In thermodynamics, the Maxwell construction refers to a set of geometrical instructions that modify a given constant temperature curve (isotherm) to produce its experimentally observed vapor-liquid phase transition section. The isotherm is usually generated by an equation of state. The method was first presented by James Clerk Maxwell in an 1875 lecture to the Chemical Society in London, and subsequently published in Nature. Maxwell used it in connection with the isotherms of the van der Waals equation to describe its phase change, in particular its vapor pressure, the liquid and vapor states that are its extremes, and the temperature dependence of these quantities. Simply stated, the Maxwell construction produces the horizontal (constant pressure) line between points B and F on the isotherm, shown dashed in Fig. 1 below. This line is the one for which the two areas, I and II, shown in the figure, are equal. Hence, it is also known as the equal area rule. A few years later, Josiah Willard Gibbs showed that the Maxwell construction was equivalent to the condition of material equilibrium given by the equality of the electrochemical potential of the two phases. As such, Gibbs' formulation is more fundamental than Maxwell's, but due to the ease with which areas could be measured with a planimeter, the equal area rule continued to be widely used for many years. Its use has declined in the present age of digital computers, which can perform complex computations rapidly; however, due to its easily understood physical basis, the Maxwell construction is still discussed whenever phase transitions are studied.
Stability criteria In thermodynamic equilibrium, a necessary condition for stability is that pressure, p {\displaystyle p} , does not increase with volume, or molar volume, v = V / N {\displaystyle v=V/N} , in which N = N p / N A {\displaystyle N=N_{p}/N_{A}} is the number of moles of the substance; this is expressed mathematically as ∂ v p | T < 0 {\displaystyle \partial _{v}p|_{T}<0} , where T {\displaystyle T} is the temperature. This basic stability requirement, and similar ones for other conjugate pairs of variables, is violated in analytic models of first-order phase transitions. The most famous case is the van der Waals equation,
p = R T / ( v − b ) − a / v 2 {\displaystyle p=RT/(v-b)-a/v^{2}}
where a , b , R {\displaystyle a,b,R} are dimensional constants. This violation is not a defect; rather, it is the origin of the observed discontinuity in properties that distinguishes a liquid from a vapor, and defines a first-order phase transition.
Figure 1 shows an isotherm drawn for v > b {\displaystyle v>b} , as a continuously differentiable solid black, dotted black, and dashed gray curve. The decreasing part of the curve to the right of point C in Fig. 1 describes a gas, while the decreasing part to the left of point E describes a liquid. These two parts are separated by a region between the local minimum and local maximum on the curve with a positive slope that violates the stability criterion. This mathematical criterion expresses a physical condition which Epstein described as follows: "It is obvious that this middle part, dotted in our curves [dashed in Fig.1 here], can have no physical reality. In fact, let us imagine the fluid in a state corresponding to this part of the curve contained in a heat-conducting vertical cylinder whose top is formed by a piston. The piston can slide up and down in the cylinder, and we put on it a load exactly balancing the pressure of the gas. If we take a little weight off the piston, there will no longer be equilibrium, and it will begin to move upward. However, as it moves, the volume of the gas increases and with it its pressure. The resultant force on the piston gets larger, retaining its upward direction. The piston will, therefore, continue to move and the gas to expand until it reaches the state represented by the maximum of the isotherm. Vice versa, if we add ever so little to the load of the balanced piston, the gas will collapse to the state corresponding to the minimum of the isotherm." This situation is similar to a body exactly balanced at the top of a