Pressure Clausius-Clapeyron 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... 

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 ... 

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. 

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 ... 

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. 

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... 

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 ... 

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 ... 

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,... 

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. 

Referring to Figure 8, temperature Tc is the chamber temperature and Ts is the surface temperature at the salt solution/vapor interface. The temperature of the chamber is well defined and is an experimental variable, whereas Ts must be higher than Tc due to condensation of vapor on the saturated solution surface. We can determine Ts by applying the Clausius-Clapeyron equation to the problem. Assume that the vapor pressures of the surface and chamber are equal (no pressure gradients), indicating that the temperature must be raised at the surface (to adjust the vapor pressure lowering of the saturated solution) to Pc (at Tc) = Ps (at Tc). However, there is a difference in relative humidity between the surface and the chamber, where RHC is the relative humidity in the chamber and RH0 is the relative humidity of the saturated salt solution, and we obtain... 

If the vapor-phase temperature is to be evaluated from the Clausius-Clapeyron equation, the pressure in the two-phase tubular contactor must be known at each axial position. This need once again illustrates the necessity of obtaining an understanding of the hydrodynamics of two-phase systems in order to carry out the design of heat-transfer contactors. 

The vapor pressure of the liquid at the surface Pg can be evaluated from an integrated from of the Clausius-Clapeyron equation if the surface temperature Ts is known. 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

---