Clausius-Clapeyron relationship

The vapor pressure for the soHd at 25°C has been calculated from the value for the Hquid at 70°C and the heats of vaporization and fusion using the Clausius-Clapeyron relationship. 

For some reacting mixtures, it is difficult to find physical property data. An alternative version of Leung s method331 makes use of the Clausius-Clapeyron thermodynamic relationship to give a formula-in which all. the data required can be measured experimentally. The Clausius-Clapeyron relationship (T(dPydT) = hfg/vfg) only holds, for ideal single-component systems, and so its use introduces the following additional conditions of applicability ... 

These conditions are most likely to be met at relatively low pressure (less than, say, 5 bar). If there is any doubt over the applicability of the Clausius-Clapeyron relationship, it is suggested that a different method be used (see 6.3.2 and 6.4). 

Fauske s method14,51 (given below) is based on emptying the reactor (or achieving vapour/ liquid disengagement) before the pressure has risen from the relief pressure to the maximum accumulated pressure in a vented system. The method incorporates the simplified equilibrium rate model, ERM, for saturated liquid inlet (see 9.4.2) together with the Clausius-Clapeyron relationship (discussed in 6.3.3). The method makes use of adiabatic experimental rate data for the runaway, whose measurement is described in Annex 2. 

For wide boiling multi-component mixtures, the Clausius-Clapeyron relationship, used in deriving equations (A8.5)-(A8.10), is invalid. The definitions of Omega can be rewritten without the use of the Clausius-Clapeyron relationship. For example, equation (A8.5) becomes . ... 

Equations such as (A8.11) have wider applicability because they do not require the Clausius-Clapeyron relationship to hold and they should, therefore, be applicable for wider boiling mixtures. However, all of the approximate equations given in this section are evaluated only at the conditions in the upstream reactor. This reduces their accuracy if the pressure drop is high (e.g. if there is a piping system with substantial pressure drop). In such cases, equation (A8.4) is to be preferred. Alternatively, different calculation methods to the Omega method can be used. 

Figure 7.7 Clausius-Clapeyron relationship between water activity and temperature for native potato starch. Numbers on curves indicate water content, in g per g dry starch (from Fennema,...

Figure 24.15 shows that the martensitic transformation temperature in the In-Tl system is raised by applying a constant uniaxial compressive stress. Using the thermodynamic formalism leading to Eq. 24.11, develop a Clausius-Clapeyron relationship that relates the observed effect of applied stress on transformation temperature to thermodynamic quantities. 

This decrease in the amount adsorbed at higher temperatures follows the Clausius Clapeyron relationship,... 

Thermodynamic data regarding the adsorption of CO on Au/Ti02 catalysts with varying Au cluster sizes have been acquired with TPD using the well-known Redhead method " and with IRAS using the Clausius-Clapeyron relationship. Results for these measurements are displayed in Fig. 6. CO adsorption on Au clusters larger... 

Clausius-Clapeyron relationship - a thermodynamic equation applying to two-phase equilibrium for a pure substance which relates vapor pressure to temperature. 

If the aggregation number, m, is assumed to be independent of temperature, then the enthalpy of micelle formation, aH, can be estimated by using the Clausius-Clapeyron relationship ... 

The theory goes as follows in the stable liquid state there are two distinct structural species of the liquid, a high density liquid (HDL) and a low density liquid (LDL). Due to fluctuations each component will vary between the two species, with only the time averaged distribution of species staying constant for a specific temperature and pressure. Upon an increase in pressure the proportion of the more dense species increases, which leads to an increase in the density of the liquid above that of the corresponding crystal. This causes the melting curve to have a negative relationship with pressure, as can be seen from the Clausius-Clapeyron relationship ... 

Two empirical parameters are evident in equation 7, the heat of vaporization and the integration constant, I. Experimental data indicate that the linear relationship suggested by Clausius-Clapeyron may not be followed over a large temperature range (4) therefore additional adjustable parameters have been added to equation 7 to improve its correlating abiUty. The most prominent of these is the Antoine equation ... 

