# 5.30: Untitled Page 92

- Page ID
- 18225

## Chapter 5

sign in the definition of the thermal expansion coefficient makes positive for most liquids. There is an interesting counter example, however, and it is the density of liquid water at low temperatures. For liquid water between 4 C and the freezing point, 0 C, the coefficient of expansion is negative, i.e., 0 . If it were not for this characteristic, water in lakes and rivers would freeze from the bottom during the winter, and this would destroy most aquatic life.

The density of *ideal liquid mixtures* is computed using Amagat’s law that leads to

*A * *N*

*A * *N*

o

*V*

*V*

*m*

(5‐18)

*A*

*A*

*A*

*A * 1

*A * 1

Here *V* is the volume of the mixture, while *m * and o

represent the masses and

*A*

*A*

densities of the pure components. Equation 5‐18 can be used to compute the density of the mixture according to

*m*

*m*

1

(5‐19)

*A * *N*

*A * *N*

*V*

o

o

*m*

*A*

*A*

*A*

*A*

*A * 1

*A * 1

At low to moderate pressures this version of Amagat’s law is a satisfactory approximation. However, non‐ideal behavior of liquid mixtures is a very complex topic. When some of the components of a liquid mixture are above their boiling points, or if the components are polar, the use of Eq. 5‐19 may lead to significant errors (Reid *et al*., 1977).

**5.3 Vapor Pressure of Liquids **

If we study the *p*‐ *V*‐ *T* characteristics of a real gas using the experimental system shown in Figure 5‐2, we find the type of results illustrated in Figure 5‐3.

In the system illustrated in Figure 5‐2, a single component is contained in a cylinder immersed in a constant temperature bath. We can increase or decrease the pressure inside the cylinder by simply moving the piston. When the molar volume (volume per mole) is sufficiently large, the distance between molecules is large enough (on the average) so that molecular interaction becomes unimportant. For example, at the temperature *T *, and a large value of *V / n *, we 3

observe ideal gas behavior in Figure 5‐3. This is illustrated by the fact that at a fixed temperature we have

*Two‐Phase Systems* & *Equilibrium Stages *

165

*pV n * constant

(5‐20)

However, as the pressure is increased (and the volume decreased) for the system illustrated in Figure 5‐3, a point is reached where liquid appears and the pressure *Figure 5‐2*. Experimental study of *p*‐ *V*‐ *T* behavior remains constant as the volume continues to decrease. This pressure is referred to as the *vapor pressure* and we will identify it as *p*

. Obviously the vapor

*vap*

pressure is a function of the temperature and knowledge of this temperature dependence is crucial for the solution of many engineering problems.

In a course on thermodynamics students will learn that the *Clausius‐Clapeyron* *equation* provides a reasonable approximation for the vapor pressure as a function of temperature. The Clausius‐Clapeyron equation can be expressed as

*H*

1

1

*p*

*p*

( *T *) exp

*vap*

(5‐21)

*A*, *vap*

*A*, *vap*

o

*R*

*T*

*T*

o

in which *p*

represents the vapor pressure at the temperature *T*. We have *A*, *vap*

used *p*

( *T *) to represent the vapor pressure at the reference temperature *T *, *A*, *a*

*v p*

o

o

while *H*

represents the molar heat of vaporization. A more accurate *vap*

166