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
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:
• Raoult’s law
• Clapeyron equation
• Clausius –Clapeyron equation
• Henry’s law
• Equilibrium constants
• Distribution law
Box 10.3 Josiah Willard Gibbs
• Spent most of his working life at Yale
• May be regarded as the originator of chemical thermodynamics
• Reflected for ten years before publishing his conclusions
• Published his papers in the transactions of the Connecticut Academy of Arts and Sciences, not a well known journal
His work remained overlooked for 20 years. Roozeboom, Ostwald, van’t Hoff and others showed how this rule could be utilized in the study of problems in heterogeneous equilibria.
Definitions Phase
A phase is defined as any homogeneous and physically distinct part of a system which is separated from other parts of the system by interfaces.
A part of a system is homogeneous if it has identical physical properties and chemical composition throughout the part.
• A phase may be gas, liquid or solid.
• A gas or a gaseous mixture is a single phase.
• Totally miscible liquids constitute a single phase.
• In an immiscible liquid system, each layer is counted as a separate phase.
• Every solid constitutes a single phase except when a solid solution is formed.
• 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 10.1.1
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: CaCO3(s) CaO(s) + CO2 (g) Number of phases = 3
This system consists of 2 solid phases, CaCO3 and CaO and one gaseous phase, that of CO2.
c) Ammonium chloride undergoes thermal decomposition.
The chemical reaction is:
NH4Cl(s) ' NH3 (g) + HCl (g) Number of phases = 2
This system has two phases, one solid, NH4Cl and one gaseous, a mixture of NH3
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 0oC.
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 -4oC 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 0oC. 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 CaCO3. In this system, there are three distinct phases: solid CaCO3, solid CaO and gaseous CO2. Though there are 3 species present, the number of components is only two, because of the equilibrium:
Any two of the three constituents may be chosen as the components. If CaO and CO2
are chosen, then the composition of the phase CaCO3 is expressed as one mole of component CO2 plus one mole of component CaO. If, on the other hand, CaCO3 and CO2 were chosen, then the composition of the phase CaO would be described as one mole of CaCO3 minus one mole of CO2.
d) A system in which ammonium chloride undergoes thermal composition.
NH4Cl (s) NH3(g) + HCl (g)
There are two phases, one solid-NH4Cl and the other gas – a mixture of NH3 and HCl. There are three constituents. Since NH3 and HCl can be prepared in the correct stoichiometric proportions by the reaction:
NH4Cl → NH3+HCl
The composition of both the solid and gaseous phase can be expressed in terms of NH4Cl. Hence the number of components is one.
If additional HCl (or NH3) were added to the system, then the decomposition of NH4Cl would not give the correct composition of the gas phase. A second component, HCl (or NH3) would be needed to describe the gas phase.
Degrees of freedom (or variance)
The degrees of freedom or variance of a system is defined as the minimum number of variables such as temperature, pressure, concentration, which must be arbitrarily fixed in order to define the system completely.
Examples
Systems of different variance
a) A gaseous mixture of CO2 and N2. Three variables: pressure, temperature and composition are required to define this system. This is, hence, a trivariant system.
b) A system having only liquid water has two degrees of freedom or is bivariant. Both temperature and pressure need to be mentioned in order to define the system.
c) If to the system containing liquid water, pieces of ice are added and this system with 2 phases is allowed to come to equilibrium, then it is an univariant system. Only one variable, either temperature or pressure need to be specified in order to define the system. If the pressure on the system is maintained at 1 atm, then the temperature of the system gets automatically fixed at 0oC, the normal melting point of ice.
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 0.0075oC 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=2
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.
fl (T, P, Cl) = 0 for the liquid phase
Cl and Cv 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
µH O (P, T, Cl) =
2
ν
µH O(P, T, Cv)
This system has four variables, T, P, Cl 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=1
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, Ci) = 0
For water f (T, P, Cl) = 0 For vapour f (T, P, Cv) = 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 O2 (T, P, Ci) =2
l
µ
H O (T, P, Cl)2 l
µH O (T, P, Cl) = µvH O2 (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 =0
This is an invariant system.
d) Example 4: A homogeneous solution of sugar in water. This system has two components, sugar and water, and one phase. The phase equation for the system is:
f (T, P,
H O2
C , Csugar) = 0 Total number of variables = 4 Number of equations =1
Number of degrees of freedom, F=4-1=3.
