Phase Rules!

Applications of Gibbs's Phase Rule and Phase Diagrams


  1. Systems, Phases, Constituents and Variance
  2. Some Thermodynamics
  3. Systems With Multiple Phases
  4. Binary Systems
  5. Ice and Salt
  6. Iron
  7. References

Systems, Phases, Constituents and Variance

The system is simply the matter that we are studying, collected in one place and with definite boundaries. Our systems will generally not exchange matter with the surroundings, and so will be called closed. They may be isolated and not exchange energy with the environment, or sometimes in contact with a thermal reservoir of temperature T. A system may absorb a quantity of heat Q or do work W on the environment. Heat is the transfer of energy of disordered molecular motion, while work is a macroscopic, ordered, transfer of energy, such as the movement of a piston or an electric current. The state of a system is a specification of conditions sufficient to determine its macroscopic characteristics completely. A system is in equilibrium if there are no changes of its state with time.

Phase is derived from the Greek verb fasis, "appearance, assertion, accusation" which, as indicated, has important figurative meanings. Its use for the phases of the moon, which repeat in a regular sequence, gave rise to its use for the time sequence of a signal, particularly a sinusoidal signal, where the phase is represented as an angle that steadily increases with time. In this article, its meaning is quite different, referring to matter that appears in a homogeneous form: its "appearance."

A material phase is a homogeneous substance at an atomic level, with the same conditions existing everywhere within it. The conditions may be expressed as intensive parameters that do not depend on the amount of substance, such as the pressure P and the temperature T, or by extensive parameters that are proportional to the amount of substance, such as the entropy S or the volume V. Extensive parameters may be referred to a fixed amount of substance, either a certain mass, or, better to a gram-mole. In this case, we shall refer to them with small letters, as s or v, and call them "specific" quantities. A phase may be a pure substance or a mixture, provided the mixture is like a solution or a solid solution, where the mixing is at an atomic level. A mixture of sand and salt is not a phase, but two phases each of which can have its own macroscopic characteristics.

The system that we study may consist of one or more phases. The system also has a certain number of chemical constituents. If they can react with one another to form new substances, the new substances are not independent constituents so long as their amounts can be determined by the conditions existing in the system. For example, if Na+ and Cl- ions are present in a system, then NaCl ← Na+ + Cl- is not a new constituent. However, H and He are two constituents of a system, even though He can be formed by fusion of four hydrogen atoms. The fusion reaction will not take place under the conditions of any reasonable system, so the amounts of each will be constant. If the system is solar matter, on the other hand, we may want to talk about only one constituent.

Let us suppose our system consists of only one phase, say water vapor. We also have only one constituent, water. We can describe the state of the system uniquely by specifying a pressure P and a temperature T, which can be established independently. If the vapor is considered to be an ideal gas, then PV = NkT, where k is Boltzmann's constant and N is the number of molecules. From this, it is clear that the molar volume is v = RT/P, where R = NAk. It cannot be specified at will, but is determined once the pressure and temperature are fixed. A system consisting of a single phase of one component is said to have variance f = 2. This is not the same variance as in statistics, and is, in fact, a bad choice of words, but a conventional one. This conclusion is valid even if the ideal gas assumption is dropped, and an arbitrary equation of state is assumed instead. The Phase Rule will give the variance for more complicated systems.

Some Thermodynamics

Thermodynamics is unavoidable when studying phase changes, equilibrium, chemical reactions, electrolytic cells, and similar subjects. Here is a brief synopsis of the subject to serve as a handy reference that includes the most-used relations and concepts.

The internal energy U of a system is the total energy of its molecules, kinetic and potential, referred to an arbitrary zero. The specific internal energy is the internal energy per unit mass: u = U/M. The modifier "specific" will mean "per unit mass." For a simple system, u may be a function of the macroscopic variables s and v, the specific entropy and specific volume. We write u = u(s,v). Changes to U can be divided into heat and work, so that ΔU = Q - W, or, for convenience, dU = dQ - dW, or du = dq - dw. This is the First Law of Thermodynamics. dq and dw represent infinitesimal changes, not the differentials of functions. du is, of course, the differential of a function. The First Law says that u is a function of state that depends on energy transfers with the system.

