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Physical chemistry
Phase Equilibrium
Dr. R. Usha
Miranda House, Delhi University
Delhi
CONTENTS
Phase equilibrium Phase The phase rule What is a phase diagram? The phase diagram for water Equilibrium between solid and vapour (sublimation curve) Equilibrium between liquid, vapour and solid water (ice) Equilibrium between solid and liquid (fusion curve) Metastable equilibrium involving liquid and vapour phases The phase diagram for carbondioxide The phase diagram for sulphur Enantiotropy Monotropy Metastable equilibria in the sulphur system Phase equilibria of two component systems Thermal analysis Saturation or solubility method The bismuth-cadmium system The Lead-Silver system The Magnesium-Zinc system The Sodium chloride-water system The Ferric chloride - Water system Efflorescence and deliquescence Liquid-Liquid mixtures – ideal liquid mixtures Raoult’s law Effect of temperature on the solubility of gases Effect of pressure on the solubility of gases Konowaloff’s rule The Duhem-Margules equation Fractional distillation of non ideal solution Partially miscible liquids Phenol-water system Triethylamine-water system Nicotine-water system Steam distillation
Phase equilibrium
Various heterogeneous equilibria (Box 10.1) have been studied by methods suitable to that type of equilibrium, such as vaporization by using Raoult’s law and Clausius-Clapeyron equation; distribution of solutes between phases by using the distribution law etc. A principle called the phase rule can be applied to all heterogeneous equilibria. The number of variables to which heterogeneous equilibria are subjected to under different experimental conditions may be defined using this principle. The phase rule is able to fix only the number of variables involved. The quantitative relation between the variables is obtained by using the various laws and equations (Box 10.2).
This rule was first put forward by J. Willard Gibbs, an American Chemist, in the year 1876 (Box 10.3). The full implications of this rule was understood by chemists only when Roozeboom, Ostwald and van’t Hoff applied it to some well known physical and chemical equilibria in a language that could be easily followed. After this its value as a fundamental generalization was fully realized.
Certain terms like phases, components, degrees of freedom, true and metastable equilibrium (Box 10.4) need to be explained in some detail before the phase rule is stated.
Box 10.1 Examples of heterogeneous equilibria:
- Liquid – vapour (vapourization)
- Solid – vapour (sublimation)
- Solid – liquid (fusion)
- Solid 1 – solid 2 (transition)
- Solubility of solids, liquids and gases in each other
- Vapour pressure of solutions
- Chemical reaction between solids or liquids and gases
- Distribution of solutes between different phases
Box 10.2 Laws and equations to study heterogeneous equilibria:
- A solid solution is considered as a single phase.
- Each polymorphic form constitutes a separate phase.
The number of phases does not depend on the actual quantities of the phases present. It also does not depend on the state of subdivision of the phase.
Examples 10.1.
Counting the number of phases
a) Liquid water, pieces of ice and water vapour are present together.
The number of phases is 3 as each form is a separate phase. Ice in the system is a single phase even if it is present as a number of pieces.
b) Calcium carbonate undergoes thermal decomposition.
The chemical reaction is: CaCO 3 (s) CaO(s) + CO 2 (g)
Number of phases = 3
This system consists of 2 solid phases, CaCO 3 and CaO and one gaseous phase, that of CO 2.
c) Ammonium chloride undergoes thermal decomposition.
The chemical reaction is:
NH 4 Cl(s) ' NH 3 (g) + HCl (g)
Number of phases = 2
This system has two phases, one solid, NH 4 Cl and one gaseous, a mixture of NH 3 and HCl.
d) A solution of NaCl in water
Number of phases = 1
e) A system consisting of monoclinic sulphur, rhombic sulphur and liquid sulphur
Number of phases = 3
This system has 2 solid phases and one liquid. Monoclinic and rhombic sulphur, polymorphic forms, constitute separate phases.
Box 10.4 True and metastable equilibrium
- True equilibrium is obtained when the free energy content of a system is at a minimum for the given set of variables.
- A state of true equilibrium is said to exist in a system when the same state can be obtained by approaching from either direction.