These three degrees of freedom are temperature, pressure and concentration of one component.
e) Example 5: A system made up of 2 phases and 2 components at constant temperature and pressure. Let the concentration of component 1 in phase 1 be C11 and in phase 2 be C12. Let the concentration of component 2 in phase 1 be C12and in phase 2 be C22. The phase equations are:
for phase 1: f1 (T, P, C , C )=011 12 and for phase 2: f2 (T, P, C , C )=012 22
As the system is at equilibrium, the chemical potential of component 1 in the 2 phases is the same, as also that of component 2. Let the chemical potential of component 1 in the two phases be µ11 and µ12, that of component 2 be µ12 and µ22. The chemical potential equations are µ11 = µ12 and µ12= µ22. Total number of variables are 6; P, T, C , C , C11 12 12 andC22. Total number of equations are 4, 2 phase equations and 2 chemical potential equations. The number of degrees of freedom, F=6-4=2.
Derivation of the phase rule for the nonreactive system
Let us consider a system of P phases and C components existing in equilibrium at constant temperature and pressure.
The number of degrees of freedom or the variance F is equal to the number of intensive variables required to describe a system, minus the number that cannot be independently varied, which in turn is given by the number of equations connecting the variables.
We begin by finding the total number of intensive variables that would be needed to describe the state of the system. Let us assume that all the C components are present in all P phases. The state of a system is specified at equilibrium if temperature, pressure and the amounts of each component in each phase are specified. Since the actual amount of material in any phase does not affect the equilibrium, it is the relative amount and not the absolute amount that is important.
Therefore, mole fractions are used instead of number of moles.
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) X1 + X2 + X3 + ………..+ Xc = 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 = µ = µ =
11 12 13 ………=µ
1pP-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-1
Pressure, temperature variables = 2
Total number of variables = CP+1
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-2
Total number of equations = P+(C-1)(P-1)+(P-2) = C(P-1)-1
Number of degrees of freedom, F=total number of variables – total number of equations F=CP-1-{C(P-1)-1}
P+F = C+2
Thus we see that the effect of one of the components not present in one phase is reduction in both the number of variables and the number of equations by one.
The difference is thus the same and the phase rule remains unchanged. This shows that the phase rule is generally valid under any kind of distribution as long as the system is under equilibrium.
Derivation of the phase rule for a reactive system
Let us consider a system of C constituents and P phases under equilibrium at constant temperature and pressure. Let us assume that four of the constituents are involved in a reaction given by:
ν1A1+ν2A2
ν3A3 + ν4A4We next find the total number of variables that are needed to describe the system completely and the number of equations that are available at equilibrium.
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 find the total number of equations connecting the variables.
A phase equation relating the mole fractions for each phase X1 + X2 + X3 + ………..+ Xc = 1
There are P equations for P phases P
Chemical potential equations:P-1 equations for each constituent
C(P-1)
µ = µ = µ =11 12 13 ………=µ1p 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 ν3µ3+ ν4µ4 - ν1µ1-v2µ2=0.
Thus we get one more equation. 1
Total number of equations available P+C(P-1)+1
Variance=number of variables – number of equations
F=CP+2-{ P+C(P-1)+1}
F=(C-1)-P+2
If in a system, two independent reactions are possible, then it can be shown that F=(C-2)- P+2
Generalizing, we write F=(C-r)-P+2
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 NH4Cl in vacuum.
NH4Cl (s)
NH3 (g) + HCl (g)Additional restriction that exists in the gaseous phase is XNH3 = 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.
The additional restricting equation is XA+ = 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+2 F=C'–P+2
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, CO2 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=1
F=3-P=3-1=2
This is a bivariant system. We need to state the values of 2 variables in order to define the system completely. These are temperature and pressure. The given component may exist in any of the three phases, solid, liquid or vapour.
b) System having 2 phases, i.e. P=2 F=3-P=3-2=1
This is a univariant system and hence the value of either of the 2 variables, temperature or pressure, would define the system completely. The two phases in equilibrium with each other may be solid-solid, solid-liquid, liquid-vapour and solid-vapour.
c) System having 3 phases, i.e. P=3.
F=3-P=3-3=0
The system is invariant and three phases can exist in equilibrium only at definite values of temperature and pressure. The three phases in equilibrium with each other may be solid-liquid-vapour, solid-solid-liquid, and solid-solid-vapour.
Thus, we see that the maximum number of phases existing in equilibrium in a one component system is three. For a one component system, as the maximum number of degrees of freedom is two, the equilibrium conditions can be represented by a phase diagram in two dimensions choosing pressure and temperature variables.
What is a phase diagram?
