Clausius-Clapeyron equation, enthalpy

Enthalpy of Vaporization The enthalpy (heat) of vaporization AHv is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liqmd of a pure component i.e., at a temperature (below the critical temperature) anci corresponding vapor pressure. AHy is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation ... 

STRATEGY We expect the vapor pressure of CC14 to be lower at 25.0°C than at 57.8°C. Substitute the temperatures and the enthalpy of vaporization into the Clausius-Clapeyron equation to find the ratio of vapor pressures. Then substitute the known vapor pressure to find the desired one. To use the equation, convert the enthalpy of vaporization into joules per mole and express all temperatures in kelvins. 

As already mentioned, the system ofEqs. (8.1-8.5) is supplemented by the Clausius-Clapeyron equation, as well as by the correlation that determines the dependence of enthalpy on temperature and describes the thermohydrodynamical characteristics of flow in a heated capillary. It is advantageous to analyze parameters of such flow to transform the system of governing equations to the form that is convenient for significant simplification of the problem. 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

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,... 

From Appendix E, the molar enthalpy of vaporization of mercury at the normal boiling point is 58.6 kJ/mol. Using the Clausius-Clapeyron equation to find the vapor pressure of mercury at 25°C, we have... 

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] ... 

Since surface pressure is a free energy term, the energies and entropies of first-order phase transitions in the monolayer state may be calculated from the temperature dependence of the ir-A curve using the two-dimensional analog of the Clausius-Clapeyron equation (59), where AH is the molar enthalpy change at temperature T and AA is the net change in molar area ... 

Solution The enthalpy of vaporization (33.05 kj-mol1) corresponds to (3.305 X 104 J-moE1) 25°C corresponds to T2 = 298 K, and 57.8°C corresponds to T, = 332.0 K. We substitute these values into the Clausius-Clapeyron equation and obtain... 

The most recent confirmation of the validity of the Clausius-Clapeyron equation for hydrates was by Handa (1986a,b), who measured the heat of dissociation (via calorimetry) of the normal paraffins that form simple hydrates. Table 4.8 shows Handa s values for hydrate dissociation enthalpy compared to those calculated with the Clausius-Clapeyron equation by Sloan and Fleyfel (1992). The agreement appears to be very good for simple hydrates. 

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 problem with use of the Antoine equation is that its use can introduce unreasonable assumptions about the change in AHv with temperature. This equation tends to overestimate the increase in enthalpy of vaporization with decreasing temperature. Grain (1982) used an approximation to the somewhat more realistic Watson24 expression for this temperature dependence. To calculate the vapor pressure at temperature T, lower than the boiling point, Tb, using the Clausius-Clapeyron equation, Watson suggested the function... 

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 ... 

Another useful equation is the Clausius-Clapeyron equation. It states that, provided the ideal gas law holds and the enthalpy of vaporization, Aft, is independent of T (which is a reasonable assumption for a small temperature range), the slope of the vapor pressure curve is given by... 

After freezing, the time to sublimate the solvent is given by the drying expressions in Tables 8.3 and 8.4, where the enthalpy of vaporization for drying is replaced by the enthalpy of sublimation. The enthalpy of sublimation is often equal to the sum of the heats of fusion and vaporization [16]. The enthalpy of sublimatian is also substituted for the enthalpy of vaporization in the Clausius Clapeyron equation (8.9) required for the calculation of the solvent partial pressure. The same rate determining steps of boundaiy layer mass transfer and heat transfer as well as pore diffusion and porous heat conduction are applicable in sublimation. 

Summarizing, an attempt has been made to provide a systematic account of the thermodynamic properties of the adsorbed phase. The Gibbs adsorption equation, as an extension of the Clausius-Clapeyron equation, has played a key role in linking experimental isotherm data to the determination of molar or differential entropies and enthalpies. Similarly, calorimetric measurements can be systematically applied to obtain the same type of information. 

The Clausius-Clapeyron equation is useful in the calculation of the enthalpy of vaporisation,... 

Figure 4 Examples of the differential sorption heat calculated from the sorption data using the Clausius-Clapeyron equation. The arrows depict the most negative enthalpy value in the region of strongly (SP) and weakly (WP) bound water, the limit of the strongly bound water region (SBC), the moisture content where bound water first appeared (BWiso), and the tissue moisture range corresponding to weakly bound water (WBC) [56].

The Clausius-Clapeyron equation can also be applied to estimate the vapour pressure of a solid precursor. In this case, the enthalpy of sublimation (AHsub) should replace the enthalpy of vaporisation. 

The Clausius-Clapeyron equation relates pressure with temperature, enthalpy, and volume, and has been used to develop semi-theoretical expressions of vapor pressure ( ). Many properties, including viscosity, can be related to an energy barrier, free volume and temperature. The attempt here is to express viscosity in the form of the Clausius-Clapeyron equation. 

Salmeterol xinafoate is known to exist in two polymorphic forms, forms I and II. Form I is stable, and form II is the metastable polymorph at ambient temperature. The enthalpies of solution (AHso ) of forms I and II determined from van t Hoff solubility-temperature plots are 32.1 and 27.6 kJ/mol, respectively, and the transition temperature obtained by linear extrapolation of the van t Hoff plots is 99 °C. The enthalpy of polymorphic conversion (AHii i) calculated from the plots of log solubility ratio of polymorphs versus the reciprocal of absolute temperature is negative (—4.55 kJ/mol) (5). However, the change in molar volume (AFu i) due to the conversion is positive. Therefore, according to the Clausius-Clapeyron equation,... 

Although the heat of adsorption or enthalpy change accompanying adsorption is directly obtained by calorimetry, it can conveniently be evaluated from the adsorption isostere. According to thermodynamics, the relationship between temperature T and pressure P under a state of -(J> phase equilibrium can generally be expressed with the Clausius-Clapeyron equation ... 

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