Calculation of Fugacity (Liquids)

To introduce the calculation of fugacity for liquids, consider Fig. 9.5. The shape of an isotherm below the critical temperature differs significantly from an ideal-gas isotherm. Such an isotherm is illustrated which begins in the vapor region at low pressure, intersects the phase boundary where vapor and liquid coexist, and then extends to higher pressure in the liquid region. Point A represents a vapor state, point B represents saturated vapor, point C represents saturated liquid, and point D represents a liquid.

Figure 9.5. Schematic for calculation of Gibbs energy and fugacity changes at constant temperature for a pure liquid.

We showed in Section 9.6 on page 346 that

Note that we have designated the fugacity at points C and B equal to f^sat. This notation signifies a saturation condition, and as such, it does not require a distinction between liquid or vapor. Therefore, we may refer to point B or C as saturated vapor or liquid interchangeably when we discuss fugacity. The calculation of the fugacity at point B (saturated vapor) is also adequate for calculation of the fugacity at point C, the fugacity of saturated liquid. Calculation of the saturation fugacity may be carried out by any of the methods for calculation of vapor fugacities from the above section. Methods differ slightly on how the fugacity is calculated between points C and D. There are two primary methods for calculating this fugacity change. They are the Poynting method and the equation of state method.

Poynting Method

The Poynting method applies Eqn. 9.19 between saturation (points B, C) and point D. The integral is

Since liquids are fairly incompressible for T_r < 0.9, the volume is approximately constant, and may be removed from the integral, with the resultant Poynting correction becoming

 Poynting correction.

The fugacity is then calculated by

 Poynting method for liquids.

Saturated liquid volume can be estimated within a slight percent error using the Rackett equation

The Poynting correction, Eqn. 9.38, is essentially unity for many compounds near room T and P; thus, it is frequently ignored.

 Frequent approximation.

Equation of State Method

Calculation of liquid fugacity by the equation of state method uses Eqn. 9.24 just as for vapor. To apply the Peng-Robinson equation of state, we can use Eqn. 9.33. The only significant consideration is that the liquid compressibility factor must be used. To understand the mathematics of the calculation, consider the isotherm shown in Fig. 9.6(a). When T_r < 1, the equation of state predicts an isotherm with “humps” in the vapor/liquid region. Surprisingly, these swings can encompass a range of negative values of the pressure near C′ (although not shown in our example). The exact values of these negative pressures are not generally taken too seriously, however, because they occur in a region of the P-V diagram that is unimportant for routine calculations. Since the Gibbs energy from an equation of state is given by an integral of the volume with respect to pressure, the quantity of interest is represented by an integral of the humps. The downward and upward humps cancel one another in generating that integral. This observation gives rise to the equal area rule for computing saturation conditions to be discussed in Section 9.10 where we show that the shaded area above line  is equal to the shaded area below, and that the pressure where the line is located represents the saturation condition (vapor pressure). With regard to fugacity calculations, it is sufficient simply to note that these humps are in fact integrable, and easily computed by the same formula derived for the vapor fugacity by an equation of state.

Fig. 9.6(b) shows that the molar Gibbs energy is indeed continuous as the fluid transforms from the vapor to the liquid. The Gibbs energy first increases according to Eqn. 9.14 based on the vapor volume. Note that the volume and pressure changes are both positive, so the Gibbs energy relative to the reference value is monotonically increasing. During the transition from vapor to liquid, the “humps” lead to the triangular region associated with the name of van der Waals loop. Then the liquid behavior takes over and Eqn. 9.14 comes back into play, this time using the liquid volume. Note that the isothermal pressure derivative of the Gibbs energy is not continuous. Can you develop a simple expression for this derivative in terms of P, V, T, C^P, C^V, and their derivatives? Based on your answer to the preceding question, would you expect the change in the derivative to be a big change or a small change?

Figure 9.6. Schematic illustration of the prediction of an isotherm by a cubic equation of state. Compare with Fig. 9.5 on page 350. The figure on the right shows the calculation of Gibbs energy relative to a reference state. The fugacity will have the same qualitative shape.

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Example 9.5. Vapor and liquid fugacities using the virial equation

Determine the fugacity (MPa) for acetylene at (a) 250 K and 10 bar, and (b) 250 K and 20 bar. Use the virial equation and the shortcut vapor pressure equation.

Solution

From the back flap of the text for acetylene: T_c = 308.3 K, P_c = 6.139, ω = 0.187, Z_c = 0.271. For each part of the problem, the fluid state of aggregation is determined before the method of solution is specified. At 250 K, using the shortcut vapor pressure equation, Eqn 9.11, the vapor pressure is P^sat = 1.387 MPa.

We will calculate the virial coefficient at 250 K using Eqns. 7.7–7.9:

T_r = 250/308.3 = 0.810, B^o = –0.5071, B¹ = –0.2758, B = –233.3 cm³/mol.

a. P = 1 MPa < P^sat so the acetylene is vapor (between points A and B in Fig. 9.5). Using Eqn 7.10 to evaluate the appropriateness of the virial equation at 1 MPa, P_r = 1/6.139 = 0.163, and 0.686 + 0.439P_r = 0.76 and T_r = 0.810, so the correlation should be accurate.

Using Eqn. 9.31,

(f = ϕ P = 0.894 (1) = 0.894 MPa

b. P = 2 MPa > P^sat, so the acetylene is liquid (point D of Fig. 9.5). For a liquid phase, the only way to incorporate the virial equation is to use the Poynting method, Eqn. 9.39. Using Eqn. 7.10 to evaluate the appropriateness of the virial equation at the vapor pressure, P_r^sat = 1.387/6.139 = 0.2259, and 0.686 + 0.439P_r^sat = 0.785, and T_r = 0.810, so the correlation should be accurate.

At the vapor pressure,

f^sat = ϕ^sat P^sat = 0.8558(1.387) = 1.187 MPa

Using the Poynting method to correct for pressure beyond the vapor pressure will require the liquid volume, estimated with the Rackett equation, Eqn. 9.40, using V_c = Z_cRTc/P_c = 0.271(8.314)(308.3)/6.139 = 113.2 cm³/mol.

The Poynting correction is given by Eqn. 9.38,

Thus, f = 1.187(1.015) = 1.20 MPa. The fugacity is close to the value of vapor pressure for liquid acetylene, even though the pressure is 2 MPa.