A phase diagram is one in which the relationships between the states or phases of a substance can be summarized. The diagram shows the various phases present at different temperatures and pressures. They are also called equilibrium diagrams.
We can see important properties like melting point, boiling point, transition points and triple points of a substance on the phase diagram. Every point on the phase diagram represents a state of the system as it described T and p values. The lines on the phase diagram divide it into regions.
These regions may be solid, liquid or gas. If the point that describes a system falls in a region, then the system exists as a single phase. On the other hand, if the point falls on a line, then the system exists as two phases in equilibrium. The liquid-gas curve has a definite upper limit at the critical temperature and pressure as liquid and gas become indistinguishable above this temperature and pressure.
The phase diagram for water
The simplest case of a one component system is one in which there is only one solid phase. In a system having more than one solid phase, there are a number of possible equilibria and the phase diagram gets quite complicated. In the case of “water” system, above –20oC and below 2000 atm pressure, there is only one solid phase, namely, ordinary ice. We will discuss the phase
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 = – Hm,Subl 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 0.0075oC 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 m,v m,l l v
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
The Clausius – Clapeyron equation:
lnp = Hm,vap 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.
lower the pressure. This continues till the entire vapour phase disappears and there is only liquid water in the system. These changes lead us to conclude that the regions above and below the curve OA represent liquid and vapour phases, respectively.
Equilibrium between solid and liquid (fusion curve)
If solid water (ice) at some high pressure is heated slowly, it starts melting after a certain temperature is reached. Temperature of the system then remains constant till the entire solid phase is converted to liquid. The temperature at which the solid melts to give liquid depends on the pressure on the solid phase. The line OC (Fig.10.2.2) depicts the various conditions of temperature and pressure at which ice and water are in equilibrium with each other. This line OC is known as the fusion curve and the slope of this line is given by the Clapeyron equation:
m,fus m,l m,s s l
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
As V -Vm,l m,s is small, the slope of the line OC is comparatively large and hence the line OC is almost vertical. For water, Vm,l-Vm,s is negative and hence the line OC is slightly tilted towards the pressure axis (Box 10.2.2).
Metastable equilibrium involving liquid and vapour phases
If a system represented by a point on the curve AO, liquid water in equilibrium with vapour, is cooled rapidly, ice may fail to form at the triple point and the vapour pressure of the liquid may continue along OA'. This represents metastable equilibrium involving liquid and vapour phases.
At the triple point, the slope of the solid-vapour curve, OB is greater than that of the liquid- vapour curve, OA. This can be shown by using the Clapeyron equation as follows:
m,sub1 m,v m,s s v
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
m,vap m,v m,l l v
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
Vm,v – Vm,s
Vm,v – Vm,l∆Hm,sub1= ∆Hm,fus+ ∆Hm,vap since∆Hm,subl >∆Hm,vap
s v l v
dp dp
dT dT
⎛ ⎞ >⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
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
1) BO Sublimation
curve Solid
vapour
P=2, F=1 T or P
2) OA Vaporization
curve Liquid
vapour
P=2, F=1 T or P
3) OC Fusion curve Solid
liquid
P=2, F=1 T or P
4) Area left of BOC Solid phase P=1, F=2 T and P
5) Area AOC Liquid phase P=1, F=2 T and P
6) Area below AOB Vapour phase P=1, F=2 T and P
7) Point O (0.0075oC, 4.6mmHg) Triple point, Solid
Liquid
vapour
P=3, F=0
8) Point A (374oC, 217.5 atm) Critical temperature, critical pressure
9) OA' metastable
equilibrium Liquid
vapour
P=2, F=1 T or P
Box 10.2.2.1
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 100oC under a pressure of 25,000 atm. Isn’t the melting of ice hot?
Box 10.2.2.2
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 CO2 (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 m,l m,s s l
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
As Vm,l > Vm,s and Vm,l-Vm,s is small, the line OC has a large positive slope.
The triple point, O (Fig.10.2.3) occurs at -56.4oC 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 CO2 generally contain liquid and gas in equilibrium, the pressure in the cylinder is about 67 atm if the temperature is 25oC. When this gas
comes out through a fine nozzle, it cools and condenses into a finely divided snow-like solid as the outside pressure is only 1 atm (Box.10.2.3).
Table 10.2.3 Description of the phase diagram for carbon dioxide.
