Phase Equilibrium Conditions

Chapter IV: Mechanical Theory of Phase Coexistence

4.1 Phase Equilibrium Conditions

Thermodynamics provides a powerful framework to study systems at equilibrium.

At the heart of thermodynamics is the thermodynamic extremum principles which is rooted in the second law of thermodynamics. The extremum principles state that the system possesses a minimum free energy at equilibrium. In some cases, the

minimum free energy state is achieved by having multiple phases in a system: phase equilibrium.

At phase equilibrium, the temperature𝑇, pressure 𝑝, and chemical potential 𝜇 of each species are uniform in all phases — the thermal, mechanical, and chemical equilibria. To see this, let us consider an isolated 𝐶-component system with 𝑁_𝑖 particles of species𝑖 (∈ {1,2, . . . , 𝐶}), volume𝑉, and energy of the system 𝐸 and suppose the system has two phases 𝛼 and 𝛽 at contact. We denote variables for each phase with superscripts e.g. 𝑉^𝛼 is the volume occupied by𝛼phase. Since the interface of the two phases is diathermal, movable, and permeable to the chemical species, the energy, volume, and number of particles of each phase are unconstrained.

Using the additivity of the entropy𝑆 =𝑆^𝛼+𝑆^𝛽and the fundamental thermodynamic relation, the variation in the system entropy to the first orders is given by

𝛿 𝑆 = 1

𝑇^𝛼

− 1 𝑇^𝛽

𝛿 𝐸^𝛼+

𝑝^𝛼 𝑇^𝛼

− 𝑝^𝛽 𝑇^𝛽

𝛿𝑉^𝛼−

𝐶

∑︁

𝑖=1

𝜇^𝛼

𝑖

𝑇^𝛼

− 𝜇

𝛽 𝑖

𝑇^𝛽

! 𝛿 𝑁^𝛼

𝑖 . (4.1)

From the second law of thermodynamics, the entropy of an isolated system reaches a maximum at equilibrium and thus the first differential of the system entropy 𝑆 with respect to any macrostate variable𝑋vanishes at an equilibrium state: 𝛿 𝑆/𝛿 𝑋 = 0. Consequently 𝛿 𝑆 = 0 for any small variations in unconstrained variables at equilibrium and therefore from eq. (4.1)

𝑇^𝛼 =𝑇^𝛽, 𝑝^𝛼 = 𝑝^𝛽, 𝜇^𝛼

𝑖 = 𝜇

𝛽

𝑖 ∀𝑖 ∈ {1,2, . . . , 𝐶}. (4.2) When more than two phases are present, the phase equilibrium condition (4.4) is satisfied by every pair of phases by the zeroth law of thermodynamics. Hence,𝑇, 𝑝, and 𝜇 are the same in all coexisting phases at equilibrium. We note that the Gibbs phase rule is an immediate consequence of a degree of freedom analysis on the phase equilibrium condition.

One can also find thermodynamic stability conditions by considering the second order variation of the entropy. For example, we consider variations in the energy:

𝛿²𝑆= 1 2

𝜕

𝜕 𝐸^𝛼 1

𝑇^𝛼

𝑉^𝛼, 𝑁^𝛼

+ 𝜕

𝜕 𝐸^𝛽 1

𝑇^𝛽

𝑉^𝛽, 𝑁^𝛽

(𝛿 𝐸^𝛼)². (4.3) Now we let the two phases to be identical — the system is homogeneous — and treat the interface as an imaginary surface dividing the system into two regions 𝛼

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v ^b

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Thermodyanmically

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unstable

Figure 4.1: (a) A schematic description of the double-tangent construction on an isotherm in the diagram of the molar Helmholtz free energy 𝐹 versus the molar volume𝑣 at a fixed temperature𝑇. The points at contact with the common tangent line (blue) correspond to the coexisting phases. The slope and 𝑦-intercept of the common tangent represents the the coexistence pressure (𝜕 𝐹/𝜕 𝑣)𝑇 = −𝑝 and the coexistence chemical potential 𝐹− 𝑣(𝜕 𝐹/𝜕 𝑣)𝑇 = 𝜇. The red region of the curve indicates unstable region from the stability condition (4.5). (b) A schematic description of the Maxwell construction on an isotherm in the𝑝−𝑣diagram. A horizontal line that cuts equal areas from the isotherm above (shaded blue) and below (shaded red) gives the coexistence pressure. The intersections of the horizontal line and isotherm correspond to the coexisting phases. The red region of the curve indicates unstable region from the stability condition (4.5).

