Clapeyron equation approximations

Known as the Clapeyron equation, this is an exacl thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation... 

Figure 26-65 illustrates that Eq. (26-90) provides a linear approximation to the nonlinear relationship between two-phase specific volume and reciprocal pressure (v vs. P or vs. T ). For single components, me initial slope of the curve is found using me Clapeyron equation to give ... 

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. 

Line db in Figure 8.1 represents the equilibrium melting line for C02. Note that the equilibrium pressure is very nearly a linear function of T in the (p, T) range shown in this portion of the graph, and that the slope of the line, (d/ /d7 )s ], is positive and very steep, with a magnitude of approximately 5 MPa-K-1. These observations can be explained using the Clapeyron equation. For the process... 

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... 

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

Since 7 , is not an experimentally measurable quantity, it is useful to insert the solution for Ts (from the Clausius-Clapeyron equation) and solve for W h as an explicit function of RH0 and RHC. VanCampen et al. showed (using sample algebraic approximations and conversion factors) that substituting for Ts in Eq. (35) gives the useful solution... 

The vapour pressure can be approximately be calculated from the Clausius-Clapeyron equation ... 

A rate of reaction usually depends more strongly on temperature than on concentration. Thus, in a first-order (n = 1) reaction, the rate doubles if the concentration is doubled. However, a rate may double if the temperature is raised by only 10 K, in the range, say, from 290 to 300 K. This essentially exponential behavior is analogous to the temperature-dependence of the vapor pressure of a liquid, p, or the equilibrium constant of a reaction, K. In the former case, this is represented approximately by the Clausius-Clapeyron equation,... 

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 ... 

First, we rewrite the Clapeyron equation in response to approximation 2 ... 

If the surface temperature does not differ greatly from the surrounding temperature, the highly nonlinear surface boundary condition may be simplified by linearizing the expression for the radiation flux and the Clausius-Clapeyron equation to yield the approximation... 

Some of these ambiguities can be partially solved using a simple approach recently proposed by Gamier et al. [62], The sublimation pressure of a solid can be estimated using experimental fusion properties and the vaporization enthalpy derived from the equation of state. Using the Clapeyron equation P b can be approximated by ... 

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. 

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 ... 

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... 

Equation (9) is valid for evaporation and sublimation processes, but not valid for transitions between solids or for the melting of solids. Clausius-Clapeyron equation is an approximate equation because the volume of the liquid has been neglected and ideal behaviour of the vapour is also taken into account. 

These calibration curves also reveal the huge increase of deposition rate with temperature, as expected from the Clausius-Clapeyron equation. For small flows the deposition rate increases linearly with source flow whereas at flows > 500 seem the increase is sub-linear. With higher flows the deposition rate is dominated by the flow restriction from the process chamber to the source container. If the evaporation temperature is increased from 306 to 312 °C the deposition rates increases approximately 47%, from 19.2 to 28.2 A s-1. 

This approximate equation, known as the Clausius/Clapeyron equation, relates the. 

Considering a liquid-vapor transition and I v ip Fliq. we get the approximate Clapeyron equation... 

Calculate the number n of moles of HCl in the solution dispensed. Give S and 5 for the initial and final volumes, and give a limit of error (95 percent confidence) for n. The heat of vaporization of a liquid may be obtained from the approximate integrated form of the Clausius-Clapeyron equation. 

If Vs is assumed to be negligible in comparison to Vg and if the vapor is assumed to be an ideal gas, an approximate form of the Clapeyron equation is obtained,... 

Employing the second-law approach, calculate tJi at 66 K, 70 K, and 76 K from your experimental data, using both the exact and approximate Clapeyron equations. Estimate the experimental error in tJ and comment on any differences between the values obtained with Eqs. (2) and (3). 

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. 

A similar argument can be made to show that there will be a critical size bubble of vapor in a liquid which has been superheated or put under tension. The pressure P in such a critical size bubble of vapor must exceed the external pressure on the system pQ by an amount given by P = Po + 2external pressure (or tension) applied to the liquid, while P is determined by the vapor pressure of the liquid and for rough calculations may be approximated by the Clausius-Clapeyron equation. 

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. 

The indifferent line is thus given approximately by a simple equation analogous to the integrated form of the Clausius-Clapeyron equation. 

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