1) BO sublimation curve solid vapour P=2, F=1
T or P
2) OA Vaporization curve Liquid vapour P=2, F=1
T or P
3) OC Fusion curve Solid Liquid P=2,F=1 T or P
4) Area left of BOC
Solid phase P=1,
F=2
T & P
5) Area AOC Liquid phase P=1,
F=2
T & P
6) Area below AOB
Vapour phase P=1,
F=2,
T & P
7) Point O (-56.4oC,
~5 atm)
Triple point SolidLiquidvapour P=3,F=0
8) Point A (31.1oC, 73 atm)
Critical temperature, critical pressure
T
Figure 10.2.3: The phase diagram for carbondioxide
Box 10.2.3 Super critical carbondioxide
Super critical carbondioxide is obtained by heating compressed carbondioxide to temperatures above its critical temperature. The critical constants of CO2 are: Tc=304.1K and Pc=73.8 bar which are not far from ambient conditions. It is inexpensive and easily available in large quantities. It is non toxic, nonflammable and inert to most materials. It has good dissolving properties and hence used as a super critical solvent. It is thus an ideal eco friendly substitute for hazardous and toxic solvents.
It is used for extracting flavours, decaffeination of coffee and tea, recrystallization of pharmaceuticals etc. It is also used in supercritical fluid chromatography, a form of chromatography in which the supercritical fluid is used as the mobile phase. This technique can be used to separate lipids and phospholipids and to separate fuel oil into alkanes, alkenes and arenes.
The phase diagram for sulphur
Sulphur exists in two solid modifications, the rhombic form stable at ordinary temperatures and the monoclinic form at higher temperatures. Substances that can exist in more than one crystalline form, each form having its own characteristics vapour pressure curve, are said to exhibit the phenomenon of polymorphism. Two types of polymorphism are observed, enantiotropy (Greek:
opposite change) and monotropy (Greek: one change).
vapour liquid
solid
p
Enantiotropy
Two crystalline modifications of a substance are said to be enantiotropic (or to exhibit enantiotropy) when each has a definite range of stability and conversion from one modification to the other takes place at a definite temperature in either direction. This temperature is the transition point and it is the only temperature at which the two modifications can coexist in equilibrium at a given pressure. A change in this temperature results in the complete transformation of one modification into the other, one being stable above the transition point and the other below it.
We represent, say, the two enantiotropic forms by α and β and we assume that the α form is stable at lower temperatures while the β form at higher temperatures.
α β
Figure 10.2.4.1 gives the vapour pressure-temperature curve of this enantiotropic system. Each form has its own vapour pressure curve, AB is the curve of the α form and BC is that of the β form. B is the transition point where both α and β forms are at equilibrium with the vapour.
Figure 10.2.4.1: Vapour pressure temperature curve of an enantiotropic system
A system represented by a point on the curve AB is heated slowly such that the α form continues to stay in equilibrium with vapour. The system moves along AB and at B, the α form is transformed into the β form, the temperature and vapour pressure remain constant. The heat supplied goes towards converting the α into the β form. After all the α form gets converted into the β form, the vapour pressure of the system changes along the curve BC. Slopes of the sublimation curves AB and BC at any point are given by the Clapeyron equation:
m,subl m,v m,s s v
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
Where the variables refer to the α and β forms respectively. At the triple point, we can write
B
E C
D
A
T
p
∆
Hm,subl (solid α) >∆
Hm, subl (solid β)As a result of the above relation, the slope of the vaporization curve of the α form (AB) is greater than that of the β form (BC) at the triple point B.
The β form melts at C and CD is the vaporization curve. If the α form is heated rapidly, no β form appears at B, the vapour pressure of the system continues along BE and a metastable equilibrium exists between the α form and vapour.
Similarly, if a system represented by a point on the curve CD is cooled rapidly, no β form separates at C, cooling of the liquid continues along CE and a metastable equilibrium exists between liquid and vapour. The two curves meet at the point E, which represents the melting point of the α form.
This explains why an enantiotropic substance melts at different temperatures depending upon whether the solid is heated slowly or rapidly. The two melting points are those of the β and α forms respectively.
Enantiotropy, a more common form of polymorphism is exhibited by sulphur, tin, ammonium nitrate, carbon tetrachloride among other substances.
Monotropy
If one crystalline form is stable and the other form metastable over the entire range of their existence, the substance is said to exhibit monotropy. The transformation from one form into another takes place in one direction only, that is, from metastable to stable form.