For a system whose volume can change, dw = Pdv, or P = (∂u/∂v)q. This partial derivative means the same thing, and the reader should not be put off by the notation. The subscript on the partial derivative means that dq = 0. By analogy, we can write dq = Tds, or T = (∂u/∂s)v. A new function S or s is introduced, called the entropy. It can be found in principle by integrating ds = dq/T at constant volume, and in fact this is done in elementary thermodynamics. However, entropy cannot be properly understood on a macroscopic basis alone (though many texts make a valiant attempt to do so), and we shall say more about it below. Let us assume that entropy s is a function of the state of a system that can be determined independently the same way that v can be. The Second Law of Thermodynamics is that s is a function of state, that the heat absorbed in a reversible process is Tds, and that for any process in an isolated system, ds ≥ 0. There is a lot in this statement, and it is the core of thermodynamics. Note that entropy can increase without the absorption of heat in an irreversible process. An example is the free expansion of a gas into a vacuum. More randomness is given to the gas, but no energy is absorbed or lost.

In a small reversible change of the system, dU = TdS - PdV. This is a concise statement of both the First and the Second laws in one equation. If we consider the function H = U + PV, then dH = TdS - PdV + PdV + VdP, or dH = TdS + VdP. If we consider an isobaric process, dP = 0, then dH is the heat transferred to the system, or dH = CPdT. Therefore, dS = CPdT/T - (V/T)dP, from which we can find S as a function of P and T. For an ideal gas with constant specific heat, s = cP ln T - R ln P + so. The constant of integration so should be noted. Energy is arbitrary up to a constant. Adding a constant to the energy will not change any results. Entropy is different. Unlike energy, entropy has an absolute value, and the constant so is not arbitrary. A serious fault of thermodynamics based only on macroscopic considerations is that this constant cannot be determined by its means.

The entropy has just been introduced in the most mysterious way possible, as the S in dQ = TdS. It is very difficult to understand entropy on this basis, if indeed it can be understood at all. To know what entropy means, we must consider the individual molecules of the substance in their chaotic environment. The entropy of a substance of fixed energy and volume, S = S(U,V), is S = k ln W, where W is the number of states of equal probability open to the system. For a macroscopic system with an entropy of only 1 cal/K, W is about 1010, an extremely large number. At the other extreme, a system with only two states available to it has the unbelievably low entropy of about 0.25 x 10-23 cal/K. Entropy is a help only when the system has a large W, like all macroscopic systems made up of molecules. It is not a substance, does not flow, and cannot be isolated. The expression for the molar entropy of an ideal gas shows that it increases with the molar volume, as then the molecules can be spread over an even larger volume with greater randomness.

It should be clear that we could just as well take s and v as parameters as P and T. Which two parameters we choose is unimportant, just so we choose two independent parameters that can be varied in any way. Whichever pair we choose, the other two are uniquely determined. We can now write the Phase Rule for systems of one phase and one component as f = 2. There is not much of a surprise here, but we will be able to generalize this expression.

Let's make a little excursion here to see what the thermodynamic conditions are for equilibrium. We are specially interested in a system held at a fixed pressure P and temperature T, typical of laboratory measurements and easy to imagine. The processes that could take place might be a chemical reaction in which the components rearrange themselves chemically, electrolysis with work done electrically, or a change of phase, say from liquid to vapor. We shall assume that Pdv alone represents the work done by the system, for simplicity. Define a function g = u + Pv - Ts, where we have used molar quantities. This is called the Gibbs function. Then, dg = du + Pdv + vdP - Tds - sdT = vdP - sdT. In any process at constant P and T, then, we must have dg = 0. The Gibbs functions of the initial and final states when such a change is made reversibly must be equal, or g(f) = g(i). Equations like this can be written for each pair of phases in equilibrium, and these were some of the equations used by Gibbs in his deduction of the phase rule.

We also find that (∂v/∂T)P = -(∂s/∂P)T. This is one of Maxwell's relations that appear to drop out of the sky on us in thermodynamics. They are no more than the mathematical consequences of our definition of certain functions.

If we wish to consider equilibrium at constant temperature and volume instead, then the function we need is F = U - TS, or f = u - Ts, per mole. F is the free energy. Then, df = -sdT - Pdv, and the free energies must be equal in final and intial states, f(f) = f(i). If we go a little farther and consider irreversible processes, we find that the conditions on the Gibbs function and the free energy are that they assume a minimum in equilibrium. The Maxwell relation that falls on us here is (∂s/∂v)T = (part;P/∂T)v.

