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 (Δφ) of coexisting phases than in terms of the difference in mass density. The difference Δφ is well fitted for ε=(Tc-T)Tc<0.006 by the expression Δφ=Bε^β, where B=1.071±0.023 and β=0.328±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, Δφ can be fitted for ε<0.2 by an extended scaling expression suggested by Wegner's work, ${Δ}{φ}=B{{ε}}^{{β}}+{B}₁{{ε}}^{{β}+{{Δ}}₂}+{B}₂{{ε}}^{{β}+2{{Δ}}₂}$. The exponent ${{Δ}}₂$ is fixed at 0.5; the fit gives ${β}=0.316±{}0.008$, $B=1.63±{}0.09$, ${B}₁=0.77±{}0.31$, and ${B}₂={-}2.43±{}0.40$. 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.