Clausius-Clapeyron equation 5.24

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined 

For a pure substance at rest with no electrical or magnetic effects, the first law of thermodynamics for a closed system is expressed for a differential change as 

Equation (6.8) is a general state equation for the pure substance. From the definition of enthalpy, h = u+ pv, Equation (6.7) can be expressed in an alternative form  

Consequently, it is permissible to equate the derivatives, as these are continuous functions, 

From Equation (6.8b), by division with dr or by a formal limit operation, we recognize that in this two-phase region, 

When applied to vaporization the Clapeyron equation can be modified to give another useful and important relation. We use the equation derived in Section 4.6, 

These equations, which relate the temperature dependence of the vapour pressure of a liquid to Atfvap, its enthalpy change per mole on vaporization, are called the Clausius-CIapeyron equations. Unlike the Clapeyron equation, they are not exact, as a number of approximations were introduced in their derivation, but they are nevertheless extremely valuable. 

Continuing to consider pure substances, the molar volume of a gas, Vm g  

Integration of both sides, assuming that AvavH is independent of temperature and pressure  

If we now take exponentials (Frame 6) of both sides of equation (26.11) we have  

Equation (26.11) can be transformed to produce yet another form of the Clausius-Clapeyron equation. 

A liquid boils, at Tb (= T2, say) when its vapour pressure (say P2) equals that of the surrounding atmosphere, (= P° = 1 atm), so that we can write (using equation (26.11)) that  

Within a mean field approximation the pressure induced changes of the bin-odal and critical temperature are determined by the corresponding changes 

At the critical temperature the interpretation of the thermodynamic parameters AV and AH is not correctly represented within FH theory because thermal fluctuations lead to a renormalization of the mean field critical temperature which in first order is determined as Tc = Tc (l - Gi) [lOj. Thermal fluctuations yield a slightly enhanced miscibility whose degree is determined by Gi. This is reasonable as thermal fluctuations also enhance the entropy. At elevated pressure this stabilization effect is reduced as is visible from Gi in Fig. 14. 

Also including contributions from thermal composition fluctuations the Clausius-Clapeyron equation for critical polymer blends can be expressed 

The first term is determined from the eflfective FH parameter F, while the second term describes the stabilization effect of the thermal fluctuations. The parameters Tc(l), Gz(l), H(l) represent the corresponding values at the ambient pressure field of 0.1 MPa. 

The effect of pressure on thermal fluctuations in PB/PS always seems to lead to a unique behavior, namely, to an increase of the critical temperature. But the situation is more complex if one considers the contribution from the pressure dependence of the FH parameter F P) alone and as already evident from the application of the Clausius-Clapeyron equation. The enthalpic term 

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Clausius-Clapeyron equation See Clapeyron-Clausius 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... 

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. 

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 ... 

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 ... 

Two estimates will be made using vapor pressure data from the CRC Handbook [63] and the integrated form of Clausius-Clapeyron equation ... 

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... 

Clausius-Clapeyron equation An equation expressing the temperature dependence of vapor pressure ln(P2/Pi) = AHvapCl/Tj - 1/T2)/R, 230,303-305 Claussen, Walter, 66 Cobalt, 410-411 Cobalt (II) chloride, 66 Coefficient A number preceding a formula in a chemical equation, 61 Coefficient rule Rule which states that when the coefficients of a chemical equation are multiplied by a number n, the equilibrium constant is raised to the nth power, 327... 

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

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... 

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 ... 

Claus process, 634 Clausius inequality, 288 Clausius-Clapeyron equation, 312 clay, 616... 

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... 

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