The functions F and G both contain the term TS, which is the reason why absolute entropies are important. Any arbitrary addition So to the energy of any component would change F and G by -TSo, which would not be a constant (and thereby unimportant to F and G) but would depend on T, and affect the equilibrium. Arbitrary additions to the internal energies of the various phases would have no effect, because of the conservation of energy and the absence of the factor T.

While we are on a roll, we might as well consider the internal energy U, where du = Tds - Pdv, an expression of the First and Second Laws of Thermodynamics, all wrapped up in one package. In a process at constant entropy and volume, the energy is conserved: u(f) = u(i). Furthermore, when dv = 0, du = cvdT = Tds = du, or ds = cv/T, and cv = (∂u/∂T)v. Maxwell's relation for this function is (part;T/∂v)s = (part;P/∂s)v.

That leaves us with only H = U + PV, called the enthalpy. In this case, dh = du + Pdv + vdP = Tds + vdP. This is conserved for processes at constant entropy and pressure, h(f) = h(i). If we heat the system at constant pressure, then dq = cPdT = Tds = dh, and cP = (∂h/∂T)P. The Maxwell's relation is (part;T/∂P)s = (∂v/∂s)P.

Also, ds = (∂s/dT)vdT + (∂s/∂v)Tdv. Using the definition for cv and Maxwell's relation from the free energy, we find Tds = cvdT + T(∂P/∂T)vdv. Doing the same thing for T and P instead of T and v, we get ds = (∂s/∂T)PdT + (∂s/∂P)TdP, or Tds = cPdT - T(∂v/∂T)PdP. In both equations, the second terms involve only derivatives of the equation of state.

Much use can be made of the functions and formulas we have just derived in relating the thermodynamic functions of a substance. They may seem complicated, but really are just mathematical deductions with little physics in them.

Systems With Multiple Phases

Now let us consider a more general system of c components and φ phases. In 1875, Josiah Willard Gibbs, by considering the conditions for equilibrium in such a system, and the number of indpendent variables necessary to specify the state of every phase, found the Phase Rule to be f = c - φ + 2. This agrees with our result, but is capable of extension to system where we could not guess the result. We should mention here that we consider systems that can do work only by PdV work--pressing on a piston. There are many other ways, such as electrical or magnetic, and the phase rule can be extended to them. Here, PdV work will serve as representing all such forms of work.

We know that H2O can exist as a vapor, steam, a liquid, water, and a solid, ice. Each substance forms a phase with a single constituent, and when it alone is present, the system has variance 2, and the state can be specified by two parameters. Let's use T and P for the sake of argument, and draw a diagram with these parameters as axes. It is easy to identify regions where each type of phase exists, and there must be conditions under which both exist at the same time. When this is the case, the Phase Rule says that f = 1 - 2 + 2 = 1. Therefore, we can at most specify one parameter. Suppose it is T. Then once we have fixed T, P and v are also determined for each phase. This means that two phases can only exist along a line in the diagram. This is no small prediction! It means that there are no areas where two phases can exist. On the line where they both exist, the pressure P is common to both, but the specific volumes v can be (and usually are) different. We also know that when water changes to steam, a large amount of energy is required (around 540 cal/g), the latent heat L. This heat is transferred into the steam at a constant temperature, raising its entropy by Δs = L/T while its specific volume increases by Δv = v(s) - v(w). A phase transition with a jump in entropy and specific volume is called a first-order phase transition.

The same thing takes place for all three of the possible changes, water→steam (vaporization), water→ice (freezing), and ice→steam (sublimation), and so there are three corresponding lines on the phase diagram. Now, should all three be present, the Phase Rule gives f = 1 - 3 + 2 = 0. This means that this can occur only at some one point, called the triple point. For water, this is at T = 0.01°C and P = 4.58 mmHg, and we have no choice in the matter. Furthermore, this point must lie on all three of the lines where two phases are present, so these three lines must meet at the triple point. There are no exceptions to this prediction by the Phase Rule.