- An example of such an equilibrium is ice and liquid water at 1 atm pressure and 0o^ C. At the given pressure, the temperature at which the two phases are in equilibrium is the same whether the state is attained by partial freezing of liquid water or by partial melting of ice.
Liquid water at -4 o^ C is said to be in a state of metastable equilibrium because this state of water can be obtained by only careful cooling of the liquid and not by fusion of ice. If an ice crystal is added to this system, then immediately solidification starts rapidly and the temperature rises to 0 o^ C. A state of metastable equilibrium is one that is obtained only by careful approach from one direction and may be preserved by taking care not to subject the system to sudden shock, stirring or “seeding” by solid phase.
Components
The number of components of a system at equilibrium is the smallest number of independently varying chemical constituents using which the composition of each and every phase in the system can be expressed. It should be noted that the term “constituents” is different from “components”, which has a special definition. When no reaction is taking place in a system, the number of components is the same as the number of constituents. For example, pure water is a one component system because all the different phases can be expressed in terms of the single constituent water.
Examples 10.1.2.
Counting the number of components
a) The sulphur system is a one component system. All the phases, monoclinic, rhombic, liquid and vapour – can be expressed in terms of the single constituent – sulphur.
b) A mixture of ethanol and water is an example of a two component system. We need both ethanol and water to express its composition.
c) An example of a system in which a reaction occurs and an equilibrium is established is the thermal decomposition of solid CaCO 3. In this system, there are three distinct phases: solid CaCO 3 , solid CaO and gaseous CO 2. Though there are 3 species present, the number of components is only two, because of the equilibrium:
d) If in the system mentioned above, a small quantity of water is allowed to evaporate and then the system is allowed to come to equilibrium, then the number of phases in equilibrium will be three. This system has no degrees of freedom or it is invariant. Three phases, ice, water, vapour can coexist in equilibrium at 0.0075o^ C and 4.6mm of Hg pressure only. A change in temperature or pressure will result in one or two phases disappearing.
Hence the degree of freedom of a system may also be defined as the number of variables, such as temperature, pressure and concentration that can be varied independently without altering the number of phases.
The phase rule
The phase rule is the relationship between the number of phases, P, the number of components, C and the number of degrees of freedom, F of a system at equilibrium at a given P and T. The rule is P+F = C+2, where 2 stands for the intensive variables pressure, P and temperature, T. This is a general rule applicable to all types of reactive and nonreactive systems. In a nonreactive system, the various components are distributed in different phases without any complications, such as reacting chemically with each other. First let us derive this rule for the nonreactive system and then show that the same rule applies to the reactive system as well.
Derivation of the phase rule
Before taking up the derivation of the phase rule, let us determine the number of degrees of freedom of some simple systems without using the phase rule.
a) Example 1 – A gaseous system having one component.
No. of phases in the system = 1
Every homogeneous phase has an equation of state or phase equation given by f(P (^) ,T,C)=0 where P stands for pressure, T for temperature and C for concentration. This phase equation has three variables P (^) ,T and C. If the values of 2 variables are known, the third can be calculated using this equation. Hence the number of variables that need to be actually specified is equal to 2.
Number of degrees of freedom = Total number of variables – number of equations connecting the variables
F = 3-1=
The above mentioned system is a bivariant one.
b) Example 2 – a system consisting of water and water vapour in equilibrium with each other.
This system has 2 phases – liquid water and water vapour. One can write one equation of state or phase equation for each phase.
f (^) l (T, P, Cl ) = 0 for the liquid phase
f (^) v (T, P, Cv ) = 0 for the gaseous phase
Cl and C (^) v are concentrations in the liquid and gaseous phases respectively. As water and vapour are in equilibrium at a definite temperature and pressure, there is a chemical potential relation equating the chemical potential of water in the 2 phases.
2
l μ H O (P, T, Cl ) = (^2) ν μ (^) H O (P, T, Cv )
This system has four variables, T, P, C (^) l and Cv , and three equations relating them, two equations of state and one chemical potential equation. Hence the number of degrees of freedom, F= number of variables – number of equations relating the variables.