smooth surface that, with the slightest disturbance, will depart from its equilibrium position and continue until it reaches a local minimum. As they are described, such states are dynamically unstable, and consequently, they are not observed. The gap v m i n ≤ v ≤ v m a x {\displaystyle v_{\rm {min}}\leq v\leq v_{\rm {max}}} is a precursor of the actual phase change from liquid to vapor. The points E ( p m i n , v m i n ) {\displaystyle (p_{\rm {min}},v_{\rm {min}})} and C ( p m a x , v m a x ) {\displaystyle (p_{\rm {max}},v_{\rm {max}})} , where ∂ v p | T = 0 {\displaystyle \partial _{v}p|_{T}=0} , that delimit the largest possible liquid and smallest possible vapor states are called spinodal points. Their locus forms a spinodal curve that bounds a region where no homogeneous stable states can exist. Experiments show that if the volume of a vessel containing a fixed amount of liquid is heated and expands at constant temperature, at a certain pressure, p s ( T ) {\displaystyle p_{s}(T)} , vapor, (denoted by dots at points f {\displaystyle f} and g {\displaystyle g} in Fig. 1) bubbles nucleate so the fluid is no longer homogeneous, but rather it has become a heterogeneous mixture of boiling liquid and condensing vapor. Gravity separates the boiling (saturated) liquid, v f = V f / N f {\displaystyle v_{f}=V_{f}/N_{f}} , from the less dense condensing (saturated) vapor, v g = V g / N g > v f , {\displaystyle v_{\text{g}}=V_{\text{g}}/N_{\text{g}}>v_{f},} that coexist at the same saturation temperature and pressure. As the heating continues, the amount of vapor, N g {\displaystyle N_{\text{g}}} , increases, and that of the liquid, N f = N − N g {\displaystyle N_{f}=N-N_{\text{g}}} , decreases. All the while the pressure, p s {\displaystyle p_{s}} and temperature, T {\displaystyle T} , remain constant and the volume V = V f + V g {\displaystyle V=V_{f}+V_{\text{g}}} increases. In this situation, the molar volume of the mixture is a weighted average of its components
v = V / N = ( V f / N f ) ( N − N g ) / N + ( V g / N g ) ( N g / N ) = v f ( 1 − x ) + v g x {\displaystyle v=V/N=(V_{f}/N_{f})(N-N_{\text{g}})/N+(V_{\text{g}}/N_{\text{g}})(N_{\text{g}}/N)=v_{f}(1-x)+v_{\text{g}}x}
where x = N g / N {\displaystyle x=N_{\text{g}}/N} , the mole fraction of the vapor, 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} , increases continuously; however, the molar volume of the substance itself has only the largest possible stable value for its liquid state, and smallest possible stable value for its vapor state at the given p ( T ) {\displaystyle p(T)} . To repeat, although the mixture molar volume passes continuously from v f {\displaystyle v_{f}} to v g {\displaystyle v_{\text{g}}} (denoted by the dashed line in Fig. 1), the underlying fluid has a discontinuity in this property, and others as well. This equation of state of the mixture is called the lever rule.
The dotted parts of the curve in Fig. 1 are metastable states. For many years, such states were an academic curiosity; Callen gave as an example, "water that has been cooled below 0°C at a pressure of 1 atm. A tap on a beaker of water in this condition precipitates a sudden, dramatic crystallization of the system." However, studies of boiling heat transfer have made clear that metastable states occur routinely as an integral part of this process. In it the heating surface temperature is higher than the saturation temperature, often significantly so, hence the adjacent liquid must be superheated. Further, the advent of devices that operate with very high heat fluxes has created interest in the metastable states, and the thermodynamic properties associated with them, in particular the superheated liquid states. Moreover, the fact that they are predicted by the van der Waals equation, and cubic equations in general, is compelling evidence of its efficacy in describing phase transitions; Sommerfeld described this as follows:It is very remarkable that the theory due to van der Waals is in a position to predict, at least qualitatively, the existence of the unstable [called metastable here] states along the branches AA′ or BB′ [BC and FE in Fig. 1 here].