The slopes dP/dT of these lines where phase changes take place are expressed by Clapeyron's Equation. One derivation uses the first TdS equation we found above. For dT = 0, it gives Tds = T(∂P/∂T)vdv. In the present case, (∂P/∂T)v = dP/dT for both phases, Tds = TΔs = L, and dv = Δv, so we have dP/dT = L/TΔv. In going from water to steam, both L and Δv are positive, while from steam to water, they are both negative. Therefore, dP/dT > 0. In going from water to ice, L is negative, but Δv is positive for water, so dP/dT < 0. Normally, dP/dT is positive for freezing as well. Another, possibly better, derivation uses the fact that g(f) = g(i) at both T and T + dT (two neighboring points on the curve), so that dg(f) = dg(i) as well. Using the expression for dg when T changes by dT and P by dP gives Clapeyron's Equation as well. All this is illustrated in the diagram at the right, which is a schematic diagram of the phase diagram, or projection of the PVT surface, of water. The specific volume v can be considered as out of the page, and each of the lines is an edge-on view of a surface. Areas are regions of f = 2, lines of f = 1, and the triple point of f = 0. The critical point is just the place where the difference of specific volumes of water and steam becomes zero, as does the latent heat. Beyond this point, the two phases cannot be distinguished and become a single phase.

Binary Systems

Now we consider systems with two constituents. If these are mixtures of two metals, they are called binary alloys. The concentration of the constituents can be specified by one number, the mole or weight fraction of one component. If the constituents do not react, then their concentrations remain constant for any given system. We still have the pressure P and temperature T, which are common to all phases. Therefore, a phase has variance 3 in general, since the Phase Rule gives f = 2 - 1 + 2 = 3. A three-dimensional model would be required to depict the state, and one phase would correspond to a volume. When we are dealing with metal alloys, the pressure is of little importance and is usually constant, so we can omit it in the diagram. The result is a two-dimensional phase diagram, as for a single substance, where the coordinates are the temperature T and the concentration c. We must remember that the pressure and the specific volume point in other directions. The Phase Rule becomes f = c - φ + 1 if we leave the pressure out of consideration. Then, for a single phase, f = 2 - 1 + 1 = 2, which are the temperature and the concentration.

A schematic phase diagram for a system in which the constituents are completely miscible in both the liquid and the solid phases is shown at the left. We may consider that it represents the copper-nickel system, which has this kind of phase diagram. Both copper and nickel crystallize in the face-centered cubic system, with lattice constants a = 0.3608 nm for Cu and a = 0.3517 nm for Ni. The two ions are very close to the same size, as well. Copper donates one electron per ion, and nickel two, so the electron density increases as we go from pure copper to pure nickel. Therefore, copper and nickel can replace one another in the lattice quite freely, and the energy difference is reflected in the difference in melting points, which shows that nickel is more tightly bound, as expected.

In the regions marked "liquid" and "α" there is one phase, and so a variance of 2. The region between the liquid and the solid is bounded by two lines, and within this region there are two phases, the liquid metal and the solid. The Phase Rule then gives f = 2 - 2 + 1 = 1, and there is only one degree of freedom. For any temperature, the concentrations of liquid and solid are determined by the ends of a horizontal tie line. The concentration in the liquid is given by the liquidus, and the concentration in the solid by the solidus, and these are not the same. Let us suppose we have a liquid of concentration c, and cool it slowly from point A to point B, where it has solidified. The overall concentration of the system cannot change, of course, and must follow the vertical line. When we reach point a', that is, of course, the concentration of the liquid. The first solid to separate is at point b', much richer in nickel than the liquid. If we cool the system a little more, the liquid is now at concentration a, while the solid that is separating is at concentration b. Further cooling gives points a" and b", connected by a tie line.

Note that solid b is poorer in nickel than solid b' that was deposited earlier, while liquid a is richer in copper than the original liquid at a'. In order for this to happen, the solid must have gained some copper and lost some nickel, while the opposite has happened to the liquid. Usually, this must happen by diffusion, which can be slow in a solid. If insufficient time is taken to reach equilibrium, then the deposited metal becomes successively richer in copper, as does the liquid, since nickel solidified preferentially at first. The average concentration remains constant, but the metals have been separated somewhat. The solid shows cores rich in nickel, the outsides rich in copper. If diffusion is allowed to do its work, the composition in the solid phase becomes uniform.

Note that every time a phase boundary is crossed, the number of phases changes by 1. On cooling, the phases increase from 1 to 2 at the liquidus, and back from 2 to 1 at the solidus. The difference in concentrations in the regions of two phases is like the difference in specific volume in the regions of two phases in the PVT surface of a single substance. In our PT diagram for water we did not see any such areas, but they would appear in a Pv diagram, where the surface is seen from another direction.

