X-ray studies of the equilibrium properties of argon near the liquid-vapor critical point

X-ray absorption and small-angle scattering have been used to study the equilibrium equation of state of argon in the neighborhood of the liquid-vapor critical point. The form of the liquid-vapor coexistence curve was determined from the absorption measurements. The quantity ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$, where $\rho$ is the number of molecules per unit volume, ${\rho }_c$ is the value of $\rho$ at the critical point, $P_c$ is the critical pressure, and $K_T$ is the isothermal compressibility, was obtained by extrapolating the scattering curves to obtain the zero-angle scattered intensity. The long-range correlation length $\xi$, which characterizes the extent of the long-range density fluctuations which occur when a fluid is near its critical point, was computed from the angular dependence of the scattered intensity. In the analysis of the scattering data, the consistency of the results obtained by different methods of analysis was used to determine the value of the critical exponent $\eta$, which is a measure of the deviation of the scattered intensity predicted by the Ornstein-Zernike theory. This analysis indicated that $\eta =0.10\mathrm{}\pm 0.05$. The temperature dependence of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ and $\xi$ was studied, both at the critical density above the critical temperature $T_c$ and also in the liquid and vapor phases on the coexistence curve below $T_c$. Along the coexistence curve, the temperature dependence of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ has the liquid-vapor asymmetry predicted by several authors. The deviation of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ from the form given by the scaling laws was found to be much greater in the liquid phase than in the vapor. The angular dependence of the scattered intensity was different above and below $T_c$, with the Fisher-Burford equation giving a good description of the scattering at the critical density above $T_c$ and with the Tarko-Fisher equation applying on the coexistence curve below $T_c$. Several critical exponents were evaluated, including $\gamma$ and ${\gamma }^{\prime }$, which give the temperature dependence of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ above $T_c$ at the critical density and below $T_c$ on the coexistence curve, respectively; $\nu$ and ${\nu }^{\prime }$, which describe the temperature dependence of $\xi$ at the critical density above $T_c$ and on the coexistence curve below $T_c$, respectively; and $\beta$, which specifies the temperature variation of the densities of the two coexisting phases below $T_c$. The short-range correlation length $R={\left(kT\rho K_T\right)}^{-\frac{1}{(2-\eta )}}\xi$, where $k$ is Boltzmann's constant, was calculated from the x-ray data, both above and below $T_c$.