The Clausius-Clapeyron Equation

When the gas phase of a substance coexists in equilibrium with the liquid or solid phase, and provided T and p are not close to the critical point, the molar volume of the gas is much greater than that of the condensed phase. Thus, we may write for the processes of vaporization and sublimation 

The further approximation that the gas behaves as an ideal gas, RT/p, then changes Eq. 8.4.5 to 

Equation 8.4.10 is the Clausius-Clapeyron equation. It gives an approximate expression for the slope of a liquid-gas or solid-gas coexistence curve. The expression is not valid for coexisting solid and liquid phases, or for coexisting liquid and gas phases close to the critical point. 

At the temperature and pressure of the triple point, it is possible to carry out all three equilibrium phase transitions of fusion, vaporization, and sublimation. When fusion is followed by vaporization, the net change is sublimation. Therefore, the molar transition enthalpies at the triple point are related by 

Since all three of these transition enthalpies are positive, it follows that Asub-f is greater than Ayap f at the triple point. Therefore, according to Eq. 8.4.10, the slope of the solid-gas coexistence curve at the triple point is slightly greater than the slope of the liquid-gas coexistence curve. 

The Clausius-Clapeyron equation The Clapeyron equation can be used to derive an approximate equation that relates the vapor pressure of a liquid or solid to temperature. For the vaporization process 

If the enthalpy of vaporization has only a small dependence on T, or if the integral is evaluated over only a small change, dr, then AvipHm can be assumed to be constant, and integration between pressure limits, p and pi, and temperature limits, 7i and 7%, yields 

Equivalent expressions can be obtained for the (solid + vapor) equilibrium to give the vapor pressure of a solid, by substituting the sublimation enthalpy, AjubT/m, for Avap//m  

Substitution of this expression into equation (8.6) gives, after integration, 

Another expression that does a reasonably good job of representing vapor pressures of liquids, if the temperature range is not too great, is the Antoine equation  

FIGURE 5.3 On a P-T diagram for a single pure chemical, we increase the temperature hy dT, which causes the pressure to increase hy dP. On the equilibrium curve 

Example 5.1 Compute the value of dP/dT for the steam-water equilibrium at 212°F using the Clapeyron equation, and compare it with the value in the steam table [5]. 

Using the nearest adjacent steam table entries for vapor pressure, we have 

As in Example 4.1, this agreement does not demonstrate the correctness of the Clapeyron equation the authors of the 

The Clapeyron equation is rigorous and exact. It applies not only to gas-liquid equilibrium, but to any two-phase equilibrium of a pure species (e.g., liquid-solid, gas-solid) or change between two different crystal forms (e.g., graphite to diamonds, see Example 4.2). By adding some simplifications, we find the Clausius-Clapeyron (C-C) equation, which is only approximate but is a surprisingly good approximation of observed behavior for gas-liquid and gas-solid equilibria (vapor pressures) of single pure species. 

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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... 

Fundamental Property Relation. The fundamental property relation, which embodies the first and second laws of thermodynamics, can be expressed as a semiempifical equation containing physical parameters and one or more constants of integration. AH of these may be adjusted to fit experimental data. The Clausius-Clapeyron equation is an example of this type of relation (1—3). 

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. 

Nomograph defined. This method assumes the application of the Clausius-Clapeyron equation, Henry s law, and... 

Better examples of shortcut design methods developed from property data are fractionator tray efficiency, from viscosity " and the Clausius-Clapeyron equation which is useful for approximating vapor pressure at a given temperature if the vapor pressure at a different temperature is known. The reference states that all vapor pressure equations can be traced back to this one. 

Unfortunately values of A// at sueh low temperatures are not readily available and they have to be eomputed by means of the Clausius-Clapeyron equation or from the equation given by Hildebrand and Scott" ... 

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

This suggests that a plot of P against 1/T should yield a line having a local slope of (-A, /R). A straight line is obtained only when is nearly constant, i.e., over a narrow range of temperatures. An integrated version of the Clausius-Clapeyron equation finds use in correlation of vapor pressure data ... 

This equation is known as the Clausius-Clapeyron equation. Rudolph Clausius (1822-1888) was a prestigious nineteenth-century German scientist B. P. E. Clapeyron (1799-1864), a French engineer, first proposed a modified version of the equation in 1834. 

In using the Clausius-Clapeyron equation, the units of AH and R must be consistent. If AH is expressed in joules, then R must be expressed in joules per mole per kehrin. Recall (Table 5.1, page 107) that... 

Strategy It is convenient to use the subscript 2 for the higher temperature and pressure. Substitute into the Clausius-Clapeyron equation, solving for Pi. Remember to express temperature in K and take R = 8.31 J/mol K. 

Use the Clausius-Clapeyron equation to relate vapor pressure to temperature. 

The Arrhenius equation can be expressed in a different form by following the procedure used with the Clausius-Clapeyron equation in Chapter 9. At two different temperatures, T2 and Tlt... 

As pointed out earlier, the equilibrium constant of a system changes with temperature. The form of the equation relating K to T is a familiar one, similar to the Clausius-Clapeyron equation (Chapter 9) and the Arrhenius equation (Chapter 11). This one is called the van t Hoff equation, honoring Jacobus van t Hoff (1852-1911), who was the first to use the equilibrium constant, K. Coincidentally, van t Hoff was a good friend of Arrhenius. The equation is... 

Corollary 3.—The Clausius-Clapeyron equation, in terms of reduced magnitudes, may be written... 

We have deduced the Clausius-Clapeyron equation for the vapor pressure of a liquid at two different temperatures ... 

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 vapor pressure of a liquid increases as the temperature increases. The Clausius—Clapeyron equation gives the quantitative dependence of the vapor pressure of a liquid on temperature. 

STRATEGY Use the Clausius-Clapeyron equation to find the temperature at which the vapor pressure has risen to 1 atm (101.325 kPa). 

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

Using the Clausius-Clapeyron Equation Living Graph on the Web site for this book, plot on the same set of axes the lines for AH = 15, 20., 25, and 30. kj-mol 1. Is the vapor pressure of a liquid more sensitive to changes in temperature if AH is small or large ... 

This simple theory is unsatisfactory, in that the rate of change of the difference in free energy of liquid and crystalline lead predicted by the Clausius-Clapeyron equation leads to a temperature scale for Fig. 8 four... 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

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. 

At large Euler numbers when AP < 1, the vapor essure may be calculated by the Clausius-Clapeyron equation. In this case Ps and ft in Eq. (9.38) correspond to the saturation parameters. 

This latent heat of evaporation, Le, also appears in the fundamental description of the dependence of the vapor pressure of water, p, on temperature, T - the Clausius-Clapeyron equation ... 

Although thermodynamically it is relatively simple to determine the amount of water vapor that enters the atmosphere using the Clausius-Clapeyron equation (see, e.g.. Chapter 6, Equation (1)), its resultant atmospheric residence time and effect on clouds are both highly uncertain. Therefore this seemingly easily describable feedback is very difficult to quantify. 

The saturation vapor pressure is strictly a function of temperature as indicated by the Clausius-Clapeyron equation... 

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.-... 

Use the vapor composition and the Clausius/Clapeyron equation in reverse to roughly estimate AHvap. This requires two points on the vapor pressure chart. 

For a bulk liquid at pressure pL, the vapor pressure pG of the superheated liquid near the wall can be related to the amount of superheat, (TG — Tsat), by the Clausius-Clapeyron equation,... 

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

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