Clapeyron Equation solid-liquid equilibrium

Clapeyron Equation. Solid/Liquid Equilibrium at Temperature, T and Pressure, P... 

A confirmation of this conclusion also is provided by an examination of the solid-liquid equilibrium in the neighborhood of 0 K. As shown in Equation (8.9), a two-phase equilibrium obeys the Clapeyron equation ... 

Using the Gibbs-Helmholtz equation obtained Clapeyron equation for the equilibrium solid liquid. 

The Clapeyron-Clausius equation (1.52) for solid = liquid equilibrium cannot be integrated easily since Vs cannot be ignored in comparison with Vt. Also the laws of liquid state are not so simple as those for gaseous state. However, this equation can be used for calculating the effect of pressure on the melting point of a solid. Eq. 1.52 can also be used for calculating heats of fusion from vapour pressure data obtained at different temperatures. 

THE INTEGRATION OF THE CLAPEYRON EQUATION 12.6.1 Solid-Liquid Equilibrium... 

This equation is called the Clapeyron equation and can be applied to any two phases in equilibrium, e.g., solid and liquid, liquid and vapor, solid and vapor or two crystalline forms of the same solid. Thus for the equilibrium... 

The worst deviations from the Clapeyron equation occur when one of the phases is a gas. This occurs because the volume of a gas depends strongly on temperature, whereas the volume of a liquid or solid does not. Accordingly, the value of A Vm is not independent of temperature when the equilibrium involves a gas. 

The Clapeyron equation, Equation (5.1), yields a quantitative description of a phase boundary on a phase diagram. Equation (5.1) works quite well for the liquid-solid phase boundary, but if the equilibrium is boiling or sublimation - both of which involve a gaseous phase - then the Clapeyron equation is a poor predictor. 

The Clapeyron equation can be reduced to a particularly convenient form when the equilibrium between A and B is that of a gas (g) and a condensed (cond) phase [liquid or solid]. In this situation... 

The Clapeyron equation can be simplified to some extent for the case in which a condensed phase (liquid or solid) is in equilibrium with a gas phase. At temperatures removed from the critical temperature, the molar volume of the gas phase is very much larger than the molar volume of the condensed phase. In such cases the molar volume of the condensed phase may be neglected. An equation of state is then used to express the molar volume of the gas as a function of the temperature and pressure. When the virial equation of state (accurate to the second virial coefficient) is used,... 

If the gas phase activity of the host is controlled by the presence of a pure condensed phase, solid or liquid, the equilibrium between host and guest in a stoichiometric clathrate can be described in terms of the gas phase pressure of the guest. This is, in effect, a vapor pressure for the guest. At higher pressures the guest will condense to form clathrate, and at lower pressures the clathrate will decompose. Temperature variation of this pressure will follow the Clapeyron equation which, with the usual assumptions (ideal gas behavior of the vapor and negligible volume of the condensed phase), reduces to the Clausius-Clapeyron equation ... 

Condensed-phase equilibria are treated by Eq. (40), the Clapeyron equation. The most important type of condensed-phase equilibrium is that between solid and liquid. For melting, Ais always positive, because the solid is the lowest-energy (and enthalpy) arrangement of molecules. The direction of change of the melting temperature with pressure,... 

Solids Below the triple point, the pressure at which the solid and vapor phases of a pure component are in equilibrium at any given temperature is the vapor pressure of the solid. It is a monotonic function of temperature with a maximum at the triple point. Solid vapor pressures can be correlated with the same equations used for liquids. Estimation of solid vapor pressure can be made from the integrated form of the Clausius-Clapeyron equation... 

T and pressure P. It should be noted that equation (33.26) is the exact form of the Clausius-Clapeyron equation (27.12). If the vapor is assumed to be leal, so that the fugacity may be replaced by the vapor pressure, and the total pressuic is taken as equal to the equilibrium pressure, the two equations become identical. In this simplification the assumption is made that the activity of the liquid or solid does not vary with pressure this is exactly equivalent to the approximation used in deriving the Clausius-Clapeyron equation, that the volume of the liquid or solid is negligible. 

We have already seen that the curve for S—V ends at the melting-point. At this point, liquid and solid are each in equilibrium with vapour at the same pressure, and they must also be in equilibrium with each other and the particular value of temperature and vapour pressure must lie on the S—Y as well as on the L—Y curve. At one time it was thought that the S—Y curve passes continuously into the L—V curve, but it follows quite clearly from the Clapeyron equation,... 