Figure 10.2.4.2: Vapour pressure temperature curve of a monotropic system
A
B C
D
E F
T
p
Fig.10.2.4.2 gives the vapour pressure – temperature curve of a monotropic system. The stable α form has the lower vapour pressure curve AB and the metastable β form has the upper vapour pressure curve CD. BE is the vaporization curve. At the triple point B, α form, liquid and vapour are in equilibrium with each other. The curve EB is extended to meet the curve CD at D, the melting point of the β form. The curves CD and AB are extended to meet at a point F, a hypothetical transition temperature. This has no real existence as it is above the melting points of both the forms and only one solid form has a stable existence. The points D and F are metastable triple points.
At point D β form liquid vapour At point F β form α form vapour
As the effect of pressure on the melting and transition points is usually very small, points B, D and F may be referred to as the melting point of the α form, melting point of the β form and the transition point respectively.
The metastable β form undergoes transition to the stable α form but the reverse of this process does not take place. The β form, however, can be obtained by an indirect method from the α form. The α form is first heated to get a liquid, the liquid is then cooled rapidly when the system moves along BD instead of BA and the β form separates at D.
Monotropy is exhibited by iodinemonochloride, silica, benzophenone among other substances.
Some substances for example, phosphorous exhibit both enantiotropy and monotropy.
Discussion – the phase diagram of sulphur
Figure 10.2.4.3. gives the phase diagram of sulphur. As mentioned earlier, rhombic and monoclinic are the two enantiotropic forms of sulphur, rhombic being stable at lower temperatures. If the temperature of the system, rhombic sulphur in equilibrium with sulphur vapour, represented by the point A is raised at constant volume, the vapour pressure increases along the curve AB. The curve AB, sublimation curve of rhombic sulphur, gives the temperatures and pressures at which rhombic sulphur and its vapour exist in equilibrium.
Figure 10.2.4.3: The phase diagram for sulphur
If a system on the curve AB is allowed to expand keeping temperature constant, more of solid sulphur will sublime to keep the vapour pressure at a constant value. If this process in continued, then the solid phase will disappear and the system will be all vapour. If on the other hand the same system is subjected to an isothermal compression, then vapour will condense to keep the vapour pressure at a constant value. On the continuation of the process, vapour phase will disappear and the system will be all solid rhombic sulphur. Hence, we conclude that the phases above and below the curve AB comprise of rhombic sulphur and vapour sulphur respectively.
As the heating of the system, rhombic sulphur in equilibrium with vapour, is continued at constant volume, the vapour pressure of rhombic sulphur increases along AB and at the point B becomes equal to that of monoclinic sulphur. Then the rhombic form undergoes transition into the monoclinic form and the system at the point B has three phases, rhombic, monoclinic and vapour. This is an invariant system (P=3 and F=0) and the temperature and pressure of the system remain unchanged as long as the three phases coexist. These variables remain constant as the heating at constant volume is continued till the entire rhombic sulphur gets converted into the monoclinic sulphur. The system then has only 2 phases, monoclinic and vapour in equilibrium
A
B
C
D E
liquid
T
with each other. If the heating at constant volume is continued, then the monoclinic in equilibrium with vapour moves along BC, the sublimation curve of monoclinic sulphur. The slope of the curve AB is larger than that of the curve BC at the triple point B as can be seen by applying the Clapeyron equation at the point B.
m,rhom,subl m,v m,rhom rhom v
∆H
T(V -V )
⎛ ⎞ =
⎜ ⎟
⎝ ⎠
dp
dT
m,mono,subl m,v m,mono mono v
dp H
dT T(V V )
⎛ ⎞ = ∆
⎜ ⎟ −
⎝ ⎠
m,r hom,subl m,rhom mono m,mono,subl
H H → H
∆ = ∆ + ∆
m,rhom,subl m,mono,subl
∆H >∆H
r hom v mono v
dp dp
dT dT
⎛ ⎞ >⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
It can be shown by taking a system on the curve BC and subjecting to isothermal expansion and compression that the system above the curve BC is solid monoclinic sulphur and below BC is vapour (Fig.10.2.4.3).
The vapour pressure of monoclinic sulphur increases as heating at constant volume is continued and becomes equal to that of liquid sulphur at the point C. At C, a triple point, three phases co- exist, monoclinic sulphur, liquid sulphur and vapour. As the heating is continued, all the solid melts to give liquid, temperature and pressure remaining constant. The vapour pressure of liquid sulphur in equilibrium with vapour moves along CD with heating and reaches D, the critical temperature. It can be seen that the slope of the curve BC is larger than that of the curve CD at the point C in accordance with the Clapeyron equation. It can also be shown by subjecting a system on the curve CD to isothermal expansion and compression that the phase below CD is vapour and that above CD is liquid sulphur (Fig 10.2.4.3).
