Clausius-Clapeyron plots

A phase change scheme similar to those described above was proposed by Schwartz and Schmidt 141,142) on the basis of LEED experiments, and by Schiith and Wicke (91,101) on the basis of IR measurements for the oscillatory CO/NO reaction on Pt(lOO). The experiments of Schwartz and Schmidt demonstrated that the transition from the high- to the low-reaction-rate state was accompanied by a change from the 1 x 1 to the hex phase in LEED patterns. The position of the L-CO band in the IR spectra recorded during oscillations varied between the high- and low-reaction-rate states with a relatively high absorption band below 2050 cm present in the low-reaction-rate state, which is characteristic of CO on the Pt(lOO) hex surface. Further proof was provided by Clausius-Clapeyron plot of the conditions for the occurrence of oscillations, which yielded points near the isostere associated with the hex 1 x 1 phase transition (Fig. 14). 

Figure 3.34 Left Adsorption data for benzyl alcohol (BA, symbols) at different temperatures, 20 o, 30 , 40 o, 50 A, and 60 °C. The solid lines are the correlations given by the Flory-Huggins model. Right Clausius-Clapeyron plots for BA, with data at different loadings, 10 o, 20 , 30 o, and 40 g/1 A. Reproduced with permission from I. Quidones, J. C. Ford and G. Guiochon, Chem. Eng. ScL, 55 (2000) 909 (Figs. 7 and 8).

Figure 3.4.22 EtOH vapor adsorption istotherm curves at 5, 10, 15, 20 °C for [Cu2(bza)4(pyz)](a), Clausius-Clapeyron plot from ad- and desorption jumps (b), temperature difference between the curves at 10 and 20 °C (c), and the deduced mass-induced phase transition profile during a- 3 host crystal phase transition induced by the gas adsorption (d).

In laboratory praxis, from the Clausius-Clapeyron plot of In/ against 1/r, the enthalpy can be derived. 

CLAUSIUS-CLAPEYRON PLOT VAPOR PRESSURE I - FORMYLPIPERIOINE... 

Fig. 5. Clausius-Clapeyron plots for Ln(thd)3 complexes. For clarity complexes of metals with even atomic number are depicted with dashed lines no other significance is intended.

The basic principle of SIM follows from a fundamental phenomenological experience that stems from basic research executed in the area of physical sorption over many decades, viz., sorption isosteres may presumptively be considered as straight lines at constant sorption-phase composition, n = const., in Clausius-Clapeyron plots, In p vs. I / T.ln accordance with [41-43], this finding allows to calculate the differential molar sorption heat, Q, as difference between the molar enthalpy of the gas phase, Hg, and the partial molar enthalpy of the sorbed substance, // ... 

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 ... 

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 ... 

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. 

As we have already observed, the vapor-pressure-temperature curve is nonlinear. To reduce this curve to a linear form, a plot of log (p ) versus (1/T) can be made for moderate temperature intervals. The resultant straight line is described by the following expression, which can be derived from the Clausius-Clapeyron equation. 

As indicated by the plots in Figure 10.13a, the vapor pressure of a liquid rises with temperature in a nonlinear way. A linear relationship is found, however, when the logarithm of the vapor pressure, In Pvap, is plotted against the inverse of the Kelvin temperature, 1 /T. Table 10.8 gives the appropriate data for water, and Figure 10.13b shows the plot. As noted in Section 9.2, a linear graph is characteristic of mathematical equations of the form y = mx + b. In the present instance, y = lnPvap, x = 1/T, m is the slope of the line (- AHvap/R), and b is the y-intercept (a constant, C). Thus, the data fit an expression known as the Clausius-Clapeyron equation. ... 

The Clausius-Clapeyron equation makes it possible to calculate the heat of vaporization of a liquid by measuring its vapor pressure at several temperatures and then plotting the results to obtain the slope of the line. Alternatively, once the heat of vaporization and the vapor pressure at one temperature are known, the vapor pressure of the liquid at any other temperature can be calculated, as shown in Worked Example 10.5. 

The heat of vaporization, AHvap, of a liquid can be obtained either graphically from the slope of a plot of In Pvap versus 1 /T, or algebraically from the Clausius-Clapeyron equation. As derived in Worked Example 10.5,... 

To a fair engineering approximation AH is not only a function of the hydrogen bonds in the crystal, but also a function of cavity occupation. Because the Clausius-Clapeyron equation determines the heat of hydrate formation by the slopes of plots of In P versus 1/T, one may easily determine relationships between heats of dissociation. 