and 𝛽. By the second law, the entropy of the system holds its maximum value at equilibrium. Thus,𝛿²𝑆 ≤0 and from eq. (4.3) we should have

𝑐_𝑉 = 1 𝑁

𝜕 𝐸

𝜕𝑇

𝑉 , 𝑁

≥0, (4.4)

where 𝑐^𝛼

𝑉 is the molar heat capacity at constant volume and𝑇 = 𝑇_𝛼 = 𝑇_𝛽 from the equilibrium condition (4.4). Using the thermodynamic extremum principles, similar variational analyses can be performed on systems with different conditions.

One of the most useful stability conditions is obtained by the Helmholtz energy minimization principle for the system at fixed𝑁 , 𝑉 ,and𝑇:

𝜅_𝑇 =−1 𝑉

𝜕𝑉

𝜕 𝑝

𝑇 , 𝑁

≥ 0, (4.5)

where 𝜅_𝑇 is the isothermal impressibility and 𝑝 = 𝑝_𝛼 = 𝑝_𝛽 from the equilibrium condition (4.4).

A phase diagram can be constructed by directly finding states that satisfy the phase equilibrium condition. When a pressure-explicit equation of state that is valid for multiple phases (e.g. van der Waals equations state for liquid and gas) is known for a single component system, alternative graphical representations of the equilibrium condition provide more convenient and instructive methods to obtain a phase diagram: the double-tangent and Maxwell (equal-area) constructions.

In the double-tangent construction, the molar Helmholtz free energy 𝐹 curve is drawn as a function of the molar volume𝑣at a constant temperature𝑇 as shown in Fig. 4.1(a). A tangent line of the free energy curve has the slope (𝜕 𝐹/𝜕 𝑣)𝑇 = −𝑝 and𝑦-intercept𝐹−𝑣(𝜕 𝐹/𝜕 𝑣)𝑇 =𝜇. Therefore if multiple points, say(𝑣^𝛼, 𝐹(𝑣^𝛼, 𝑇)) and(𝑣^𝛽, 𝐹(𝑣^𝛽, 𝑇)), on the curve share a common tangent, the corresponding states (𝑣^𝛼, 𝑇)and(𝑣^𝛽, 𝑇)satisfy the phase equilibrium condition and are called the binodal.

When the overall molar volume of the system 𝑣 ∈ (𝑣^𝛼, 𝑣^𝛽), the total Helmholtz free energy is minimized by having the two separated phases with (𝑣^𝛼, 𝑇) and (𝑣^𝛽, 𝑇): the Helmholtz free energy minimization principle for fixed 𝑁 , 𝑉, and𝑇. The phase separation always occurs when the free energy curve is convex upward, i.e. (𝜕²𝐹/𝜕 𝑣²)𝑇 = −(𝜕 𝑝/𝜕 𝑣)𝑇 < 0 — the red segment in Fig. 4.1(a) — to meet the mechanical stability condition (4.5). The points at the onset the instability, i.e. (𝜕²𝐹/𝜕 𝑣²)𝑇 = 0 are called the spinodal. The states between the binodal and spinodal is called metastable. In the metastable region, the system is allowed to exist as a homogeneous state without a phase separation since the stability condition (4.5) is satisfied. However, a phase separation would occur whenever possible in order to minimize the total Helmholtz free energy of the system.

