Liquids Clapeyron equation

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... 

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 ... 

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 ... 

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. 

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) ... 

This equation is called the Clapeyron equation and can be applied to any two phases in equilibrium, e.g., solid and liquid, liquid and vapor, solid and vapor or two crystalline forms of the same solid. Thus for the equilibrium... 

It is of interest to consider the variation of vapor pressure with temperature. The vapor pressure of a liquid is constant at a given temperature. It increases with increasing temperature upto the critical temperature of the liquid. The liquid is completely in the vapor state above the critical temperature. The variation of the vapor pressure with temperature can be expressed mathematically by the Clapeyron-Clausius equation. Clausius modified the Clapeyron equation in the following manner by assuming that the vapor behaves like an ideal gas. 

If the temperature is not near the critical temperature, the volume of a liquid can be considered to be negligibly small compared with the volume of the vapor. The Clapeyron equation then becomes... 

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,... 

Since the molar volume of a gas is much larger than the molar volume of the liquid phase, the Clapeyron equation can be reduced to... 

It is possible, however, to simplify the calculation of the energy transfer by assuming that the vapor phase is always a saturated vapor. O Connor (Ol) has shown that the rate of approach of a superheated vapor to saturated conditions is extremely rapid when the superheated vapor is in direct contact with its liquid phase. If the vapor phase is assumed to be saturated, the temperature of the phase can be calculated from an integrated form of the Clausius-Clapeyron equation instead of from the vapor-phase energy-transfer equation. 

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. 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

The third method is a variation of the second a heat pulse dg is supplied, keeping both the pressure and temperature constant by means of a feedback system. In this process, some moles of liquid 3He are solidified the latent heat of solidification is the measured dg. Since from 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,... 

This is known as the Clausius-Clapeyron equation. It is a state relationship that allows the determination of the saturation condition p = p(T) at which the vapor and liquid are in equilibrium at a pressure corresponding to a given temperature. 

Recall 7F o = 1 for a pure fuel condensed phase, and Ff(0) is not known. For a liquid fuel, Ff(0) is found from the Clausius-Clapeyron equation provided we know 7 (0). 

We are now able to use this model for the Gibbs energy of the liquid to calculate the melting line for four-coordinated Si by using the Clapeyron equation (eq. 2.10) ... 

We are permitted to assume that dp is directly proportional to dT because AH and AV are regarded as constants, although even a casual inspection of a phase diagram shows how curved the solid-gas and liquid-gas phase boundaries are. Such curvature clearly indicates that the Clapeyron equation fails to work except over extremely limited ranges of p and T. Why ... 

The worst deviations from the Clapeyron equation occur when one of the phases is a gas. This occurs because the volume of a gas depends strongly on temperature, whereas the volume of a liquid or solid does not. Accordingly, the value of A Vm is not independent of temperature when the equilibrium involves a gas. 

We saw from the Clapeyron equation, Equation (5.1), how the decrease in freezing temperature dT is proportional to the applied pressure dp, so one of the easiest ways of avoiding the lethal conversion of solid ice forming liquid water is to apply a smaller pressure - which will decrease dT in direct proportion. 

Having qualitatively discussed the way a pressure cooker facilitates rapid cooking, we now turn to a quantitative discussion. The Clapeyron equation, Equation (5.1), would lead us to suppose that dp oc dT, but the liquid-gas phase boundary in Figure 5.12 is clearly curved, implying deviations from the equation. Therefore, we require a new version of the Clapeyron equation, adapted to cope with the large volume change of a gas. To this end, we introduce the Clausius-Clapeyron equation ... 

Worked Example 5.3 The Clausius-Clapeyron equation need not apply merely to boiling (liquid-gas) equilibria, it also describes sublimation equilibria (gas-solid). 

The Clapeyron equation, Equation (5.1), yields a quantitative description of a phase boundary on a phase diagram. Equation (5.1) works quite well for the liquid-solid phase boundary, but if the equilibrium is boiling or sublimation - both of which involve a gaseous phase - then the Clapeyron equation is a poor predictor. 

A sublimation process is controlled primarily by the conditions under which phase equilibria occur in a single-component system, and the phase diagram of a simple one-component system is shown in Figure 15.30 where the sublimation curve is dependent on the vapour pressure of the solid, the vaporisation curve on the vapour pressure of the liquid, and the fusion curve on the effect of pressure on the melting point. The slopes of these three curves can be expressed quantitatively by the Clapeyron equation ... 

The Clapeyron equation can be reduced to a particularly convenient form when the equilibrium between A and B is that of a gas (g) and a condensed (cond) phase [liquid or solid]. In this situation... 

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