The above equations are variously labelled as Clausius-Clapeyron equations. Subject to the satisfactory nature of the assumptions made, a plot (Figure 26.1(a)) of the variation of the natural logarithm of the vapour pressure, In(P/P°), over a liquid measured at various temperatures against the reciprocal of temperature (1 /T) should be linear and have a gradient equal to — Avap H°/R so provides a means of measuring Avap H° for a variety of liquids (Figure 26.1(b)). Also from vapour pressure data for solids at two or more different temperatures one can measure AsubH°. 

The vapor pressure values obtained from the data of Table I by letting a — 1 and M = 66 (the average for hydrazine and perchloric acid -(vide infra) are tabulated in Table II and plotted as the Clausius-Clapeyron expression in Figure 1. 

To determine APsub from the approximate Clausius-Clapeyron equation [Eq. (35)], plot In p against 1/T and determine the slope of the best straight-line fit to the data by graphical or least-squares methods. 

Considering that dissociation occurs upon volatilization, the temperatures can be correlated extremely well on a In P vs (1/rd.voi) plot, where P is the total system pressure and T a.voi decomposition temperature, as the case dictates. Such a plot is shown in Fig. 11. Since the Clausius-Clapeyron relation for vapor pressure of pure substances shows an exponential dependence on temperature, TVoi was considered a pseudo-boiling point at the respective system pressure. For a substance that vaporizes congruently to its gaseous state, the slope of lines on a In P vs (l/Tvoi) plot represents the enthalpy of vaporization. Indeed, the enthalpy of vaporization calculated from the slope on a In P vs (l/TVoi) plot for the B-O2 system (360 kJ/mol) agrees exactly with the value calculated by using... 

This approximate equation, known as the Clausius/Clapeyron equation, relates the latent heat of vaporization directly to the vapor pressure curve. Specifically, it shows j that ah " is proportional to the slope of a plot of In vs. 1/ T. Experimental data for many substances show that such plots produce lines that are nearly straight. According to the Clausius/Clapeyron equation, this implies that AH " is almost constant, virtually independent of T. This is not true AH " decreases monotonically with increasing temperature from the triple point to the critical point, where it becomes zero. The assumptions on which the Clausius/Clapeyron equation are based have approximate validity only at low pressures. 

Why would it be preferable to use a Cox chart to plot and extrapolate the data of question 2 rather than the Clausius-Clapeyron equation ... 

Use a semilog plot based on the Clausius-Clapeyron equation to derive an equation for / (mm Hg) as a function of T ( C). From the plot, estimate the heat of vaporization of ethylene glycol in kJ/mol. (Remember to use absolute temperatures in the Clausius-Clapeyron equation.)... 

Equation (22-19) is useful particularly for pairs of chemically similar liquids. If Raoult s law holds, relative volatility is equal to pjpi- Therefore, it is possible to plot liquid-vapor composition diagrams for closely similar liquids, such as benzene-toluene, without further ado. Note that the value of iJFis not strictly constant over the whole composition range, even for such a mixture, because pi and P2 do not necessarily vary similarly with temperature (Clausius-Clapeyron equation). 

Stamm and Loughborough (46) first calculated and Ql as functions of wood moisture content M by applying the Clausius-Clapeyron equation to the moisture sorption isotherms for wood at several temperatures. For example, can be obtained by replotting sorption isotherms, such as are shown in Figure 9, into the form of isosteres of constant moisture content, of In h against the reciprocal of Kelvin temperature. These plots yield a family of essentially straight lines, each at a different moisture content. The magnitude of Qi at any moisture content is calculated from... 

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. 

From the form of the Clausius-Clapeyron equation given by equation (2.6), the slope of a plot of log P against 1/T is... 

The indefinite integral, Eq. (4.16), is known as the Clausius-Clapeyron equation. Unfortunately, a plot of In p versus l/T over a significant range of l/T does not give a straight line. Consequently, Eq. (4.16) often is modified one result is the Antoine equation discussed in Sec. 3.3. A definite integral of Eq. (4.15) is... 

If you plot the temperature and vapor pressure data given in Table 1, you reconstruct the liquid-vapor equilibrium line in the phase diagram of that liquid (Fig. 174). The equation of this line, and you might remember this from your freshman chemistry course, is the Clausius-Clapeyron equation ... 

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