Boiling Clapeyron equation

Use the Clausius-Clapeyron equation to estimate the vapor pressure or boiling point of a liquid (Examples 8.1 and 8.2). 

If one measures the boiling points at several pressures, including that of atmospheric pressure, one can then extrapolate to obtain the vapor pressure of a material at ambient temperature. This is done using the Clausius-Clapeyron equation, i.e.-... 

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... 

The Clausius-Clapeyron equation quantifies the way a boiling temperature changes as a function of the applied pressure. At the boiling points of 7i and T2, the external pressures pi and p2 are the same as the respective vapour pressures. 

The intercept of a Clapeyron graph is not useful its value may best be thought of as the pressure exerted by water boiling at infinite temperature. This alternative of the Clausius-Clapeyron equation is sometimes referred to as the linear (or graphical) form. 

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

The Clapeyron equation, Equation (5.1), yields a quantitative description of a phase boundary on a phase diagram. Equation (5.1) works quite well for the liquid-solid phase boundary, but if the equilibrium is boiling or sublimation - both of which involve a gaseous phase - then the Clapeyron equation is a poor predictor. 

Remember that the temperature of boiling water 7"(b0ii) is itself a function of the external pressure, according to the Clausius-Clapeyron equation (see Section 5.3). 

Any one of Equations (8.14), (8.15), or (8.16) is known as the Clausius-Clapeyron equation and can be used either to obtain AH from known values of the vapor pressure as a function of temperature or to predict vapor pressures of a hquid (or a solid) when the heat of vaporization (or sublimation) and one vapor pressure are known. The same equations also represent the variation in the boiling point of a liquid with changing pressure. 

Raoult s law works for small polymers as well as small molecules. Determination of M is based for both ebulliometry (boiling point elevation) and cryometry (freezing point lowering) on the Clausius-Clapeyron equation ... 

In this case, this equation is commonly referred to as Clausius-Clapeyron equation (e.g., Atkins, 1998). We can integrate Eq. 4-7 if we assume that Avap/7, is constant over a given temperature range. We note that AvapHt is zero at the critical point, Tc, it rises rapidly at temperatures approaching the boiling point, and then it rises more slowly at lower temperatures (Reid et al., 1977). Hence, over a narrow temperature range (e.g., the ambient temperature range from 0°C to 30°C) we can express the temperature dependence of p by (see Eq. 3-51) ... 

In a very similar way as discussed above for estimating pi from boiling point data, one can treat the vapor pressure curve below the melting point. Again we use the Clausius-Clapeyron equation ... 

Values of the heat of concentration and heat capacity of sea water near room temperature have been measured experimentally. The heat of concentration values compare favorably with those calculated from the vapor pressure data given by Arons and Kientzler by use of the Clapeyron equation. The heat capacity agrees with tne values reported by Cox and Smith. Calculated values for the heat of concentration and boiling point elevation from 77° to 302° F. at salinities up to 9% are presented in both tabular and graphical form. 

The boiling point elevations were obtained by extrapolation with the Clapeyron equation ... 

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... 

The normal boiling point of benzene = 353.26K. Use the Clausius Clapeyron equation to get AHV... 

The Clapeyron equation relates pressure to temperature, and hence boiling or melting points can be calculated with changing pressure. By using Eq. (1.156), we can equate the Gibbs-Duhem equation for two phases... 

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. 

Estimate the vapor pressure of a pure substance at a specified temperature or the boiling point at a specified pressure using (a) the Antoine equation, (b) the Cox chart, (c) the Clausius-Clapeyron equation and known vapor pressures at two specified temperatures, or... 

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). 

For one of the following liquids, detenu ine the heat of vaporization at its nonnal boiling point by application of tlie Clapeyron equation to the given vapor-pressine equation. Use generalized correlationsfrom Chap. 3 to estimate AV. 

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... 

The Clausius-Clapeyron equation can be used to estimate the vapor pressure of dipropylene glycol as a function of temperature, with the boiling temperature as a reference. 

The boiling point of a liquid is the temperature at which the pressure of the vapor in equilibrium with it is equal to the external pressure hence, in the form of (27.9), the Clapeyron equation gives the variation of the boiling point T of a liquid with the external pressure P. 

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