1. Heterogeneous Systems and Chemical Equilibrium

The preceding section involved only single phase systems. For it to be in thermodynamic equilibrium, a homogeneous system must be in thermal equilibrium (inﬁnitesimal difference in temperature between system and environment) and in mechanical equilibrium (inﬁnitesimal difference in pressure). A heterogeneous system can involve more than one phase and, for it, thermodynamic equilibrium requires an additional criterion, the system must also be in chemical equilibrium (no conversion of mass between one phase and the other).

Thermodynamic equilibrium between phases introduces additional constraints that actually reduce the degrees of freedom of a heterogeneous sytem below those of a homogeneous system.

1a. Chemical Equilibrium Let’s consider a two-component mixture of dry air and water with the latter ex- isting in vapor and possibly one condensed phase. For the two phases to be in equilibrium, in addition to thermal and mechanical equilibrium, the phases must also be in chemical equilibrium. The GIbbs function is used because here T and P are independent variables and the process of condensation is an isobaric and isothermal process. The change in Gibbs function is: (We will now move away from the use of speciﬁc quantities and multiply by mass) The total Gibbs function is Gtot = Gg + Gc where g, c, and tot denote the gaseous phase, condensed phase and total system. Now the mass can change so: For the gas phase, the change in the Gibbs function is:

∂Gg ∂Gg ∂Gg ∂Gg dGg = dT + dp + dnd + dnv ∂T p,n ∂p T,n ∂nd T,p,n ∂nv T,p,n (1) For the condensed phase ∂Gc ∂Gc ∂Gg dGc = dT + dp + dnc (2) ∂T p,n ∂p T,n ∂nc T,p,n

∂G The quantity µi = is deﬁned as the chemical potential of com- ∂ni T,p,n6=ni ponent i. Adding the two components:

1 ∂Gg ∂Gc ∂Gg ∂Gc dGtot = + dT + + dp+µddnd +µvdnv +µcdnc ∂T ∂T p,n ∂p ∂p T,n (3) Let’s consider a virtual displacement at constant T and p of the closed system (nd + nv + nc = constant) assuming nd = constant. Then dnd = dT = dp = 0 and dnc = dnv.

dGtot = (µv − µc)dnv (4)

The condition for stale equilibrium is dGtot ≥ 0 for an arbitrary dnv, since dnv can be positive or negative, chemical equilibrium requires that:

µv = µc (5)

2. Thermodynamic Degrees of Freedom

A system comprised of c components and σ phases. Each phase of each component we have temperature Ti and pressure pi - how Consider a single-compnent system involving two phases (water and water vapor) we have respectively p, T and p0,T 0. For mechanical, thermal and chemical equilibrium between the two phases we must have:

p = p0 (6) T = T 0 µ = µ0

Since there are four potential independent variables and three relations among them, there is only one independent variable in general A one-component system involving two phases at equilibrium with one another possesses only one thermodynamic degree of freedom. Such a system must possess an equation of state of the form:

p = p(T ) (7) If three phases (solid, lilquid and gas) are present simultaneously in equilib-

2 rium, then we have:

p = p0 = p00 (8) T = T 0 = T 00 (9) µ = µ0 = µ00 (10)

Since there are six unknowns and six relations among them, there are no independent variables. All values are ﬁxed and deﬁne a triple point where all three phases are in equilibrium.

In general, the number of thermodynamic degrees of freedom possessed by a heterogeneous system is described by the following principle i. Gibbs’ Phase Rule The number of state variables for a heterogeneous system involving c dynamically distinct by nonreactive components and σ phases is given by:

n = c + 2 − σ (11)

3. Thermodynamic Characteristics of Water

Water is a pure substance so its equation of state is:

p = p(α, T ) (12)

1. If only vapor Eq. 12 is the equation of state.

2. If two phases are present Eq. 12 reduces to p = p(T )

In heterogeneous state, when different phases coexist at equilibrium, the in- dividual phases are said to be saturated. The net ﬂux of mass from one state to another vanishes. If one of the phases is vapor, the pressure of the heterogeneous system represents the equilibrium vapor pressure with respect to water of ice pw or pi respectively. According to the Gibbs’ phase ruel, there exists a single state at which all three phases coexist at equilibrium, the triple point is deﬁned by:

3 pT = 6.1mb (13)

TT = 273K (14) 5 3 −1 αT v = 2.06 × 10 m kg (15) −3 3 −1 αT w = 1 × 10 m kg (16) −3 3 −1 αT i = 1.09 × 10 m kg (17)

4. Equilibrium Phase Transformations

The speciﬁc latent heat of a transformation is deﬁned as the heat absorbed by the system (per unit mass) during an isobaric phase transformation:

l = dq = dh − αdp (18) = dh (19)

6 −1 ◦ lv latent heat of vaporization (liquid-gas) =2.5 × 10 Jkg at 0 C

5 −1 ◦ lf latent heat of fusion (solid-liquid) =3.34 × 10 Jkg at 0 C

6 −1 ◦ ls latent heat of sublimation (solid-vapor) =2.83 × 10 Jkg at 0 C ls = lf + lv

However, the latent heat is a property of the system and depends on the thermodynamic state (generally expressed as a function of temperature)

4a. Clausius-Clapeyron Equation In states involving two phases, the system of pure water possesses only one thermodynamic degree of freedom (specifying temperature determines its pressure). We can derive the equation of state using fundamental relations subject to condi- tions of chemical equilibrium. Consider two phases a and b, and a transformation between them that occurs reversibly. The heat transfer during such a process equals the latent heat of transformation...for speciﬁc quantities: l ds = (20) T

4 For a system of dry air and water inwhich the latter appears only in trace abundance, the Chemical Potential (which is the partial molar Gibbs function) is approximately equal to the speciﬁc Gibbs function:

∂G µk = = gk (21) ∂nk pT n Consequently, for chemical equilibrium to occur:

ga = gb (22)

dgq = dgb (23)

applying the fundamental relation:

−(sb − sa)dT + (αb − αa)dp = 0 (24) or dp ∆s = (25) dT ∆α Incorporating Equation 20 yields:

dp l = (26) dT T ∆α Where l is the latent heat appropriate to the phases present. This is the Clausius- Clapeyron equation and related the equilibrium vapor pressure to the temperature of the heterogeneous system. It constitutes an equation of state for the heterogeneous system when two phases are present. i. Water-Ice l is the latent heat of fusion. It is more convenient to express the equation as: dT T ∆α = (27) dp l Because the change in volume during fusion is negligible, the equation of state becomes:

5 dT ≈ 0 (28) dp So the surface of water and ice in a T-p diagram is vertical. ii. Water-Vapor or Ice-Vapor The change of volume is approximately equal to that of the vapor produced: R T ∆α ≈ v (29) p dp lp = 2 (30) dT T Rv which can also be expressed as:

dlnp l = 2 (31) dT vaporization or sublimation RvT for l constant p l 1 1 ln 2 = − (32) p1 Rv T1 T2

Using the respective lv for vaporization and ls for sublimation yields (pw, pi in mb).

2.354 × 103 log p = 9.4041 − (33) 10 w T 2.667 × 103 log p = 10.55 − (34) 10 i T (35)

For a better approximation we can use an equation to express the change in latent heat with temperature. (HW)

These equations describe the surfaces that correspond to vapor being in chemical equilibrium with a condensed phase and to the system pressure equaling the equilibrium vapor pressure. I