with the functional forms obtained from renormalization-group calculations and with asymptotic exponents which are like those presently calculated for the Ising model. The range of asymptotic behavior seems to be larger for a liquid-liquid critical point than for a liquid-gas critical point.