F=4–3=
This system is thus a univariant one.
c) Example 3 – a system consisting of ice, liquid water and vapour in equilibrium at constant temperature and pressure.
There are 3 phases and hence 3 phase equations
For ice f (T, P, C (^) i ) = 0
For water f (T (^) , P, Cl ) = 0
For vapour f (T, P, C (^) v ) = 0
Ci,^ Cl and Cv are the concentrations of ice, liquid water and water vapour respectively.
When these three phases coexist in equilibrium at a definite temperature and pressure, the chemical potential of water is the same in each phase. The chemical potential equations are:
μ iH O 2 (T, P, Ci ) = μlH O 2 (T, P, Cl )
2
l μ (^) H O(T, P, Cl ) = (^2) v μH O (T, P, Cv )
Total number of variables = 5. These are T, P, Ci , Cl and Cv.
Total number of equations relating these variables = 5, 3 equations of state and 2 chemical potential equations. The number of degrees of freedom, F=5–5 =
This is an invariant system.
Number of concentration variables (C mole fractions to describe one phase; P×C to describe P phases)
P × C
Temperature, pressure variables 2
Total number of variables PC + 2
Let us next find the total number of equations connecting the variables.
A phase equation for each phase (For each phase, the sum of mole fractions equals unity)
X 1 + X 2 + X 3 + ………..+ X (^) c = 1
P equations for P phases P
Chemical potential equations (At equilibrium the chemical potentialof each component is the same in every phase.) Equations for component 1in P phases
μ 11 = μ 1 2 = μ 13 =……………= μ 1 p
P-1 equations for each component
C(P-1) equations for C components (^) C(P–1)
Total number of equations P+C(P–1)
Number of degrees of Freedom, F=Total number of variables – total number of equations
F = P × C + 2 – {P+C(P-1)}
F=PC + 2- P-CP+C
F=C+2-P
P+F = C+2, which is the Gibb’s phase rule.
This rule gives the number of variables, F that need to be specified in order to define the system completely and unambiguously.
It was assumed in this derivation that each component is present in every phase. It can be shown that the phase rule remains unaltered even if all the components are not present in all the phases.
Derivation of the rule taking that one of the components is present only in P-1 phases.
We consider, as in the earlier case, a system consisting of C components and P phases under equilibrium at constant temperature and pressure. One of the components is missing from one phase and hence is present in only P-1 phases.
Let us first find out the total number of intensive variables that are needed to describe the state of the system. As one component is excluded from one phase, the number of concentration variables will be CP-1.
Number of concentration variables = CP-
Pressure, temperature variables = 2
Total number of variables = CP+
Let us next find the total number of equations connecting the variables.
Number of phase equations = P
Number of chemical potential equations for C-1 components in P phases
= (C-1) (P-1)
for one component in P-1 phases = P-
Total number of equations = P+(C-1)(P-1)+(P-2) = C(P-1)-
Number of degrees of freedom, F=total number of variables – total number of equations
F=CP-1-{C(P-1)-1}
μ 11 = μ 21 = μ 13 =……………= μp 1
For C constituents in P phases, there are C(P-1) equations
For a reactive system, there is another condition that has to be satisfied. At equilibrium, the reaction potential, ∆rG is zero. This gives ν 3 μ 3 + ν 4 μ 4 - ν 1 μ 1 -v 2 μ 2 =0. Thus we get one more equation. 1
Total number of equations available P+C(P-1)+
Variance=number of variables – number of equations
F=CP+2-{ P+C(P-1)+1}
F=(C-1)-P+
If in a system, two independent reactions are possible, then it can be shown that F=(C-2)- P+
Generalizing, we write
F=(C-r)-P+
Where r is the number of independent reactions that are taking place in a system.
Sometimes a chemical reaction takes place in such a manner that requires additional equations expressing further restrictions upon the mole fractions to be satisfied. One such reaction is the thermal decomposition of solid NH 4 Cl in vacuum.
NH 4 Cl (s) NH 3 (g) + HCl (g)
Additional restriction that exists in the gaseous phase is
XNH 3
= XHCl
The number of such equations as this one which impose additional restrictions should also be included in the total number of equations.