It is more common to have systems of two metals that are miscible in the liquid state, but crystallize in different systems in the solid state, or have ions significantly different in size, so they cannot occupy the same lattice sites without considerable strain. Usually each crystal lattice can accommodate a few foreign ions and make a homogeneous solid solution phase, but when the concentration of the foreign ions reaches a certain limit that depends on the temperature, the phase may refuse to accept any more. Then we have a liquid state that is one phase, while the solid state is a two-phase mixture of the terminal phases. This happens in the tin-lead system. The terminal tin phase, α, can hold less that 3% lead, while the terminal lead phase, β, can hold no more than 20%, and that at an elevated temperature. The solid must, in general, be a mixture of the two phases, nearly pure tin, and nearly pure lead. After a while, they may diffuse into one another to approach homogeneity.

The phase diagram for the tin-lead system is shown schematically at the right. Lead melts at 327°C, and tin at 232°C. Adding one to the other lowers the melting point in either case. This is largely a result of the increase in entropy produced by the mixing. There are three possible phases, α, β and the liquid. The Phase Rule says that f = 2 - 3 + 1 = 0, so they can all three be present only at one point, which is point E. This point is at 183°C and a lead concentration of 38% by weight. It is called a eutectic, and represents the minimum melting point of the system. Three large regions correspond to areas with two phases: liquid + α, liquid + β, and α + β. The liquidus is composed of the two upper lines meeting at E. The solidus is represented by the horizontal line through E, that is terminated by triple points at each end, together with the lower lines from the melting points. The final two lines represent the limits of stability of the terminal phases.

Suppose a tin-rich liquid solidifies rather quickly. The first metal to solidify is nearly pure tin α phase. This makes the liquid richer in lead, so its concentration moves to the right. This continues until 183°C is reached. Then we have α phase solid, and a liquid of the eutectic composition. Now this liquid solidifies, depositing characteristic thin layers of alternating α and β phases as it departs from equilibrium one way and then the other, correcting itself every time. If we annealed the solid and gave time for diffusion to be complete, everything would be uniform at the original composition.

The liquid begins to solidify as soon as the liquidus is reached. Past this point, we have some solid, and some liquid, which are mixed like ice and water. If we are close to pure tin or pure lead, the liquid solidifies to the corresponding terminal phase, plus a little of the other. We can reach a phase supersaturated in the other constituent, which may then precipitate as small crystals. At the eutectic, nothing happens until we reach 183°C. Then this temperature remains constant until all the metal has solidified, as the characteristic eutectic mixture.

Another behavior that is seen in some important alloys is that the crystal structure of the solid solution changes with composition. This is seen in brass, the Cu-Zn alloy, and bronze, the Cu-Sn alloy. For discussions of this important case, see Zinc and Tin. In brass, we also have the interesting phenomenon of an order-disorder transition, which is a second-order transition without a discontinuity in the entropy and specific volume.

Ice and Salt

Alloys are not the only binary systems for which phase diagrams are useful. A portion of the water-salt phase diagram is shown at the right below. At the left is pure water, in this case ice below 0°C. On the right, but not shown, is pure salt, NaCl above 0.15°C and NaCl·H2O below. Below the horizontal line at -21°C is solid ice + salt. We presume that ice and salt are immiscible, so that in the solid state neither includes any of the other, which is a good approximation. Above the horizontal line, there are three regions, the middle of which represents a homogeneous solution phase, the other two mixtures of solution with solid ice and solid salt. The point E is a eutectic, where three phases are present, so that f = 0. The vertical line above point B represents a saturated solution of salt in water. The solubility of salt does not depend much on the temperature. More accurate figures from a handbook give a solubility of 35.7% at 0°C, 39.8% at 100°C. The heat of solution of salt is -1180 cal/mole or -20.2 cal/g. The minus sign indicates that heat is absorbed when salt dissolves. Since this happens at constant pressure here, it is an enthalpy change, ΔH.

The latent heat of fusion of ice, another enthalpy change, is 79.67 cal/g. This means that when a gram of ice melts to water, the heat required could cool that water to -80°C. In equilibrium, of course, the water cannot cool and the heat must come from outside, say from the drink the ice cube floats in. This is the familiar property of ice in maintaining a low temperature, which it does effectively by the large latent heat. The ice will get your drink down to 0°C or die trying. Usually, it dies trying and melts continuously no matter how well you have insulated the glass. A mixture of crushed ice and water can usually get down to 0°C and provide a fixed reference temperature for a thermocouple. This has to be done properly, for water has maximum density at 3.8°C, and the ice is usually floating on 3.8° water (the increase in density is very slight, however, only about 1 part in 10,000). The reference junction must be kept surrounded by ice. Incidentally, the equivalent of a reference junction can be supplied in a fairly accurate and drier fashion with electronics.