When a system consisting of rhombic sulphur at some high pressure is gradually heated, a temperature is reached when rhombic gets converted to monoclinic sulphur. This temperature, known as the transition point, remains constant till all the rhombic form gets converted to the monoclinic form. The transition temperature depends on the pressure of the system and the transition line BE (Fig.10.2.4.3) gives the dependence. The line BE has a positive slope because rhombic sulphur is more dense than monoclinic. Rhombic sulphur exists to the left of the line BE and monoclinic sulphur to the right.
As the monoclinic form is heated, a temperature is reached when it starts melting and the system, monoclinic S liquid S, is represented by a point on the line CE. The temperature remains constant till the change of phase is completed. Along the line CE, the equilibrium between monoclinic and liquid sulphur exists whereas only solid exists to the left of the line and only the liquid to the right of the line. The two lines BE and CE meet at E, a triple point where rhombic, monoclinic and liquid sulphur coexist in equilibrium. If the pressure of the system is higher than
EF.
Metastable equilibria in the sulphur system
Heating a system on the curve AB rapidly may not result in the conversion of rhombic to monoclinic at B and the vapour pressure curve may continue along BG. There exists a metastable equilibrium between rhombic and vapour along BG. Similarly cooling rapidly a system on DC may not result in the formation of solid monoclinic form at C and the system may continue along CG. Liquid and vapour sulphur coexist in a state of metastable equilibrium along CG. The point G where the curves BG and CG meet is a triple point (metastable) where rhombic, liquid and vapour sulphur coexist in equilibrium.
If a system consisting of rhombic sulphur at some high pressure is heated rapidly, then transition to monoclinic form may not occur on the line BE. Rhombic form may continue until the system meets the dotted line GE when it would melt to give liquid sulphur. Along the line GE, rhombic sulphur would exist in a state of metstable equilibrium with liquid sulphur. In the area BGEB, rhombic sulphur exists in a metastable state. Similarly in the area CGEC, liquid sulphur exists in a metastable state. These metastable states are formed only if rhombic form fails to undergo transition to monoclinic form on the line BE and liquid sulphur does not pass over to monoclinic form on the line CE. As the monoclinic form is the stable form in this region BCEB, any other form has a metastable existence and has a tendency to spontaneously change over to the monoclinic form. Table 10.2.4 describes, in brief, the phase diagram of sulphur.
Table10.2.4 Description of the phase diagram for sulphur
AB sublimation curve of rhombic sulphur r v P=2 F=1
BC sublimation curve of monoclinic sulphur mv P=2 F=1
CD vaporization curve of liquid sulphur lv P=2 F=1
BE Transition line of rhombic to monoclinic rm P=2 F=1
CE Fusion line of monoclinic sulphur ml P=2 F=1
EF Fusion line of rhombic sulphur rl P=2 F=1
BG Metastable sublimation of
rhombic S rv P=2 F=1
CG Metastable vaporization curve of liquid
S lv P=2 F=1
GE Metastable fusion line of rhombic to
liquid rl P=2 F=1
B Triple point (95.5oC, 0.01mmHg) rmv P=3 F=0
C Triple point (119.2oC, 0.025mmHg) mlv P=3 F=0
E Triple point (151oC, 1290atm) rml P=3 F=0
G Metastable triple point (114.5oC,
0.03mmHg) rlv P=3 F=0
Area to the left of ABF
Rhombic sulphur P=1 F=2
Area above CD and right of CEF
Liquid sulphur P=1 F=2
Area BCEB Monoclinic sulphur P=1 F=2
Area below ABCD
Vapour sulphur P=1 F=2
Area BGEB Metastable rhombic S P=1 F=2
Area CGEC Metastable liquid S P=1 F=2
Phase equilibria of two component systems
Applying the phase rule to two component systems we have the degrees of freedom, F=C–
P+2=4–P. when a single phase is present in a two component system, the number of degrees of freedom, F=3, means that three variables must be specified to describe the phase and these are temperature, pressure and composition of the phase. When two phases are present, the number of degrees of freedom, 4–P, is reduced to 2, temperature and composition of the liquid phase. The values of the other variables get automatically fixed. If there are three phases present, then F=4–
3=1 which means the value of only one variable needs to be stated to describe the phases.