Equal area rule The discontinuity in v {\displaystyle v} , and other properties, e.g. internal energy, u {\displaystyle u} , and entropy, s {\displaystyle s} , of the substance, is called a first-order phase transition. To specify the unique experimentally observed pressure, p s ( T ) {\displaystyle p_{s}(T)} , at which it occurs another thermodynamic condition is required, for from Fig.1 it could clearly occur for any pressure in the range p m i n ≤ p ≤ p m a x {\displaystyle p_{\rm {min}}\leq p\leq p_{\rm {max}}} . Such a condition was first enunciated in a clever thermodynamic argument by Maxwell at a lecture he delivered to the British Chemical Society on Feb 18, 1875 (Fig.1, including the letters B C D E F, is the curve he described):
The portion of the curve from C to E represents points which are essentially unstable, and which cannot therefore be realized. Now let us suppose the medium to pass from B to F along the hypothetical curve B C D E F in a state always homogeneous, and to return along the straight line path F B in the form of a mixture of liquid and vapor. Since the temperature has been constant throughout, no heat can have been transformed into work. Now the heat transformed into work is represented by the excess of the area F D E over B C D. Hence, the condition which determines the maximum pressure of the vapor at given temperature is that the line B F cuts off equal areas from the curve above and below.
On a temperature--molar entropy, T {\displaystyle T} - s {\displaystyle s} , plane, the area under any curve is the heat transfer to the substance per mole, positive going from left to right and negative from right to left; moreover, in a cyclic process, the net heat transfer to the substance is the area enclosed by the cycle's closed curve. Since the cycle Maxwell considered is composed of the two gray dashed isothermals at the same temperature, on the T {\displaystyle T} - s {\displaystyle s} plane they appear as two horizontal lines, one proceeding from B to F (through C D and E) and the other directly back from F to B, at the same T {\displaystyle T} . Hence, the two lines are identical, just traversed in reverse; there is zero area enclosed, and thus q = 0 {\displaystyle q=0} . Furthermore, the area under these curves when plotted on a pressure—molar volume, p {\displaystyle p} - v {\displaystyle v} , plane (see Fig. 1), is the work done by the substance, positive going from left to right, and negative from right to left. Likewise, the net work done in a cycle is the area enclosed by the closed curve. Since the first law of thermodynamics yields in the special case of a cycle w = q {\displaystyle w=q} , for the cycle envisioned by Maxwell w = q = 0 {\displaystyle w=q=0} ; then since the area enclosed is I + II = 0, see Fig.1, with I positive and II negative, the transition pressure must be such that the two areas are equal. Written as a mathematical equation in terms of the work done in each process, this is
∫ v g v f p d v + ∫ v f v g p s d v = − ∫ v f v g p d v + p s ( v g − v f ) = 0 for T = constant {\displaystyle \int _{v_{\text{g}}}^{v_{f}}\,p\,dv+\int _{v_{f}}^{v_{\text{g}}}\,p_{s}\,dv=-\int _{v_{f}}^{v_{\text{g}}}\,p\,dv+p_{s}(v_{\text{g}}-v_{f})=0\quad {\mbox{for}}\quad T={\mbox{constant}}}
This equation, together with the equation of state written for each of the states f {\displaystyle f} and g {\displaystyle g}
p s = p ( v f , T ) p s = p ( v g , T ) {\displaystyle p_{s}=p(v_{f},T)\qquad p_{s}=p(v_{\text{g}},T)}
are three equations for the four variables, p s , T , v f , v g {\displaystyle p_{s},T,v_{f},v_{\text{g}}} , so given any one of them, say T {\displaystyle T} , the other three are determined. In other words, there is a unique value of p s ( T ) {\displaystyle p_{s}(T)} , as well as v f ( T ) {\displaystyle v_{f}(T)} and v g ( T ) {\displaystyle v_{\text{g}}(T)} , at which the phase transition can occur.
Gibbs criterion At the end of his lecture, after complimenting van der Waals by referring to his work as "an exceedingly ingenious thesis", Maxwell finished it by saying:
I must not, however, omit to mention a most important American contribution to this part of thermodynamics by Prof. Willard Gibbs of Yale College U.S., who has given us a remarkably simple and thoroughly satisfactory method of representing the relations of the different states of matter by means of a model. By means of this model, problems which had long resisted the efforts of myself and others may be solved at once.