So we see that ice is a powerful cooling engine because of its latent heat, but can only go to 0°C, which is not enough to freeze things. Also, ice is not always welcome in the winter, and it would be good to have some way to melt it so the water would flow off and not make things slippery. Both demands can be met with the help of salt, NaCl. Suppose the temperature corresponds to point c, and we sprinkle salt on the ice. The ice melts, by means of the heat supplied by the environment, and some salt solution is formed in the puddle, because the ice is not in equilibrium with all that salt. The salt solution becomes more concentrated as the salt lying around dissolves in the water from the melting ice. The ice tries to dilute the solution, while the salt wants to concentrate it, and when point a is reached, all the ice has melted and we have a salt solution only. If more salt is present, the solution becomes more concentrated and may reach point b, where the solution is saturated, and any more salt would just precipitate out. Therefore, the system ends up beyond b, with saturated solution and some solid salt, waiting for more ice.

Now suppose we mix ice and salt on purpose, to get a freezing mixture. The environment is not a heat reservoir now. In fact, we make insulate it so that it must supply all the latent heat if the ice is to melt. Ice and solution are both present, so we must be in the corresponding region to the left of line AE in the phase diagram. Suppose the solution has a concentration corresponding to point a. If a little more salt dissolves, the concentration may correspond to point e, to the right of the equilibrium concentration, so ice dissolves to dilute the solution. The provision of the latent heat required cools the system to point f, again on the equilibrium curve, but at a lower temperature. Therefore, with ice, salt and solution all present, there cannot be an equilibrium until point E is reached, where equilibrium can be established and maintained while all three phases are present. The concentration at this point is 23% (some sources say 29%), from which the amount of salt required can be estimated. The temperature is -21°C or -5.8°F, which is sufficient to freeze ice cream.

The melting of ice extracts most of the heat, but the dissolving of salt contributes its part as well. Magnesium chloride forms a eutectic at -33.6°C (-28°F) and 21.6%, and calcium chloride at -55°C (-67°F) and 29.8%. Calcium chloride can be used to freeze mercury at -38°C. The hydrated salts should be used for making freezing mixtures, and everything should be kept cold, insulated, and mixed.

Aqueous solutions of alcohol, and other substances, such as ethylene glycol, depress the freezing points of the solutions, as is well-known from motor-vehicle antifreezes. These can make freezing mixtures and melt ice as well. I made a skull-and-crossbones Martini from denatured alcohol and ice cubes, and got it down to -6°C in a few minutes. A 30% ethyl alcohol solution, with a gravity of 0.954, gets you down to -19°C (-2°F). Glycerine gives a minimum at -37.8°C, in a 70% glycerine solution.

Carbon dioxide sublimates at -78.5°C. Carbon dioxide snow, compressed into blocks, is "dry ice" which is drier than freezing mixtures. It is often mixed with liquids like alcohol, acetone, chloroform or ether to make a cooling bath. The temperature, however is not lower than the sublimation temperature of CO2, and no eutectic is formed. Liquid nitrogen boils at -196°C and is convenient to use, leaving no clean-up. With all these means at one's disposal, temperatures down to 77K are available without machinery.


Sometimes the constituents of a mixture may react to form a new substance, which has significant effects on the phase diagram. When carbon is dissolved in iron, the hard, brittle, white compound cementite, Fe3C, may be formed. From the atomic weights of Fe and C, 55.85 and 12.00, respectively, we can find that cementite corresponds to 6.7% C by weight. Cementite melts at 1837°C and has a density of 7.4 g/cc. On the phase diagram, cementite corresponds to a vertical line at 6.7% C, and divides the phase diagrams into two independent sections, one with less carbon, the other with more. Engineering is interested only in the section with less carbon.

Pure iron has a body-centered cubic crystal structure at room temperature, called α-iron or ferrite. At 910°C the structure changes to face-centered cubic, and this material is called γ-iron or austenite. At 1403°C, the structure reverts to bcc, and the material is called δ-iron, though it is really the same as α-iron. At 1535°C, the iron melts. Austenite is much softer and more easily worked than ferrite, which explains the blacksmith's heating of iron to red heat for working it.