A system containing a salt solution is another example in which an additional restricting equation relating the mole fraction of ions exists.
AB → A+^ + B-
The additional restricting equation is
X A+ = XB-
In general if there are r independent reactions and Z independent restrictive conditions in a system, then the total number of equations is given by:
Total number of equations = Number of phase equations + number of chemical potential equations + number of equations due to chemical reactions + number of equations due to restricting conditions.
Total number of equations = P+C(P–1)+r+Z
Variance, F=(CP+2)–{P+C(P–1)+r+Z}
F=(C–r–Z)–P+
F=C'–P+
Where C '=C-r-Z and is known as the number of components of the system.
Thus, the number of components of a reactive system is equal to the total number of constituents present in the system less than the number of independent chemical reactions and the number of independent restricting equations.
This equation has the same form as that for a nonreactive system with C '^ in place of C.
Phase rule gives information only about the number of degrees of freedom of a system at equilibrium. If a variable is altered, and the equilibrium is disturbed, then information regarding the direction and extent of change that will follow is not provided by the rule. This is its limitation.
For the application of phase rule to study different heterogeneous systems under equilibrium, it is convenient to classify all systems according to the number of components present. We will discuss one and two component systems in that order in the following sections.
Phase equlibria of one component systems – water, CO 2 and S systems
Applying the phase rule to a one component system, we write
F = C-P+2 = 1-P+2 = 3-P
Three different cases are possible with P taking values 1, 2 and 3.
a) System having only one phase, i.e., P=
diagram for water under moderate pressure (Fig.10.2.2) with only ordinary ice forming the solid phase (Box 10.2.2.1).
Figure 10.2.2: The phase diagram for water
Equilibrium between solid and vapour (sublimation curve)
At the point B (Fig.10.2.2) ice is in equilibrium with its vapour. The pressure at B is the vapour pressure of ice at the temperature at B. If this temperature at B is gradually raised keeping the volume constant, vapour pressure of ice also increases. If the vapour pressure of ice is plotted against temperature, the curve BO, the sublimation curve is obtained. Along the curve BO, ice and water vapour are in equilibrium with each other. The slope of the curve at any point as given by the Clapeyron equation is:
m,Subl m,v m,s
dp H dT T(V V )
⎛ ⎞^ ∆ ⎜ ⎟ = ⎝ ⎠ −
The variation of sublimation pressure with temperature is given by the Clausius-Clapeyron equation as:
lnp = – m,Subl
H I RT
∆
Where I is the constant of integration
For each temperature of this solid-vapour system, there exists a certain definite pressure of the vapour given by the curve BO.
If the system represented by point B is expanded isothermally, then this will decrease the pressure of the vapour phase. As at a given temperature, the solid-vapour system has a fixed vapour pressure, some ice will sublime to maintain the pressure. If the isothermal expansion is continued, more and more ice will sublime till the solid phase disappeared.
If, on the other hand, the system represented by point B is compressed isothermally, then some vapour will condense to form ice in order to maintain the pressure and prevent its increase. If the isothermal compression is continued, then the entire vapour phase will disappear leaving only a solid phase in the system. These show that the regions above and below the curve BO represent solid and vapour phases, respectively.
Equilibrium between liquid, vapour and solid water (ice)
The system at point B (Fig.10.2.2) is gradually heated keeping the volume constant when the vapour pressure of ice increases. A temperature is reached at which the vapour pressure of ice becomes equal to that of liquid water maintained at the same temperature. Then the solid water starts melting and the system consists of three phases, ice, water and vapour, in equilibrium with each other. This is an invariant system (F=0) and the temperature and pressure of the system remain unchanged as long as all the three phases are present together. This is the system at point O in the figure 10.2.2. and is known as triple point. This point for water lies at 0.0075o^ C and 4.6mmHg.
