Clausius-Clapeyron equation derivation

Clausius-Clapeyron Equation. Derived from equation 1, the Clapeyron equation is a fundamental relationship between the latent heat accompanying a phase change and pressure—volume—temperature (PVT) data for the system (1) ... 

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). 

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

The Clausius-Clapeyron equation The Clapeyron equation can be used to derive an approximate equation that relates the vapor pressure of a liquid or solid to temperature. For the vaporization process... 

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,... 

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... 

One of the critical issues in vapor pressure methods is the choice of the procedure to calculate the vaporization enthalpy. For instance, consider the vapor pressures of ethanol at several temperatures in the range 309-343 K, obtained with a differential ebulliometer [40]. The simplest way of deriving an enthalpy of vaporization from the curve shown in figure 2.4 is by fitting those data with the integrated form of the Clausius-Clapeyron equation [1] ... 

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 the case of second-order transitions, A F and AS have zero values, and equation (4.3) will have a 0/0 indeterminacy. We can, however, derive analogues of the Clausius-Clapeyron equation for these transitions, such as equation (4.4) ... 

As we have already observed, the vapor-pressure-temperature curve is nonlinear. To reduce this curve to a linear form, a plot of log (p ) versus (1/T) can be made for moderate temperature intervals. The resultant straight line is described by the following expression, which can be derived from the Clausius-Clapeyron equation. 

The heat of vaporization, AHvap, of a liquid can be obtained either graphically from the slope of a plot of In Pvap versus 1 /T, or algebraically from the Clausius-Clapeyron equation. As derived in Worked Example 10.5,... 

Derive the Clausius-Clapeyron equation [Eq. (44)] from Eq. (40) by neglecting the volume of the condensed phase and using the ideal gas law for the vapor. 

Problem 6 Give the thermodynamic derivation of Clapeyron equation and Clausius-Clapeyron equation. Discuss their applications also. 

Equation (9) is sometimes known as Clausius-Clapeyron equation and is generally spoken to as first latent heat equation. It was first derived by Clausius (1850) on the thermodynamic basis of Clapeyron equation. 

The effect of hydrostatic pressure p on Te can be simply determined with the aid of the Clausius-Clapeyron equation. Kittel derives... 

Derive the Clausius-Clapeyron equation for the adsorbate in the form dP/dT - (Hg - Hg)/T(Vg - Vs) at constant T and give a written description of the information provided by this expression. 

The Clausius-Clapeyron equation gives the variation of the vapour pressure p of a liquid with absolute temperature T. To derive the relationship involves integration of the expression... 

Now, these calorimetric results allow, in principle, to recalculate adsorption isotherms at other temperatures than 30°C by simple use of the isosteric method derived from the Clausius-Clapeyron equation, for instance in the following form ... 

This is known as the Clausius-Clapeyron equation. If the molar heat of vaporization and the vapor pressure at some temperature are known for a liquid, the vapor pressure at other temperatures can be calculated, provided the assumptions made in the derivation of this equation are valid. Since the normal boiling point of a liquid is defined as the temperature at which tlie vapor pressure equals one aianosphere, it is apparent that only the molar heat of vaporization and the normal boiling point of a liquid need to be known in order to calculate the vapor pressure at other temperatures. 

Use a semilog plot based on the Clausius-Clapeyron equation to derive an equation for / (mm Hg) as a function of T ( C). From the plot, estimate the heat of vaporization of ethylene glycol in kJ/mol. (Remember to use absolute temperatures in the Clausius-Clapeyron equation.)... 

This expression, known as the Clausius-Clapeyron Equation, is of great historic significance, being a very early derivation that links seemingly unrelated variables. This was considered to be a noteworthy example of the power of thermodynamic theory and may be considered a precursor to later theoretical developments. 

Phase trcmsitions in monolayers may be treated thermodynamically analogously to those in three-dimensional systems. As will be derived in sec. 3.4, the Clausius-Clapeyron equation, relating the variation of pressure with temperature for a two-dimensional situation, reads... 

A new thermodynamic derivation of eq. (4.8) has been proposed making use of a modified Clausius-Clapeyron equation. The derivation of this equation is based on the assumption that plastic deformation involves a partial melting of the polymer crystals (Hirami et al, 1999). 

An expression for the boiling point elevation may readily be derived using the Clausius-Clapeyron equation (see Box 2.1). This expression allows the calculation of the increase of boiling point, AT, from the molality, m, of the solution using... 

Equation (3.31) is derived from the Clausius-Clapeyron equation (see Chap. 4). Empirical correlations of vapor pressure are frequently given in the form of the Antoine equation (refer to Appendix G for values of the constants) ... 

Since vapor pressure is related to the kinetic energy of the molecules, vapor pressure is a function of temperature. A derivation of tine Clausius-Clapeyron equation relates vapor pressure and temperature to the heat of vaporization ... 

This expression is sometimes referred to as the Clausius-Clapeyron equation, for it was first derived by R. Clausius (1850), in the course of a comprehensive discussion of the Clapeyron equation. Although the Clausius-Clapeyron equation is approximate, for it neglects the volume of the liquid and supposes ideal behavior of the vapor, it has the advantage of great simplicity. In the calculation of dp/dT (or dT/dP) from a knowledge of the heat of vaporization, or vice versa, it is not necessary to use the volumes of the liquid and vapor, as is the case in connection with equations (27.9) and (27.10). However, as may be expected, the results are less accurate than those derived from the latter expressions. 

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. 

The evaluation of the viscosity of mold fluxes has shown that the viscosity is primarily controlled by the concentration of network forming oxides, particularly the silica content. It has also been demonstrated that the temperature dependence of viscosity can be expressed by the relation, nri = Cj + C2/T + C3 nT, derived from the Clausius-Clapeyron Equation. This relation produces a better description of viscosity vs. temperature than the more familiar Arrhenius Equation. 

In the previous section it was observed that the Langmuir postulates of sites of equal activity and no interaction between occupied and bare sites were responsible for nonagreement with experimental data. It might be surmised that these assumptions correspond to a constant heat of -ad-sm-pt-ion—Indeed.-it-is-p.QssibIe to derive the Langmuir isotherm by assuming that is independent of d. The heat of adsorption can be evaluated from adsorption-equilibrium data. First the Clausius-Clapeyron equation is applied to the two-phase system of gas and adsorbed component on the surface ... 

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