These things are reflected at the left and right boundaries of the iron-cementite phase diagram, which is sketched at the left. The actual lines are more or less curved, not straight, and some of the areas have been magnified for clarity. Only the part of the iron-carbon phase diagram to the left of cementite has been shown, since it is the only area of technical interest. The plain areas are those in which there is only one phase, for which the Phase Rule gives f = 2 - 1 + 1 = 2, and the dotted areas are those with two phases, where f = 2 - 2 + 1 = 1. For any temperature, the concentrations of the two phases present can be read off at the ends of a horizontal tie line. The names of the phases are given in all cases. They are ferrite, austenite, δ-iron, perlite, ledeburite and cementite. Only the first three occupy regions with f = 2; the final three are of definite composition.

Point E is easily recognized as a eutectic. Its temperature is the lowest fusing temperature of an iron-carbon mixture, 1130°C. A eutectic liquid will solidify to ledeburite, a typical layered eutectic solid of austenite and cementite. A liquid to the left of E will solidify to austenite + ledeburite, while one to the right will solidify to cementite + ledeburite. At 723°C, there will be a further change to ferrite + cementite. These changes in the solid state are not rapid. Any material whose composition is to the right of point B (greater than 1.7% carbon) is called cast iron. When iron is reduced in the liquid state with carbon, cast iron is the usual result. In early times, when iron men were trying to get a low-carbon iron by directly reducing the iron ore with charcoal, their work would be ruined if the furnace temperature rose above 1130°C, and the iron absorbed carbon and melted. The cast iron could not be hammered or worked; it was just a hard, disappointing, inedible, useless lump. Using cast iron to make castings was a much later development, since for several reasons iron is not as easy to cast, as bronze is.

The blast furnace just goes ahead and melts the iron in the interests of rapid mass production, so it comes out with 3% or 4% carbon, which forms cementite and graphite on cooling. This is obviously not an equilibrium phase, but is quite stable. The iron was once led through channels into ingot moulds dug in the cinders of the casting floor, which suggested a sow and her piglets, and gave the name "pig iron" to the product. It is produced because methods of removing the excess carbon are available, first the puddling furnace that made wrought iron, and later the Bessemer converter, open-hearth furnace, and basic oxygen converter. There is quite an interesting history, for which the reader is referred to books on the history of technology.

Any material with a composition to the left of B is called a steel. For such a material, heating will cause the steel to become a uniform austenite phase. This is a lucky circumstance that is responsible for the ability to heat-treat steel to suit its application. A steel of composition corresponding to point D will become a layered mixture of ferrite and cementite of average composition 0.83% carbon, called perlite. It is so-called because it can have a pearly lustre in the microscope, a play of colors produced by the interference of light in its regular layers, which act as a diffraction grating. This part of the phase diagram looks just like a eutectic, except that the upper material is solid austenite, not a liquid. It is called a eutectoid, "like a eutectic" though there is no melting involved. A steel that at room temperature is a structural mess of different phases and crystals, but of the average composition of the eutectoid, will, on heating above 723°C, become a uniform solid solution after a while, that will cool to a nice pearlite structure throughout. This normalization can be applied to any steel, and does not require melting. The steel must be heated to a high enough temperature that it all becomes austenite, however--a temperature that is certainly below 1130°C.

There is a funny bit at high temperatures near the δ-iron region. This looks like an inverted eutectoid, but it isn't a eutectoid. If we consider a melt with a composition just to the left of point C, δ-iron will freeze out and the liquid will become richer in carbon. When the temperature of C is reached, the carbon-rich liquid will dissolve some of the carbon-poor δ-iron until all the liquid freezes to a mixture of δ-iron and austenite of the composition of point C. Further cooling will decrease the amount of δ-iron and increase the amount of austenite until the solid phase is uniform austenite. What happens at point C is called a peritectic reaction, as δ-iron is dissolved and austenite formed. The carbon-cementite phase diagram illustrates most of the interesting things that can happen!

Austenite cooled quickly does not become a tough ferrite-pearlite mixture, but a difficult structure called martensite that is very hard and brittle. It is as if the carbon, trying to escape as the bcc crystals of austenite shift to fcc, are trapped in the attempt, and hinder the structure change by wedging themselves in and preventing the formation of a stress-free crystal. If you are making a sword or a knife, this may be just what you want. The steel is first cooled slowly, so ferrite and pearlite are formed, and the item can be shaped and worked. Then it is heated above the transition temperature to austenite, and finally quenched by plunging it into water, or into the body of a slave on the advice of local priests. The temper can be drawn by gentle heating to allow some of the martensite to transform into ferrite, and toughen the object. In most cases, the hard temper is only on the surface anyway, since the interior cannot be quickly quenched. The possibility of heat treatment makes steel a far superior metal for tools and weapons than any other available alloy.

