Pressure and Chemical Potential

When j = j_a, it is given by the following formula (see Section 5.10.1.2):

where, u[p_ja+1, P) = u(p_ja + Ap_a, p) = c„iui + c_u2u₂ + c_u3u₃.

Pressure Derived from Compressibility Equation

In Section 7.1, the pressure is obtained from the energy equation, but here the pressure is derived from the compressibility equation. From Eqs. (1.21) and (5.122), we get

Integrating the above equation for p in both ranges from p* to pj and Pj+1 yields

From the above two equations and the trapezoidal formula for integration, the following equation is obtained:

For any p, if Pp(p*, P) is given for a certain density p*, then using the above formula, Pp[pj, P) is obtained for any density pj. If you use the energy equation, you can get p on the spinodal line for any Pi, so if you set it as the initial value, you can use the above pressure equation derived from the compressibility equation to get pp{Pj, pi) at any point (pj, Pi).

If you use p(ps,. Pi) as the initial value in the gas phase region below the critical temperature (p > p_c), the quantity Pp[pj, Pi) is obtained for any step in the range p < pj < ps, • Here, ps, represents the density of the spinodal point on the low-density side. In the liquid phase region, if p(ps₂, Pi) is used as an initial value, the value of Pp(pj, Pi) is obtained for any step in the region of ps₂ < pj < pj. Where ps₂ is the density of the spinodal point on the high-density side, and p₍ is the density at the boundary point on the high-density side.

In the Case of T ≥ Tc

Put p, = 0 on the initial value Pp[p, p*), apply Carnahan-Staring formula (5.76), and set Pp(p, P* = 0) = (1 + p + p² — p³)/ A³. Using the pressure equation derived from the energy equation obtained in the previous section, we can find Pp[pj, pi) for any step between p and critical temperature p_c for any density pj.

Spinodal Curve

If only the pressure equation obtained from the energy equation is used, inconvenience arises in obtaining the liquefaction point and the vaporization point. In order to solve this problem, it is

Figure 7.1 Spinodal curve (black dashed line) on p-p plane for intermolec- ular interaction 0_bs(r) and isotherm (solid line) for critical temperature. Black circle: critical point. Black solid line: isothermal curve obtained from the energy equation. Red solid line: isothermal curve obtained from the compressibility equation. Since the red solid line and the black solid line coincide with each other on the scale in the figure, the red solid line is hidden behind the black solid line and cannot be seen.

necessaiy to recalculate using the pressure equation derived from the compressibility equation. These matters are explained using the calculation results obtained for the interparticle interaction

First, find the spinodal curve that satisfies A{p, p) = 0 using the energy equation. The result is shown by a broken line in the p-p diagram of Fig. 7.1. In this figure, black filled circle indicates critical point. The black and red solid lines show the p-p curves for the critical temperatures obtained from the energy equation and the compressibility equation, respectively. However, since they are the same on the scale in the figure, the red solid line is hidden behind the black solid line and cannot be seen. From this, it can be seen that the pressures derived based on the SCOZA theory, from the two different paths of the energy equation and the compressibility equation agree with high accuracy. However, strictly speaking, there is a minute mismatch between the two. In order to clarify this, an enlarged map near the critical point is shown in Fig. 7.2 for P = p_c (= 1.11664). An enlarged view of the p-p curve for ft = 1.12041

Figure 7.2 Isotherm (/8_C = 1.11664) for intermolecular interaction 0bs(r). Black circle: critical point. Black solid line: p-p curve for the critical temperature obtained from the energy equation. Red solid line: p-p curve for the critical temperature obtained from the compressibility equation.

Figure 7.3 Isotherms {p = 1.12041) and spinodal curves for intermolecu- lar interaction 0_bs(r). Black solid lines: p-p curve obtained from the energy equation. Red solid lines: p-p curve obtained from the compressibility equation. Dashed lines: spinodal lines.

is also shown in Fig. 7.3. In this enlarged view, it can be seen that there is a slight deviation between the two. The pressure (black line) obtained from the energy equation does not satisfy A[p, p) = 0, that is, (dp/dp)p = 0 at the critical point. The points that satisfy A[p, p) = 0 appear in two places on the low-density side and the high-density side that are a little away from the critical point. On the other hand, of course, such a contradiction does not occur in the pressure (red line) obtained from the compressibility equation.

Therefore, in order to obtain the pressure consistently, calculate the pressure on the spinodal curve using the formula obtained from the energy equation. Using the result as the initial value, it is necessaiy to recalculate the pressure using the formula obtained from the compressibility equation. In this way, the above contradiction is resolved. From now on, pressure will be calculated according to such a procedure. As will be described later, the same operation is performed for the chemical potential.

Figure 7.4 Spinodal curve on p-p plane for intermolecular interaction Фп{г).

Revert story to ф_с. The p-p diagram of Fig. 7.4 shows the spinodal curve with a broken line obtained using the energy equation. In addition, a part of the p-p diagram in Fig. 7.5 is indicated by a broken line. In the figures, the point C (p_c = 0.0841, p_c = 0.21935) represents the critical point. On the lower density side than the critical point C, the pressure p on the spinodal curve increases monotonously with the increase of the density p. On the other hand, on the higher density side than the critical point C, the pressure on the spinodal curve decreases as the density increases, reaching a minimum at the point Pi (p = 0.1157, p = -0.6021). After that, on the spinodal curve, the pressure increases and reaches the point P₂ [p = 0.1157, p = -0.002525) while keeping the density constant.