This remark proved prescient because in 1876-1878 Gibbs published his definitive work on thermodynamics in which he showed that thermodynamic equilibrium of a heterogeneous substance requires that, in addition to mechanical equilibrium (the same pressure for each component) and thermal equilibrium (the same temperature for each component), there must also be material equilibrium (the same chemical potential for each component). In the present instance of one substance and two phases in addition to p f = p g = p s {\displaystyle p_{f}=p_{\text{g}}=p_{s}} and T f = T g = T {\displaystyle T_{f}=T_{\text{g}}=T} , material equilibrium requires g f = g g {\displaystyle g_{f}=g_{\text{g}}} (for the special case of one substance its chemical potential is the molar Gibbs function, μ = g ≡ G / N {\displaystyle \mu =g\equiv G/N} where g = u + p v − T s {\displaystyle g=u+pv-Ts} ). This condition can be deduced by a simple physical argument as follows: the energy required to vaporize a mole is from the second law at constant temperature q v a p = T ( s g − s f ) {\displaystyle q_{\rm {vap}}=T(s_{\text{g}}-s_{f})} , and from the first law at constant pressure q v a p = h g − h f {\displaystyle q_{\rm {vap}}=h_{\text{g}}-h_{f}} , then equating these two and rearranging produces the result since h = u + p v {\displaystyle h=u+pv} . The conditions of material equilibrium lead to the famous Gibbs phase rule, D = n − r + 2 {\displaystyle D=n-r+2} , where n {\displaystyle n} is the number of substances, r {\displaystyle r} the number of phases, and D {\displaystyle D} the number of independent intensive variables required to specify the state. In the case of one substance and two phases discussed here, this gives D = 1 {\displaystyle D=1} , the experimentally observed number. Now g ( p , T ) {\displaystyle g(p,T)} is a thermodynamic potential function, its differential is
d g = ∂ p g | T d p + ∂ T g | p d T = v d p − s d T {\displaystyle dg=\partial _{p}g|_{T}\,dp+\partial _{T}g|_{p}\,dT=v\,dp-s\,dT}
Integrating this at a constant temperature produces
g ( p , T ) = g A ( T ) + ∫ p A p v ( p ¯ , T ) d p ¯ {\displaystyle g(p,T)=g_{A}(T)+\int _{p_{A}}^{p}\,v({\bar {p}},T)\,d{\bar {p}}}
Here g A {\displaystyle g_{A}} is a constant of integration, but the constant is different for each isotherm, hence it is written as a function of T {\displaystyle T} . In order to evaluate g {\displaystyle g} one must invert p = p ( v , T ) {\displaystyle p=p(v,T)} to obtain v = v ( p , T ) {\displaystyle v=v(p,T)} . However, it is the nature of the phase transition phenomenon that this inversion is not unique; for example, the van der Waals equation written for v {\displaystyle v} is,
p v 3 − ( p b + R T ) v 2 + a v − a b = 0 , {\displaystyle pv^{3}-(pb+RT)v^{2}+av-ab=0,}
a cubic with either 1 or, in this case, 3 real roots. Thus, there are three curves, as seen in Fig. 2, consisting of stable (shown in solid black), metastable (shown in dotted black), and unstable (shown in dashed gray) states. Actually, the figure was not produced by solving the cubic and integrating, rather g ( v , T ) {\displaystyle g(v,T)} was obtained from its definition by first obtaining u ( v , T ) {\displaystyle u(v,T)} and s ( v , T ) {\displaystyle s(v,T)} , which is easily done analytically for the van der Waals equation, and plotting it parametrically with p ( v , T ) {\displaystyle p(v,T)} , using v {\displaystyle v} as the parameter. Considering only its stable states g ( p , T ) {\displaystyle g(p,T)} is continuous with discontinuous partial derivatives, ∂ p g | T = v {\displaystyle \partial _{p}g|_{T}=v} and ∂ T g | p = − s {\displaystyle \partial _{T}g|_{p}=-s} , at the phase transition point. In the Ehrenfest classification, a first-order phase transition refers to the discontinuity of the first partial derivatives of g {\displaystyle g} while a second-order phase transition would involve discontinuities of the second partial derivatives.