The Maxwell construction is obtained by considering the first derivatives with respect to the molar volume𝑣in the double-tangent construction. We now consider the pressure 𝑝 = −(𝜕 𝐹/𝜕 𝑣)𝑇 instead of 𝐹 and draw a 𝑝−𝑣 isotherm as shown in Fig. 4.1(b). Then one seeks the coexistence pressure 𝑝 = 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇) such that

𝑝(𝑣^𝛼, 𝑇) = 𝑝(𝑣^𝛽, 𝑇) = 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇),

∫ ^𝑣^𝛽

𝑣^𝛼

[𝑝(𝑣 , 𝑇) − 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇)] 𝑑 𝑣 =0, (4.6) then the intersections of the horizontal line and the isotherm gives the coexisting states. The first condition in (4.6) is trivially required to have the same pressure in the two states (𝑣^𝛼, 𝑇) and (𝑣^𝛽, 𝑇). The integral in (4.6) is required for the same chemical potential in the coexsiting phases. To see this, we integrate by parts to

obtain

[𝑣(𝑝−𝑝^{𝑐𝑜 𝑒𝑥})]^𝑣_𝑣^𝛽^𝛼−

∫ ^𝑣^𝛽

𝑣^𝛼

𝑣 𝜕 𝑝

𝜕 𝑣

𝑇

𝑑 𝑣 =0. (4.7)

Due to the first condition in (4.6), the bracketed term in (4.7) vanishes. The integral in (4.7) represents the difference in the chemical potentials at 𝑣^𝛼 and 𝑣^𝛽 since 𝑑𝜇 = 𝑣 𝑑 𝑝 = 𝑣(𝜕 𝑝/𝜕 𝑣)𝑇 𝑑 𝑣 at constant temperature. Thus, the second integral condition enforces chemical potentials of the two coexisting states to be identical.

From the Maxwell construction, it is clearly seen that the coexistence pressure𝑝^{𝑐𝑜 𝑒𝑥} depends on the temperature since the isotherms depends on the temperature of the system. The relationship between 𝑝^{𝑐𝑜 𝑒𝑥} and 𝑇 — the Clausius equation — can be directly derived from the equilibrium condition (4.1). Suppose that a single- component system at the temperature 𝑇 has two phases 𝛼 and 𝛽 at equilibrium:

𝑇 =𝑇_𝛼 =𝑇_𝛽, 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇) = 𝑝^𝛼 = 𝑝^𝛽, and 𝜇^𝛼(𝑇 , 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇)) = 𝜇^𝛽(𝑇 , 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇)). Then we imagine that the temperature varies slightly by𝛿𝑇and a new phase equilibrium is achieved. At the new temperature𝑇+𝛿𝑇, the equilibrium condition (4.1) is satisfied again: 𝑇+𝛿𝑇 =𝑇_𝛼=𝑇_𝛽, 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇+𝛿𝑇) = 𝑝^𝛼 = 𝑝^𝛽, and𝜇^𝛼(𝑇+𝛿𝑇 , 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇+𝛿𝑇))= 𝜇^𝛽(𝑇+𝛿𝑇 , 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇+𝛿𝑇)). Taylor expanding the new chemical equilibrium condition about the original temperature, we obtain

𝛿𝑇

−Δ𝑠+ 𝑑 𝑝 𝑑𝑇

𝑐𝑜 𝑒𝑥

Δ𝑣

+𝑂(𝛿𝑇²) =0, (4.8)

where𝑠is the molar entropy andΔrepresents the difference between the two phases e.g. Δ𝑣 = 𝑣^𝛼−𝑣^𝛽. Now by taking a limit𝛿𝑇 → 0 the temperature dependence of the coexistence pressure is obtained as

𝑑 𝑝 𝑑𝑇

𝑐𝑜 𝑒𝑥

= Δ𝑠 Δ𝑣

. (4.9)

The entropy change due to the phase transition at constant temperature and pressure can be written as Δ𝑠 = Δℎ^𝐿/𝑇, where Δℎ^𝐿 is the molar latent heat involved in transition from phase 𝛽to𝛼. Replacing theΔ𝑠in eq. (4.9), we obtain the Clausius equation:

𝑑 𝑝 𝑑𝑇

𝑐𝑜 𝑒𝑥

= Δℎ^𝐿 𝑇Δ𝑣

. (4.10)

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 70-74)

Chapter IV: Mechanical Theory of Phase Coexistence

4.1 Phase Equilibrium Conditions

v

v

p

v a

v b

v a

v b

(a) (b)

p

F

v ^a

v ^b

v ^a

v ^b