Gases 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). 

Vapor Pressures and Adsorption Isotherms. The key variables affecting the rate of destmction of soHd wastes are temperature, time, and gas—sohd contacting. The effect of temperature on hydrocarbon vaporization rates is readily understood in terms of its effect on Hquid and adsorbed hydrocarbon vapor pressures. For Hquids, the Clausius-Clapeyron equation yields... 

By comparison of Eq. (2-9) and the Clausius-Clapeyron equation with the perfect gas approximation,... 

The Clausius-Clapeyron equation implies that if we plot the natural log of the pressure of the gas phase versus inverse temperature, the slope of the resulting line is the heat of vaporization divided by the gas constant (R). A plot of In P (vapor pressure of water) versus inverse temperature is given in Figure 3. The calculated heat of vaporization (determined by multiplying the slope by R) is 10,400 cal/mol. The important aspect of Eq. (10) with regard to moisture sorption is the fact that increasing the temperature also increases the vapor pressure. 

The introduction of the perfect gas law to the Clausius-Clapeyron equation (Equation (6.14)) allows us to obtain a more direct approximation to p p(T) in the saturation region. We use the following ... 

Having qualitatively discussed the way a pressure cooker facilitates rapid cooking, we now turn to a quantitative discussion. The Clapeyron equation, Equation (5.1), would lead us to suppose that dp oc dT, but the liquid-gas phase boundary in Figure 5.12 is clearly curved, implying deviations from the equation. Therefore, we require a new version of the Clapeyron equation, adapted to cope with the large volume change of a gas. To this end, we introduce the Clausius-Clapeyron equation ... 

We need to understand that the Clausius-Clapeyron equation is really just a special case of the Clapeyron equation, and relates to phase changes in which one of the phases is a gas. 

Worked Example 5.3 The Clausius-Clapeyron equation need not apply merely to boiling (liquid-gas) equilibria, it also describes sublimation equilibria (gas-solid). 

Another important example involves water vapor in the atmosphere. Water vapor is the most important greenhouse gas, and its concentration in the atmosphere is a function of temperature as given by the Clausius-Clapeyron equation ... 

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... 

For pressures below one atmosphere, a number of simplifications can be made. The volume change on formation of vapor can be approximated reasonably by the volume of the vapor. The vapor is assumed to behave like an ideal gas. In particular, AV = VM = RT/ P. Substituting this value for AV into Equation (1) yields the Clausius-Clapeyron equation ... 

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

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... 

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 ... 

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. 

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 ... 

If we use the ideal gas equation to approximate the volume of vapor as Fvap = RT/P, we obtain the Clausius-Clapeyron equation... 

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. 

Although we shall not be concerned experimentally with measuring heats of adsorption, it is appropriate to comment that Mi for the physical adsorption of a gas is always negative, since the process of adsorption results in a decrease in entropy. The isosteric heat of adsorption (the heat of adsorption at constant coverage 6) can be obtained by application of the Clausius-Clapeyron equation if isotherms are determined at several different temperatures the thermodynamics of adsorption have been fully discussed by Hill. ... 

Equations 22-29 for the gas-phase profiles, temperature gradient, flame position, and vaporization rate depend only on the temperature at the surface of the droplet, Tg. (The Clausius-Clapeyron equation relates to Tg.) An iterative numerical procedure for satisfying the continuity of Tg at the liquid/gas interface is described in the Appendix. 

As in the development of tlie Clausius/Clapeyron equation (Example 6.5), if for low pressures one assumes tliat tlie gas pliase is ideal and tliat tlie adsorbate is of negligible volume... 

The influence of temperature on the vapour pressure of these dmgs is plotted according to equation (2.6) in Fig. 2.6. The vapour above the dmgs behaves as an ideal gas because of the low quantity of dmg transferred to the gaseous phase and the Clausius-Clapeyron equation is obeyed in all cases. The vapour pressure of carmustine is about 10-100 times greater than that of the other antineoplastic agents and approaches that of mercury (1.0 Pa at TO C) at elevated temperamre, with implications for occupational safety when handling this dmg. 

Gaseous anaesthesia in mice is an equilibrium process between the anaesthetic gas and the phase in which the gas exerts its effect (the biophase). As such, it should be amenable to treatment by the Clausius-Clapeyron equation. Modification of this equation is required when the distribution of a series of gases is to... 

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