Isotherms

When T < T_c, the pressure must be calculated separately for the

gas region and the liquid region under the boundary condition

Figure 7.5 Isotherms and gas-liquid coexistence region. The blue, orange, purple, green, and black lines show the isotherms for p = p_Q = 0.107891, 0.1154, 0.1414, 0.2684, and 0.5954, respectively. Black circle C is a critical point. The black circle P₂ represents the intersection of the spinodal curve and the black solid line. The red solid line and the black broken line represent the binodal curve and the spinodal curve, respectively.

set on the spinodal curve. Using Pp[p, P) on the spinodal curve obtained from the energy equation as the initial value, you can get Pp[pj, Pi) for any step below the critical temperature from the pressure equation obtained from the compressibility equation according to the methods explained in Sections 7.1 and 7.3.1.

The isotherms of pressure obtained in this way are shown in Fig. 7.5 for p = p_c (= 0.107891), 0.1154,0.1414,0.2684, and 0.5954, respectively, by blue, orange, purple, green, and black curves. These temperatures are the same as the representative ones mentioned in Section 6.2.4. The black circle C in the figure represents the critical point. The black dashed line represents the spinodal curve. The solution of Eq. (5.124) is unstable in the region inside the spinodal curve. Of note is the behavior of the isotherm of the liquid. The isotherm moves to the low pressure side until the temperature drops to reach p = 0.2684 (green line), just like normal substances. However, when it is further cooled below this temperature, the isotherm reverses and moves to the high pressure side, and behaves in the opposite manner to a normal substance. That is, the low temperature black line is located on the higher pressure side than the higher temperature purple line and green line. This strange behavior produces negative thermal expansion, as explained in Section 2.2.

In the gas region, the isotherm monotonously moves to the low pressure side as the temperature decreases.

Chemical Potential Derived from the Energy Equation

Integrating both sides of the energy equation (1.20) with respect to p in the range between Д, (<к f}{) and Д (/ > 1), we get the following equation:

Chemical Potential at β = β∗

If /1, _cl(r), so the fluid can be considered as an ideal gas. In this case, the first term and the second term on the right-hand side of Eq. (7.2) can be obtained as follows: Consider an ideal gas consisting of A^r monatomic molecules with a mass of m. The Hamiltonian of this system is

Here, an orthogonal coordinate system was adopted and serial numbers were assigned to the coordinate components. Considering the canonical distribution, the partition function is

In this case, Helmholtz’s free energy is calculated using Stirling’s formula In ЛП ~ Af In Af — Af,

From this, the chemical potential of the ideal gas is

Here, A is the de Broglie thermal wavelength and is defined by the following equation:

The chemical potential obtained by Carnahan-Staring for the hard sphere fluid (see Fig. 7.6) is given by

In this book, we replace In p in Eq. (7.5) with Ppcs, and propose the following equation as a chemical potential for a high- temperature gas:

Substituting Eq. (7.8) into Eq. (7.2) gives the following expression:

Figure 7.6 Carnahan-Staring's chemical potential Pucs for hard sphere fluid.

Figure 7.7 The same as Fig. 7.1 except for x!.

The second term on the right-hand side depends on temperature but not on density, so it has no effect on the determination of liquefaction and vaporization points (see Section 8.1). So we introduce the next pi'.

Using the trapezoidal formula for integration, the difference of pp'[p, pi+1) and pp'{p, fy) yields the following:

where,

Chemical Potential Derived from Compressibility Equation

From Eqs. (1.23) and (5.122), we get

If the above equation is integrated in both ranges from p to pj and pj+i with respect to p, the following equations are obtained:

From the above two equations and the trapezoidal formula for integration, the following equation is obtained:

Using p' derived from the energy equation, find the value on the spinodal curve, use it as an initial value, and use the above equation obtained from the compressibility equation to get p' at any other point (pj, Pi). The calculation method to obtain p’ is the same as in the case of pressure.

Figure 7.8 The same as Fig. 7.2, except for ц'.

Figure 7.9 The same as Fig. 7.5, but for ц'.

Also, д' obtained from each of the energy equation and the compressibility equation agrees with high accuracy, but there is a slight inconsistency between the two as in the case of pressure. We confirm these matters by taking the calculation results obtained for _b5. The chemical potential д'(р, /3) for the critical temperature obtained from each of the energy equation and compressibility equation is shown, respectively, by the black line and the red line in Fig. 7.7. On the scale of the figure, the slight deviation between the two is negligibly small, so the red solid line is hidden behind the black solid line. Figure 7.8 shows an enlarged view near the critical point. As explained with regard to pressure, the critical point must be (9p'/dp)p = 0, but the former case is not so and there is a contradiction. However, the latter does not cause such a contradiction. The same is true for other temperatures.

Returning to the case of ф_сi, the isotherm and spinodal curve of the chemical potential д' obtained through the above calculation procedure are shown in Fig. 7.9. This figure is the same as Fig. 7.5 except that the pressure p is replaced by д'.