Equilibria Clausius-Clapeyron equation

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

As pointed out earlier, the equilibrium constant of a system changes with temperature. The form of the equation relating K to T is a familiar one, similar to the Clausius-Clapeyron equation (Chapter 9) and the Arrhenius equation (Chapter 11). This one is called the van t Hoff equation, honoring Jacobus van t Hoff (1852-1911), who was the first to use the equilibrium constant, K. Coincidentally, van t Hoff was a good friend of Arrhenius. The equation is... 

Clausius-Clapeyron equation An equation expressing the temperature dependence of vapor pressure ln(P2/Pi) = AHvapCl/Tj - 1/T2)/R, 230,303-305 Claussen, Walter, 66 Cobalt, 410-411 Cobalt (II) chloride, 66 Coefficient A number preceding a formula in a chemical equation, 61 Coefficient rule Rule which states that when the coefficients of a chemical equation are multiplied by a number n, the equilibrium constant is raised to the nth power, 327... 

Equation 3 is analogous to the Clausius-Clapeyron equation for equilibrium of a substance in the vapor and condensed phases (4). 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

A rate of reaction usually depends more strongly on temperature than on concentration. Thus, in a first-order (n = 1) reaction, the rate doubles if the concentration is doubled. However, a rate may double if the temperature is raised by only 10 K, in the range, say, from 290 to 300 K. This essentially exponential behavior is analogous to the temperature-dependence of the vapor pressure of a liquid, p, or the equilibrium constant of a reaction, K. In the former case, this is represented approximately by the Clausius-Clapeyron equation,... 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

This is known as the Clausius-Clapeyron equation. It is a state relationship that allows the determination of the saturation condition p = p(T) at which the vapor and liquid are in equilibrium at a pressure corresponding to a given temperature. 

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

Equation 6.56 is known as the equation of lowering of freezing point and is valid for solid mixtures crystallizing from multicomponent melts. Like the Clausius-Clapeyron equation, it tells us how the system behaves, with changing T, to maintain equilibrium on the univariant curve. However, whereas in the Clausius-Clapeyron equation equilibrium is maintained with concomitant changes in 7) here it is maintained by appropriately varying the activity of the component of interest in the melt and in the solid mixture. 

Because H fusion is difficult to measure as a result of the high value of 7), it may be derived indirectly through calculations involving the vitreous state (see Berman and Brown, 1987) or through the Clausius-Clapeyron equation for the crystal-melt equilibrium (cf equation 6.48 and section 6.3). 

In chemisorption where severe surface perturbations can occur, the Clausius-Clapeyron equation cannot be applied, since equilibrium pressures are low and often unobtainable. Nonetheless, a differential heat analogous to the isosteric heat can be obtained from heats of immersion without recourse to pressure data where the amounts adsorbed prior to immersion can be measured gravimetrically (Sec. VII,A). 

The vapor pressure, pv, is the pressure exerted by fluids and solids at equilibrium with their own vapor phase. The vapor pressure is a strong function of T, as expressed in the Clausius-Clapeyron equation [1] ... 

Table 4.10 shows the literature values for hydrate numbers, all obtained using de Forcrand s method of enthalpy differences around the ice point. However, Handa s values for the enthalpy differences were determined calorimetrically, while the other values listed were determined using phase equilibrium data and the Clausius-Clapeyron equation. The agreement appears to be very good for simple hydrates. Note also that hydrate filling is strongly dependent on... 

The Clausius-Clapeyron equation" is an integrated version of the Clapeyron equation that applies to equilibrium between an ideal gas vapor phase and a condensed phase, with the conditions that the volume of the... 

The fundamental relationship that allows the determination of the equilibrium vapor pressure, P, of a pure condensed phase as a function of temperature is the Clausius-Clapeyron equation... 

In general, the molar enthalpy of vaporization is obtained from the Clausius-Clapeyron equation, representing the difference per mole of the enthalpy of the vapour and of the liquid at equilibrium with it ... 

Gas and condensed phase equilibrium the Clausius-Clapeyron equation... 

The integration of Equation (11.22) to determine the equilibrium constant as a function of the temperature or to determine its value at one temperature with the knowledge of its value at another temperature is very similar to the integration of the Clausius-Clapeyron equation as discussed in Section 10.2. The quantity AHB must be known as a function of the temperature. This in turn may be determined from the change in the heat capacity for the change of state represented by the balanced chemical equation with the condition that all substances involved are in their standard states. 

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 ... 

It is evident that Equation (2.68) is analogous to the well-known Clausius-Clapeyron equation for a one component gas-liquid system. Integration of Equation (2.68) between the limits of equilibrium pressures and temperatures of p,/72 and Tlt T2 gives ... 

The requirement of thermodynamic reversibility also applies to the chromatographic method, but in this case it is necessary to work at very low surface coverage (at zero coverage ) in the Henry s law region. Values of the specific retention volume, Vs, determined at different temperatures are inserted in the Clausius-Clapeyron equation in place of the equilibrium pressures to obtain A h. Provided that a number of conditions are observed, the method is capable of providing a fairly easy and rapid assessment of the adsorbent—adsorbate interaction energy. 

Note that any of the equations in (2.3.5) or (2.3.4) specifies P as a function of T or vice versa. It is not always recognized that the equilibrium constraints manifested in the Clausius—Clapeyron equation are commonly employed to fix the temperature of a helium bath in the range 0.3 to 4.2 K, by adjusting the vapor pressure of the helium gas above the liquid phase to correspond to the desired temperature. 

The heat of adsorption may be calculated from a knowledge of the sorption isotherm at equilibrium by use of the Clausius-Clapeyron equation. 

The Clausius-Clapeyron equation for the effects of pressure on an equilibrium temperature is... 

As a useful thermodynamic property, the isosteric heat of adsorption has been generally applied to characterize the adsorbent surface. The isosteric heat of adsorption is evaluated simply by applying the Clausius-Clapeyron equation if one has a good set of adsorption equilibrium ta obtained at several temperatures. 

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