Analytical Representations of the Phase Equilibrium Curve of the Water–Vapor System

Abstract

A brief critical analysis of the theoretical P(T) liquid–vapor phase equilibrium curves is given. Based on the Clapeyron–Clausius equation and data on the heat capacity of water, a two-parameter expression of the P(T) curve is obtained for water in the temperature range from the triple point to 389 K. The parameters of the P(T) curve are determined from data on the pressure and heat of evaporation of water at the triple point. The accuracy of the P(T) function is comparable to the actual accuracy of standard thermodynamic tables for water and water vapor.

APPENDIX

To determine the dependence of the condensation heat Q on temperature, we turn to the phase diagram of the water–vapor system on the TV plane (Fig. 1).

Fig. 1.

Phase diagram: (I) liquid, (II) vapor, (III) two-phase region, and (C) critical point.

Let us consider the change in the enthalpy ΔH of water and vapor when moving clockwise along a closed loop consisting of four lines: (1) interfaces of liquid with a two-phase mixture, (2) evaporation isotherms at T₂, (3) interfaces of gas (dry vapor) with a two-phase mixture, and (4) condensation isotherms at T₁. On the PT plane, curves 1 and 2 are projected onto the P(T) curve, and curves 2 and 4 are projected into points.

In further calculations, we use the general integral expression for the increment of enthalpy ΔH at the transition from the point (P₁, T₁) to the current point (P, T) in the single-phase region

$$\begin{gathered} \Delta H = H(P,T) - H({{P}_{1}},{{T}_{1}}) \\ = \int\limits_{{{P}_{1}}}^P {[V - T{{{({{\partial V} \mathord{\left/ {\vphantom {{\partial V} {\partial T}}} \right. \kern-0em} {\partial T}})}}_{P}}]dP + \int\limits_{{{T}_{1}}}^T {{{C}_{P}}dT} } . \\ \end{gathered} $$

((14))

The first integral in Eq. (14) is calculated at T₁, and the second is calculated at pressure P.

In accordance with the above numbering of the four lines, we denote the change in enthalpy during passage from the beginning to the end of the curve as ΔH₁, ΔH₂, etc.

The first and second integrals in Eq. (14) are calculated, respectively, along the vertical and horizontal straight lines on the PT plane (Fig. 2).

Fig. 2.

Integration diagram: (1) phase equilibrium curve; (2 and 3) directions of integration.

Calculation of ΔH ₁

It is easy to verify that the first integral is much smaller than the second. The volume of the liquid and, in general, the integrand function of the first integral remain almost unchanged when the pressure changes by an amount on the order of 1 kg/cm². By the order of magnitude, the first integral is 18 cm³/mol × 0.7 kg/cm² ≈ 13 kg cm/mol = 1.3 J/mol at P = 0.7 kg/cm². The heat capacity of water in the considered temperature range is 18 cal/(mol deg) = 9R, and the second integral at the same point on the P(T) curve is 18 × 90 cal/mol = 6772 J/mol.

In the range T₁ ≤ T ≤ T₁ + 90 K, ΔH₁ = 9R(T – T₁) with an accuracy up to a small value of the first integral.

Calculation of ΔH ₃

Over curve 3 (ideal gas), the first integral in Eq. (14) is zero. Up to the contribution of the small heat capacity of intramolecular oscillations, C_P = 4R and

$$\Delta {{H}_{3}} = \int\limits_T^{{{T}_{1}}} {{{C}_{P}}dT} = - 4R\left( {T - {{T}_{1}}} \right).$$

During evaporation along curve 2 and condensation along curve 4, we obtain, respectively, ΔH₂ = Q(T), ΔH₄ = –Q(T₁), where Q(T) and Q(T₁) are the molar condensation heat at T and T₁, respectively.

When circuiting along a closed loop, $\sum\nolimits_1^4 {\Delta {{H}_{i}} = 0} .$

From the expressions for ΔH₁–ΔH₄, we obtain

$$Q(T) = {{Q}_{1}}-5R(T-{{T}_{1}}).$$

((15))

Up to notation, Eq. (15) is identical to Eqs. (9) and (10).