Relationship between the Gibbs and Maxwell criteria Evaluating the integral expression for g ( p , T ) {\displaystyle g(p,T)} given previously between the saturated liquid and vapor states and applying the Gibbs criterion of material equilibrium to this phase change process requires writing it as
g g − g f = ∫ p s p m i n v l d p + ∫ p m i n p m a x v u d p + ∫ p m a x p s v v d p = 0 {\displaystyle g_{\text{g}}-g_{f}=\int _{p_{s}}^{p_{\rm {min}}}\,v_{l}\,dp+\int _{p_{\rm {min}}}^{p_{\rm {max}}}\,v_{u}\,dp+\int _{p_{\rm {max}}}^{p_{s}}\,v_{v}\,dp=0}
Here the integral has been split into three parts using the three real roots of the cubic corresponding to the liquid, v l {\displaystyle v_{l}} , unstable, v u {\displaystyle v_{u}} , and vapor, v v {\displaystyle v_{v}} , states respectively. These integrals can best be visualized by viewing Fig. 1 rotated 90 ∘ {\displaystyle 90^{\circ }} counterclockwise in the paper plane then 180 ∘ {\displaystyle 180^{\circ }} about the v {\displaystyle v} axis so that v {\displaystyle v} appears on the left side ordinate of the curve as shown in the accompanying graph. In this view, the function v ( p , T ) {\displaystyle v(p,T)} clearly is multi-valued; this is the reason it requires three real functions to describe its behavior between p m i n {\displaystyle p_{\rm {min}}} and p m a x {\displaystyle p_{\rm {max}}} . Now, on splitting the middle integral into two
g g − g f = ∫ p s p m i n v l d p + ∫ p m i n p s v u d p + ∫ p s p m a x v u d p + ∫ p m a x p s v v d p = 0 {\displaystyle g_{\text{g}}-g_{f}=\int _{p_{s}}^{p_{\rm {min}}}\,v_{l}\,dp+\int _{p_{\rm {min}}}^{p_{s}}\,v_{u}\,dp+\int _{p_{s}}^{p_{\rm {max}}}\,v_{u}\,dp+\int _{p_{\rm {max}}}^{p_{s}}\,v_{v}\,dp=0}
The first two integrals here are area I, while the second two are the negative of area II. The two areas add to zero, hence their magnitudes are equal according to this Gibbs criterion. This is again the equal area rule of Maxwell, the Maxwell construction, and it can also be shown analytically. Since d ( p v ) = p d v + v d p {\displaystyle d(pv)=pdv+vdp} ,
d g = v d p − s d T = − p d v + d ( p v ) − s d T . {\displaystyle dg=v\,dp-s\,dT=-p\,dv+d(pv)-s\,dT.}
Integrating this for constant temperature from state f {\displaystyle f} to g {\displaystyle g} with the Gibbs condition produces
g g − g f = − ∫ v f v g p ( v , T ) d v + p s ( v g − v f ) = 0 {\displaystyle g_{\text{g}}-g_{f}=-\int _{v_{f}}^{v_{\text{g}}}\,p(v,T)\,dv+p_{s}(v_{\text{g}}-v_{f})=0}
which is Maxwell's result. This equal area rule can also be derived by making use of the Helmholtz free energy. In any event, the Maxwell construction derives from the Gibbs condition of material equilibrium. However, even though g f = g g {\displaystyle g_{f}=g_{\text{g}}} is more fundamental, it is more abstract than the equal area rule, which is understood geometrically.
Common tangent construction
Another method to determine the coexistence points is based on the Helmholtz potential minimum principle, which states that in a system in diathermal contact with a heat reservoir T = T R {\displaystyle T=T_{R}} , D F = 0 {\displaystyle DF=0} and D 2 F > 0 {\displaystyle D^{2}F>0} , namely at equilibrium the Helmholtz potential is a minimum. Since, like g ( p , T ) {\displaystyle g(p,T)} , the molar Helmholtz function f ( v , T ) {\displaystyle f(v,T)} is also a potential function whose differential is,
d f = ∂ v f | T d v + ∂ T f | v