The maximum number of degrees of freedom for a two component system we see from the preceding discussion is three. In order to represent the variation in three variables graphically, we require a three dimensional diagram, a space model, which is difficult to construct on paper. To overcome this difficulty, a common practice that is adopted is to keep one of the variables constant. There are various types of equilibria, that are generally studied at constant external pressure. Thus, out of the three variables (F=3), one is already stated and the variation in the other two can be represented on a two dimensional diagram. Equilibria such as solid-liquid equilibria are such systems in which the gas phase is absent and hence are hardly affected by small changes in pressure. Systems in which the gas phase is absent are called condensed systems.
Measurements in these systems are generally carried out at atmospheric pressure. As these systems are relatively insensitive to small variation in pressure, the pressure may be considered constant. The phase rule takes the form
P+F=C+1
For such systems and in this form it is known as the reduced phase rule. For a two component system, this equation becomes F=3-P where the only remaining variables are temperature and composition. Hence solid-liquid equilibria are represented on temperature – composition diagrams. In the following sections we will be discussing systems involving only solid-liquid equilibria.
Determination of solid-liquid equilibria
Many experimental methods are used for the determination of equilibrium conditions between solid and liquid phases. The two most widely used are the thermal analysis and saturation or solubility methods. Whenever required additional data are obtained by investigating the nature of the solid phases occurring in a system.
Thermal analysis
Thermal analysis method involves a study of cooling curves (temperature time plots) of various compositions of a system. A system of known composition is prepared, heated to get a melt,
allowed to cool on its own and then its temperature noted at regular time intervals (say, half a minute). A cooling curve is obtained by plotting temperature versus time.
Information regarding the initial and final solidification temperatures is obtained from the breaks and halts in the cooling curves. This analysis method is applicable under all temperature conditions but is especially suitable for investigations at temperatures quite above and below room temperature.
Saturation or solubility method
In this method the solubilities of one substance in another are determined at different constant temperatures, then the solubilities are plotted against temperature.
To determine the solubility of A in B, excess of A is added to B (molten B, if B is a solid), kept at a desired constant temperature, stirred well until equilibrium is reached. The undissolved solid A is filtered off and the saturated solution of A in B is analysed for both constituents. Similarly where possible saturated solutions of B in A at different temperatures are analyzed.
This is the principal method of analysis of systems containing water and similar solvents as one of the constitutents. This method is very difficult to carry out at temperatures below -50oC and also above 200oC. Then the thermal analysis method is preferred.
Determination of the nature of solid phases
It is important to know the nature and composition of the solid phases which appear during crystallization and in the final solid for the complete interpretation of a phase diagram. These solid phases may be pure components, compounds formed by reaction between pure constituents, solid solutions and mixture of solids.
It is possible to arrive at the nature of the solid phases from the shape of the phase diagram quite often. However, in some cases, a more careful study may be required. To do this, the solid mass may be inspected under a microscope or may be studied using X rays.
Classification of two component solid-liquid equilibria
Phases diagrams of some two component solid liquid equilibria are simple while those of others are quite complicated. The complex ones may be considered to be made up of a number of simple types of diagrams. The classification of these equlibria is done according to the miscibility of the liquid phases. These are further divided into various types based on the nature of the solid phases crystallizing from the solution.
The classification is as follows:
Class A The two components are completely miscible in the liquid phase
Type 1. Only the pure components crystallize from the solution
Type 2 A solid compound stable upto its melting point is formed by the two constituents
Type 3 A solid compound which decomposes before it reaches its melting point is formed by the two constituents
Type 4 The two constituents are completely miscible in the solid phase. This results in the formation of a series of solid solutions
Type 5 In the solid state, the two constituents are partially miscible and they form stable solid solutions
Type 6 Solid solutions formed by the two constitutents are stable only upto a transition temperature
Class B In the liquid phase, the two components are partially miscible
Type 1 Only pure components crystallize from the solution
Class C In the liquid phase, the two components are immiscible
Type 1 Only pure components crystallize from the solution
Cooling curve of a pure component
A pure component, say A, is taken and heated to get a melt. The liquid A is allowed to cool on its own and the temperature of the melt is noted at, say, every half a minute. A cooling curve is obtained by plotting temperature versus time as shown in the fig. 10.3.3.1.
T
Figure 10.3.3.1: Cooling curve of a pure component
Cooling of liquid A takes place along ac, when at c solidification starts and the system has two phases in equilibrium becoming an invariant system (F=C+1-P=1+1-2=0). The temperature remains constant till the entire liquid A solidifies along cd. The system is cooling, losing heat to the surroundings, yet the temperature remains constant during solidification. This is due to the fact that heat is released during solidification. Cooling of solid A takes place along de. The system represented by any point on either ac or de has only one phase and hence is univariant.