Univariant Systems.—Equilibrium between liquid and vapour. Vaporisation curve. Upper limit of vaporisation curve. Theorems of van t Hoff and of Le Chatelier. The Clausius-Clapeyron equation. Presence of complex molecules. Equilibrium between solid and vapour. Sublimation curve. Equilibrium between solid and liquid. Curve of fusion. Equilibrium between solid, liquid, and vapour. The triple point. Complexity of the solid state. Theory of allotropy. Bivariant systems. Changes at the triple point. Polymorphism. Triple point Sj—Sg— V. Transition point. Transition curve. Enantiotropy and monotropy. Enantiotropy combined with monotropy. Suspended transformation. Metastable equilibria. Pressure-temperature relations between stable and metastable forms. Velocity of transformation of metastable systems. Metastability in metals produced by mechanical stress. Law of successive reactions. 

Section 8.2.1 was concerned with equilibrium between a condensed phase and the vapour. It is often necessary, however, to estimate the effect of pressure on equilibria between two condensed phases. For example, the melting point of sodium at one atmosphere pressure is 97.6°C. Can it be used as a liquid heat transfer medium at 100°C, at a pressure of 100 atm, or will it solidify It is known that the liquid is less dense than the solid, and this argues that high pressures will encourage solidification. This is another aspect of Le Chatelier s work, which we can now quantify. This and similar problems may be solved by the Clapeyron equation, which we shall now derive. 

Clausius-Clapeyron equation - An approximation to the Clapeyron equation applicable to liquid-gas and solid-gas equilibrium, in which one assumes an ideal gas with volume much greater than the condensed phase volume. For the liquid-gas case, it takes the form d(lnp)/dT = A HIRV- where R is the molar gas constant and A H is the molar enthalpy of vaporization. For the solid-gas case, A H is replaced by the molar enthalpy of sublimation, A H. 

Thermodynamic analysis of the equilibrium between a condensed phase (solid or liquid) and the vapor is summarized by the Clausius-Clapeyron equation ... 

The criteria for equilibria involving solid phases are exactly those given in 7.3.5 for any phase-equilibrium situation phases in equilibrium have the same temperatures, pressures, and fugacities. Moreover, pure-component solid-fluid equilibria obey the Clapeyron equation (8.2.27). This means the latent heat of melting is proportional to the slope of the melting curve on a PT diagram and the latent heat of sublimation is proportional to the slope of the sublimation curve. In the case of solid-gas equilibria, the Clausius-Clapeyron equation (8.2.30) often provides a reliable relation between temperature and sublimation pressures, analogous to that for vapor-liquid equilibria. 

At thermodynamic equilibrium the chemical potential of each component i in both liquid and solid phases has to be equal. For simple systems and certain simplifications, like pure crystalline solid phase of component b (see Walas 1985), thermodynamic considerations lead to the well-known Clausius-Clapeyron equation... 

Vapour pressure p represents the partial pressure of a compound above the pure solid or liquid phase at thermal equilibrium it corresponds to a steady state with a continuous exchange, but no net transfer, of molecules between the two phases. From thermodynamic considerations, the vapour pressure of a chemical is determined by its enthalpy of vaporization (A/f ) and the temperature (7) as described by the Clausius-Clapeyron equation ... 

In general, when a solution freezes there is a separation of solute and solvent. That leads to the creation of a new triple point where the solvent, in solution, is in equilibrium with both its own pure vapor and pure solid. The relationship between the decrease in the freezing temperature and the lowering of the vapor pressure is given by the Clausius-Clapeyron equation for solid-liquid equilibria ... 

Substance A must not dissociate in the liquid state and be fully immiscible with the other component in the solid state. Derived from the formulation of the fundamental equilibrium condition, for such solid-liquid equilibria a special form of the generally valid Clausius-Clapeyron equation applies ... 

The phase diagram shows the location of the phase boundaries between the solid, liquid, and gas phases. All three phases are in equilibrium at the triple point. If the temperature and pressure exceed the so-called critical values, the phase boundary between liquid and vapor vanishes. For this supercritical state, a change of pressure and temperature no longer leads to a change of the state of aggregation. The influence of pressure on a phase transition is given by the Clapeyron equation. 

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