Equilibrium between liquid and vapour (vaporization curve)
The system at the triple point is gradually heated at constant volume, the temperature and pressure do not change till the entire solid melts to give liquid water. There are only 2 phases in the system – liquid water and vapour. If the heating is continued at constant volume, the temperature and vapour pressure of the system vary along the curve OA (Fig.10.2.2). The curve OA is known as the vaporization curve and along the curve OA liquid water and vapour are in equilibrium with each other. The slope of the curve OA at any point is given by the Clapeyron equation:
m,Vap l v m,v m,l
dp H dT T(V V )
⎛ ⎞ ∆ ⎜ ⎟ = ⎝ ⎠ (^) −
The Clausius – Clapeyron equation:
lnp = m,vap
H I RT
−∆
(I=Integration constant) gives the variation of vapour pressure with temperature, i.e., the curve OA. This curve OA has an upper limit at the critical pressure and temperature, i.e., the point A.
If a system represented by any point on the curve OA is subjected to isothermal expansion, then the pressure of the vapour phase decreases, a small quantity of water evaporates to raise the pressure to a value which is the vapour pressure of liquid water at that temperature. As the isothermal expansion is continued, more and more liquid water evaporates till the entire liquid phase disappears and the system is made up of only vapour.
The continuation of the AO curve, vaporization curve beyond the triple point, i.e., OA' lies above the OB curve, the curve for the stable phase in that temperature interval. Hence the vapour pressure of the system in the metastable region is more than that of the stable system, that is, ice at the same temperature.
Table 10.2.2.Description of the phase diagram for water
- BO Sublimation curve
Solid vapour
P=2, F=1 T or P
- OA Vaporization curve
Liquid vapour
P=2, F=1 T or P
- OC Fusion curve (^) Solid
liquid
P=2, F=1 T or P
Area left of BOC Solid phase P=1, F=2 T and P
Area AOC Liquid phase P=1, F=2 T and P
Area below AOB Vapour phase P=1, F=2 T and P
Point O (0.0075 o^ C, 4.6mmHg) Triple point, Solid Liquid vapour
P=3, F=
Point A (374 oC, 217.5 atm) Critical temperature, critical pressure
OA' metastable equilibrium Liquid vapour
P=2, F=1 T or P
Box 10.2.2.
In the phase diagram for water under moderate pressure (Fig.10.2.2), there is only one solid phase, namely ordinary ice. Several crystalline modifications of ice are observed when the system is studied under very high pressures (of about50,000 atmospheres). Ice I is ordinary ice. At very high pressures ices II, III, V, VI and VII are stable. Existence of ice IV was reported but was not confirmed. It was an illusion. It is reported that ice VII melts at about 100 o^ C under a pressure of 25,000 atm. Isn’t the melting of ice hot?
Box 10.2.2.
We have seen in the water system, the fusion curve OC (Fig.10.2.2) is almost vertical with a slight tilt towards the pressure axis. This indicates that an increase in pressure decreases the melting point of ice, a property that contributes to making skating on ice a possibility. The pressure exerted by the weight of the skater through the knife edge of the skate blade lowers the melting point of ice. This effect along with the heat developed by friction produces a lubricating layer of liquid water between the ice and the blade. It is of interest to note that the skating is not good if the temperature of ice is too low.
The phase diagram for carbondioxide
The system of CO 2 (Fig.10.2.3) is very similar to the water system except that the solid – liquid line OC slopes to the right, away from the pressure axis. This indicates that the melting point of solid carbon dioxide rises as the pressure increases. The slope of this line follows the clapeyron equation:
m,fus s l m,l^ m,s
dp H dT T(V V )
⎛ ⎞ ∆ ⎜ ⎟ = ⎝ ⎠ (^) −
As V (^) m,l > V (^) m,s and Vm,l -V (^) m,s is small, the line OC has a large positive slope.
The triple point, O (Fig.10.2.3) occurs at -56.4 o^ C and a pressure of about 5 atm. We must note, that as the triple point lies above 1 atm, the liquid phase cannot exist at normal atmospheric pressure whatever be the temperature. Solid carbon dioxide hence sublimes when kept in the open (referred to as “dry ice”). It is necessary to apply a pressure of about 5 atm or higher to obtain liquid carbon dioxide. Commercial cylinders of CO 2 generally contain liquid and gas in equilibrium, the pressure in the cylinder is about 67 atm if the temperature is 25 o^ C. When this gas