Ancient iron may well have contained sufficient dissolved carbon to be considered at least very mild steel, perhaps 0.1% carbon, without any special handling. Austenite does dissolve carbon, though the process would be slow. Such steel would harden and temper. Pure iron, which was seldom produced or desired except for ornaments, would be too soft for tools and weapons. Higher concentrations of carbon can be attained by packing the iron with charcoal in a closed crucible, and heating for a good while. This, apparently, was done in India at an early date, and the resulting "wootz" was obtained by the West in trade and used for making superior weapons, as at Damascus. The Huntsman crucible process, developed in Sheffield, was similar. Steel with 1% carbon, good for making cutlery, was an expensive material that could only be used for high-value objects. Cheap steel did not arrive until the Bessemer process of burning the carbon out of cast iron with an air blast was invented around 1855.

The phase diagram had no region for martensite, because martensite is not an equilibrium phase. This indicates that many important properties may not be described by the phase diagram. Another example are the effects of small amounts of impurities. Bessemer iron contained amounts of oxygen and nitrogen that were detrimental. The oxygen forms oxides that concentrate on the grain boundaries and weaken them. The iron had to be deoxidized, by adding silicon or manganese, before it was satisfactory. These impurities are beneficial even in larger amounts, silicon in magnetic alloys and manganese in steel that will work-harden and become more durable. Sulphur and phosphorus, often found in ores or coke, are extremely detrimental in any quantity. Phosphorus forms a brittle compound with iron that collects on grain boundaries and makes the steel brittle and subject to cracking when cold. Sulphur forms compounds that are fusible, and lubricate the grains so the steel crumbles when an attempt is made to hot-roll it, called hot shortness. The control of impurities governs most iron and steel processes.


A. Findlay, The Phase Rule and its Applications, 8th ed. (London: Pitman, 1938). The classic reference to applications of the phase rule.

P. W. Atkins, The Second Law (New York: W. H. Freeman, 1984). Atkins, Morse and Kittel understand thermodynamics, while most engineering authors can do the math, but have little understanding of what is going on. This book by Atkins is a valuable gem. Since it is a Freeman product, there is no mathematics, but Atkins manages an appendix with equations. It shows very well what thermodynamics says about chemical reactions.

P. M. Morse, Thermal Physics (New York: W. A. Benjamin, 1962).

C. Kittel, Thermal Physics (New York: John Wiley & Sons, 1969).

M. W. Zemansky, Heat and Thermodynamics (New York: McGraw-Hill, 1957). A solid, useful text of the classical kind. It considers only continuuum thermodynamics, so physical insights are few. In the treatment of freezing mixtures, what does the freezing is never mentioned, just "automatic cooling." Nevertheless, it is largely correct, which is a benefit.

R. B. Leighou, Chemistry of Engineering Materials, 4th ed. (New York: McGraw-Hill, 1942). pp. 205-222. Cu-Zn p. 228, Cu-Sn p. 234, Cu-Ni p. 215, Fe-C p. 315, Pb-Sb and Ag-Pt p. 219, Cu-Au p. 218, Ca-Mg and Au-Pb p. 212, Bi-Cd p. 210.

R. A. Higgins, Engineering Metallurgy, 3rd ed. (London: The English Universities Press, 1971). pp. 161-184. Bi-Cd p. 166, Cu-Ni p. 168, Cu-Zn p. 317, Cu-Sn p. 323, Cu-Be and Cu-Cr, p. 336, Sn-Pb p. 171, Ag-Pt p. 173, Pb-Mg p. 175, Fe-C pp. 205, 208, 384, Al-Si p. 345, Al-Cu p. 348, Mg-Al, Mg-Zn, Mg-Th p. 361, Sn-Sb p. 369, Pb-Sn p. 401.

T. K. Derry and T. I. Williams, A Short History of Technology (London: Oxford University Press, 1960). Chapters 4 and 16.

N. A. Lange, Handbook of Chemistry, 10th ed. (New York: McGraw-Hill, 1961). Freezing and antifreeze mixtures are listed on p. 1191-1195.

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Composed by J. B. Calvert
Created 21 November 2002
Last revised