Curve fitting to data is most successhil when the form of the equation used is based on a known theoretical relationship between the variables associated with the data points, eg, use of the Clausius-Clapeyron equation for vapor pressure. In the absence of known theoretical relationships, polynomials are one of the most usehil forms to describe a curve. Polynomials are easy to evaluate the coefficients are linear and the degree, ie, the highest power appearing in the equation, is a convenient measure of smoothness. Lower orders yield smoother fits. 

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 alternative version (obtained by using the Clausius-Clapeyron thermodynamic relationship, which only holds if the mixture behaves as a single pseudo-component) is ... 

It is recommended that the method given in equations (6.10) and (6.11) be used to obtain dPydT (see also 6.5.2). The simplified ERM can be rewritten by making use of the Clausius-Clapeyron thermodynamic relationship which introduces the following further requirement ... 

The Clausius-Clapeyron Equation expresses the relationship between vapor pressure and temperature. It is the equation for the vapor-pressure line. We will develop this equation with the Clapeyron Equation, which was developed using thermodynamic theory. 

The Clausius-Clapeyron equation is an exact thermodynamic relationship between the slope of the vapor pressure curve and the molal heat of vaporization ... 

As indicated by the plots in Figure 10.13a, the vapor pressure of a liquid rises with temperature in a nonlinear way. A linear relationship is found, however, when the logarithm of the vapor pressure, In Pvap, is plotted against the inverse of the Kelvin temperature, 1 /T. Table 10.8 gives the appropriate data for water, and Figure 10.13b shows the plot. As noted in Section 9.2, a linear graph is characteristic of mathematical equations of the form y = mx + b. In the present instance, y = lnPvap, x = 1/T, m is the slope of the line (- AHvap/R), and b is the y-intercept (a constant, C). Thus, the data fit an expression known as the Clausius-Clapeyron equation. ... 

To a fair engineering approximation AH is not only a function of the hydrogen bonds in the crystal, but also a function of cavity occupation. Because the Clausius-Clapeyron equation determines the heat of hydrate formation by the slopes of plots of In P versus 1/T, one may easily determine relationships between heats of dissociation. 

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... 

Thermodynamics and kinetics can surely be counted—along with transport phenomena, chemistry, unit operations, and advanced mathematics—as subjects that form the foundation of Chemical Engineering education and practice. Thermodynamics is of course a very old subject. For example, it was the same Rudolf Clausius, who in 1865 coined two immortal sentences (1) "The energy of the universe is constant" and (2) "The entropy of the universe tends to a maximum," that developed the famous Clausius-Clapeyron equation, one of the most basic physico-chemical relationships. Classical thermodynamics was largely complete in the 19th century, before even the basic structure of the atom was understood. 

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... 

The Clausius-Clapeyron Equation. This equation expresses the relationship between vapor pressure and temperature. In 1834 Clapey-ron, using thermodynamic iJieory, developed the following equation which will be accepted without derivaticm... 

Since we know how the solution vapor pressure varies with concentration (the relationship being given by Equation 6.5-2) and temperature (through the Clausius-Clapeyron equation. Equation 6.1-3), we can determine the relationships between concentration and both boiling point elevation and freezing point depression. The relationships are particularly simple for dilute solutions x — 0, where x is solute mole fraction). 

Therefore, a plot of In [(Vr),/(Vr)2] versus In P. should yield a straight line with either a positive or negative correlation coefficient depending on the ratio of (Vr)i/(Vr)2. The value of (L1/L2) can be calculated from the slope and thus Pi determined from equation 3. If substances 1 and 2 are the unknown and standard compounds, respectively, then the vapor pressure of the unknown at a given temperature can be determined. The relationship between vapor pressure and temperature can be simply described by the Clausius-Clapeyron equation ... 

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