Clausius-Clapeyron equation, enthalpy vaporization

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. 

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] ... 

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 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. 

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. 

The calorimeter can be used to determine directly the enthalpy of fusion of the sample. The enthalpy of vaporization can either be measured directly in the apparatus or obtained more easily from vapour pressure measurements using the Clausius-Clapeyron equation (Sections 4.7 and 4.8),... 

Note that (17.94) is similar to the Clausius-Clapeyron equation for water vapor-water equilibrium (17.8), with the enthalpy of sublimation replacing the enthalpy of evaporation. 

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. 

The freezing point depression follows from the lowering of vapor pressure. From the Clausius-Clapeyron equation and Raoulfs law, it follows enthalpy of... 

The heat of sorption is the difference in specific heat content or enthalpy between the bound moisture and that freely attached at the same temperature and total pressure. This enthalpy difference is normally derived from a form of the Clausius-Clapeyron equation on the assumption that the moisture vapor phase acts like an ideal gas and the molal volume of the condensed phase is negligible compared with that of the vapor. These considerations lead to the expression... 

The final approximation is to treat the enthalpy of vaporization as constant. Integrating from (T, P ) to (T, Ps ), we obtain the Clausius-Clapeyron equation ... 

The enthalpy of vaporization is not constant, of course, it decreases with temperature and vanishes at the critical point. Therefore, the above result should be treated as an approximation over short temperature intervals. Despite its approximate nature, the Clausius-Clapeyron equation is useful in that it allows us to calculate the saturation pressure at a temperature, if its value is known at another. According to this equation, the logarithm of the saturation pressure is a linear function of inverse temperature with slope -AHvap/i . Saturation pressure is often plotted in semi-log axes against i/T. In this form the resulting graph is approximately linear. 

For vapor pressure values given only in the form of the Clausius-Clapeyron equation a pseudo Third Law value is calculated by evaluating the enthalpy of sublimation at the temperature extremes and averaging. The value obtained gives a reasonable estimate compared to the value that would have been obtained if all the data had been available. For vapor pressure measurements, the Revised method is used to calculate the Second Law enthalpy of sublimation, but for mass spectrographic measurements fitted to the Clausius-Clapeyron equation, the Traditional method is used. 

Because the calorimetric methods of measurement of enthalpy of vapor formation are very difficult, the indirect mefliods are used, especially for less volatile substances. The application of generalized expression of the first and second laws of thermodynamics to the heterogeneous equilibrium between a condensed phase in isobaric- thermal conditions is given in the Clausius-Clapeyron equation that relates enthalpy of a vapor formation at the vapor pressure, P, and temperature, T. For one component system, the Clausius-Clapeyron equation has the form ... 

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 ... 

Volatility Solvent volatility helps in estimation of the solvent evaporation rate at temperatures below its boiling point The Knudsen, Henry, Cox, Antoine, and Clausius-Clapeyron equations are used to estimate the vapor pressure of a solvent over a liquid, its evaporation rate, and the composition of the atmosphere over the solvent. The boiling point of a solvent gives an indication of its evaporation rate but it is insufficient for its accurate estimation because of the influence of the molar enthalpy of evaporation. 

The Clausius-Clapeyron equation gives a well-founded relationship between the enthalpy of vaporization and the vapor pressure curve. 

The Clausius-Clapeyron equation is an accurate relationship between vapor pressure and enthalpy of vaporization and widely used for the estimation of enthalpies of vaporization from vapor pressure data (see Section 3.2.3). 

Concerning the caloric properties, the enthalpy of vaporization is linked to the vapor pressure curve via the Clausius-Clapeyron equation (Eq. (2.86)). Therefore, the quality is mainly determined by the ability of the ty-function in representing the vapor pressure data, which is usually sufficient (see above). The enthalpy of vaporization can easily be calculated with a cubic equation of state by subtracting the residua] parts of the enthalpy of vapor and liquid in the saturation state... 

There is a peculiarity for the estimation of enthalpies of vaporization the Clausius-Clapeyron equation... 

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