Maxwell’s rules

Analysis of motions of a structure using a and n representations has many applications beyond chemistry. For example, some interesting results come from applications of these ideas to macroscopic, engineering-scale structures. In stmctural mechanics. Maxwell s rule gives a necessary though not sufficient condition for rigidity of a 3D bar-and-joint assembly. It is... 

Figure 1. Nitrogen in a 10 (Di teniai=3.3 nm) cylindrical pore of MCM-41 at 77.4 K. The chemical potential of equilibrium transition BF, j,e- io=-1.42 kT, is obtained from the Maxwell s rule and also corresponds to the intersection point of the Grand Potential (solid line). Lines CG and EA, which bound the hysteresis loop, correspond to the spinodal condensation and desorption, respectively.

The law of equal areas (Maxwell s rule) which defines the demixtion curve can be expressed as follows [see Appendix L, eqns (L.1) and (L.5)]... 

These equilibrium conditions can be written in terms of 77(C) alone or of fi C) alone, by introducing Maxwell s rules, which will be now established. For this purpose, we assume that the free energy Fean be written in the form... 

Equations analogous to those here also hold for the H, U, G, S, and A. Equations of the form of Eq. 7.3-lc are the Maxwell s rules or lever rules first discussed in Sec. 

The function p(v) at fixed temperature (the isotherm) is shown in Fig. 5.1. The curves 1, 2, 3 correspond to different temperatures. The curve 3 corresponds to a temperature above the critical temperature (T > T ). In this state the curve changes smoothly, pressure falls with increase of o, and the substance can be in equilibrium only in the gaseous form. The second curve corresponds to the critical temperature It is the highest temperature at which liquid and vapor states can coexist in balance with each other. At temperature T < Ti (curve 1) the dependence p o) is non-monotonous. To the left of the point B (line AB) the substance is in the mono-phase liquid state, to the right of point G (line GH) the substance is in the mono-phase vapor state. The region between points B and G corresponds to the equilibrium the bi-phase state liquid - vapor. In accordance with the Maxwell s rule, squares of areas BDE and EFG are equal. From the form of isotherms it follows that in pre-critical area (T < Tc) the cubic equation... 

Maxwell s rule n. A law stating that every part of an electric circuit is acted upon by a force tending to move it in such a direction as to enclose the maximum amount of magnetic flux. 

At the temperatures and concentrations enclosed by the curve I EC FJ (see Figure 3.53b), the coexistence conditions of two gel states with different chain conformations in a closed space are realized. Eg. for T = Ty, the value of ir is determined by the ordinate of the straight line EF drawn on the isotherm (see Figure 3.53a) according to Maxwell s rule (Equation 1.2-43). 

In the region occupied by the polymer chain, solvent molecules are mixed. Let A/ro be the chemical potential of the solvent molecule measured from the value in the pure solvent. From the thermodynamic condition A/xo = (9AF/dNo)n = —(< / )(9 AF/d4>)=0 that the chemical potential of a solvent molecule inside the region occupied by the polymer should be equal to that in the outside region, we can derive Maxwell s rule of equal area for the osmotic pressure in the form... 

It is easy to show that the Maxwell s rule is equivalent to the condition of the equality of the surface energies of the phases 2 and 1 (vapor and condensate)... 

Fig. 3. Portion of a thin film chemical potential isotherm, oc -film and -film of different thickness coexist at // = /ie. hmax and hmin are the equilibrium thin film thickness corresponding to pmax and ptnin, respectively. Equality of the two hatched areas follows from Maxwell s rule.

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

The Latent Heats and Clapeyron s Equation.—There is a very important thermodynamic relation concerning the equilibrium between phases, called Clapeyron s equation, or sometimes the Clapeyron-Clausius equation. By way of illustration, let us consider the vaporization of water at constant temperature and pressure. On our P-V-T surface, the process we consider is that in which the system is carried along an isothermal on the ruled part of the surface, from the state whore it is all liquid, with volume Fz, to the state where it is all gas, with volume F . As we go along this path, we wash to find ihe amount of heat absorbed. We can find this from one of Maxwell s relations, Eq. (4.12), Chap. II ... 

Assuming the mixture of dielectrics and spherical pore with 3-0 connectivity, the dielectric constant (K) and loss quality (g) of (Pbo 5Cao.5)(Feo 5Tao.5)03 with different porosity were evaluated by the dielectric mixing rule. For the specimens with porosity, the ionic polarizabilities modified by Maxwell s Equation were closer to the theoretical values than those modified by Wiener s Equation. The predicted loss quality obtained from intrinsic ones and Maxwell s Equation agree with the observed ones. 

We may discard the last two terms. The reduced coexistence volumes i>/(P) and Vg P) for the liquid and vapor phase are in equilibrium at T < Tc equivalently, so are the corresponding quantities (pi and vapor phases with reference to Fig. 7.1.2 (where, however, the pressure vs. volume is plotted in a highly schematic manner for purely illustrative purposes) we apply Maxwell s equal area rule to require that, for a fixed value of t < 0, V dP = 0 = Vc — l)dP. The last integral follows since... 

Figure A2.3.3 P-V isotherms for van der Waals equation of state. Maxwell s equal areas rule (area ABE = area ECD) determines the volumes of the coexisting phases at subcritical temperatures.

T< Tc, isotherms show unphysical oscillations analogous to the oscillations that result from the van der Waals equation of state for real gases, which predict an LGPT [37]. The range of densities where dP/dV)r > 0 are thermodynamically unstable and indicate that the system must phase separate into LDL and HDL. The equilibrium isotherm can be obtained from the isotherms obtained in simulations by applying Maxwell s construction (see Fig. 3). At volumes V > Vldl and V < HDL. the equilibrium states are (homogeneous) LDL and HDL, respectively. At volumes Vhdl < F < Vldl> regions of HDL and LDL coexist. The fraction of the system in each phase is determined by the lever rule [37]. 

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