Cooling curve of a mixture of two components with only pure components crystallizing on cooling the system.
Solid B is added to solid A to get a mixture of known composition. This mixture is heated to get it in the liquid phase. The cooling curve of this liquid is depicted in Fig. 10.3.3.2.
a
c d
e liquid cooling
liquid
solid cooling
Temperature →
Figure 10.3.3.2: Cooling curve of a liquid mixture of two components
The liquid cools along ab and at b solid A starts solidifying. Temperature and composition of liquid phase have to be stated to define the liquid phase completely (P=1, F=3-P, F=2) when A starts solidifying, the system becomes univariant (P=2, F=3-P=1). Thus, the temperature at which solid A starts solidifying from the liquid mixture of known composition will have a definite value given by the point b. This temperature is expected to be a little lower than the freezing point of pure A as the addition of B to A lowers the freezing point of A. Solidification of a small quantity of A changes the composition of the liquid phase and hence the temperature at which A solidifies from this liquid will take place at another fixed value but lower than the previous temperature as the molality of B in the liquid increases with the solidification of A. Thus as more and more of A separates from the liquid, the temperature of the system falls along bc. The rate of cooling is affected by the heat evolved due to the solidification of A and hence a break is observed in the curve at b. The break point indicates the temperature at which A just starts solidifying. Along the curve bc, there are two phases, liquid and solid A, hence the system is univariant (P=2, F=3-P=1). As cooling continues along bc, more and more of solid A separates and the liquid gets richer in B. At the point C, the liquid becomes saturated in B and hence B starts separating along with A. Along cd both the solids A and B separate and the system becomes invariant (P=3, F=3-P=0). Solidification from a solution of fixed composition, one corresponding to the saturation solubility of B in A takes place at constant temperature. This results in a halt or complete arrest of the cooling curve (cd). As the saturation solubility of B in A has to be maintained at this temperature, the composition of the solid phase that separates will be the same as that of the liquid phase. The temperature of the systems will remain unchanged till the whole of the liquid phase solidifies. The cooling of the solid phase is represented by de and the system is univariant (P=2, F=3-P=1).
liquid cooling
solid starts separating
solid mixture cooling second solid
starts separating
Time →
Temperature→
b
c d
e
a
Fig 10.3.3.3: Cooling curve of a liquid mixture of two components in which supercooling has occurred
a
liquid cooling
solid starts separating
solid mixture cooling
second solid starts separating
Time →
Temperature
b b'
f
c d
e
In some cases, the separation of the solid phase does not occur readily at b and the liquid continues to cool along bf (super cooling occurs). As this represents an unstable state, there is a sudden increase in temperature which is followed by a normal cooling curve along b'c. The correct freezing point is then obtained by extrapolation to the point b as shown in the figure 10.3.3.3.
Various mixtures, B in A and A in B show the type of cooling curve given in Fig.10.3.3.2 except one particular composition. With this composition, on cooling the liquid, both components start solidifying at the same time. In this case the first break does not occur and the cooling curve resembles the one given in fig. 10.3.3.1.
Systems in which solid solutions and not pure components separate on cooling, the cooling curves are similar to that given in Fig.10.3.3.2 with the horizontal portion (the halt) replaced by a break.
The bismuth-cadmium system
A number of mixtures of the two metals ranging in overall composition of 100% bismuth to 100% cadmium are prepared. They may be spaced at 10% intervals and should preferably be of equal mass. Crucibles made of inert material, say fire-clay or graphite are taken, each mixture of bismuth and cadmium is placed in a crucible and heated in an electric furnace to get the melt. An inert or a reducing atmosphere is maintained by passing hydrogen, nitrogen or carbon dioxide through the furnace, to prevent oxidation of the metals. As an additional precaution, a molten flux such as borax or a layer of powdered graphite is used to cover the mixture in the crucibles. After melting the mixture and mixing thoroughly, the furnace and contents are allowed to cool slowly. Temperature and time readings are taken until the mixtures are completely solidified. Then cooling curves are plotted and break and halt temperatures are noted. If one desires to check on the composition of the mixtures prepared, then the solids are removed from the crucibles and carefully analyzed.
The equilibrium or phase diagram for this system is then constructed by plotting break and halt temperatures from the cooling curves of the mixtures on a temperature composition diagram. Smooth curves are drawn to yield the diagram shown in the figure 10.3.4
