# 5.4: Vapor Pressure of Liquids

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If we study the \(p-V-T\) characteristics of a real gas using the experimental system shown in Figure \(\PageIndex{1}\), we find the type of results illustrated in Figure \(\PageIndex{2}\). In the system illustrated in Figure \(\PageIndex{1}\), 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 \(T_{3}\), and a large value of \(V/n\), we observe ideal gas behavior in Figure \(\PageIndex{2}\). This is illustrated by the fact that at a fixed temperature we have

\[{pV / n} = { constant} \label{20}\]

However, as the pressure is increased (and the volume decreased) in the system illustrated in Figure \(\PageIndex{2}\), a point is reached where liquid appears and the pressure remains constant as the volume continues to decrease. This pressure is referred to as the **vapor pressure **and we will identify it as \(p_{vap}\). Obviously the vapor 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 you 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

\[p_{A, vap} = p_{A, vap} (T_{ o} ) \exp \left[-\frac{\Delta H_{vap} }{R} \left(\frac{1}{T} - \frac{1}{T_{ o} } \right)\right] \label{21}\]

in which \(p_{A, vap}\) represents the vapor pressure at the temperature \(T\). We have used \(p_{A, vap} (T_{ o} )\) to represent

the vapor pressure at the reference temperature \(T_{ o}\), while \(\Delta H_{vap}\) represents the molar heat of vaporization. A more accurate empirical expression for the vapor pressure is given by Antoine’s equation^{3}

\[\log_{10} (p_{A, vap}) = A - \frac{B}{\left(\theta + T\right)} \label{22}\]

in which \(p_{A, vap}\) is determined in mm Hg and \(T\) is specified in \(C\). The coefficients \(A\), \(B\), and \(\theta\) are given in Table A3 of Appendix A for a variety of compounds. Note that Equation \ref{22} is *dimensionally incorrect* and must be used with great care as we indicated in our discussion of units in Sec. 2.3.

Example \(\PageIndex{1}\): Vapor Pressure of a Single Component

In this example we wish to estimate the vapor pressure of methanol at 25 C using the Clausius-Clapeyron equation. The heat of vaporization of methanol is \(\Delta H_{vap} = 8426\) cal/mol at the normal boiling point of methanol, 337.8 K. The heat of vaporization is a function of temperature and pressure. The data given for the heat of vaporization is for the temperature T = 337.8 K = 64.6 C.

At this temperature, the vapor pressure of methanol is equal to atmospheric pressure. In order to estimate the vapor pressure at 25 C, we use the normal boiling temperature as the reference temperature. Normally we would compute the value of the heat of vaporization at 25 C using a thermodynamic relationship and then use an average value for \(\Delta H_{vap}\) in Equation \ref{21}. However, in this example, we will estimate the vapor pressure at 25 C using the heat of vaporization at 64.6 C. All variables can be converted into SI units as follows:

\[ \text{ Temperature: } 25 \ C \ + 273.16 \ K \ = 298.16 \ K \label{1}\tag{1}\]

\[ T_{ o} = 337.8 \ K \label{2}\tag{2}\]

\[ p_{M, vap} (T_{o} ) = 1 \text{ atm } = 101,300 \ Pa \label{3}\tag{3}\]

\[ \Delta H_{vap} = (8426 \ cal/mol) (4.186 \ J/cal) = 35,271 \ J/mol \label{4}\tag{4}\]

Substitution of these results into Equation \ref{21} gives

\[\begin{align}\nonumber p_{M, vap} & = \left(101,300 \ Pa \right) \exp \left[- \frac{35271 \ J/mol}{8.314 \ m^3 \ Pa/mol \ K} \left( \frac{1}{298.2 K} - \frac{1}{337.8 K} \right) \right] \\ & = 19,112 \ Pa \label{5}\tag{5} \end{align}\]

Vapor pressures estimated using the Clausius-Clapeyron equation can exhibit substantial errors with respect to experimental values of vapor pressure. This is caused by the fact that the assumptions made in the development of this equation are not always valid. The semi-empirical equation known as Antoine’s equation has the advantage that it is based on the correlation of experimental values of the vapor pressure.

Example \(\PageIndex{2}\): Vapor Pressure of Single Components using Antoine’s Equation

In this example, we determine the vapor pressure of methanol at 25 C using Antoine’s equation, Equation \ref{22}, and compare the result with the vapor pressure computed in Example \(\PageIndex{1}\). The numerical values of the coefficients in Antoine’s equation are obtained from Table A3 of Appendix A and they are given by

\[A = 8.07246, \quad B = 1574.99, \quad \theta = 238.86\]

Once again we note that Antoine’s equation is a dimensionally incorrect empiricism and one must follow the rules of application that are given above and in Table A3 of Appendix A. Substitution of the values for \(A\), \(B\) and \(\theta\) into Equation \ref{22} gives

\[\log p_{{M},vap} = 8.07246 - \frac{1,574.99}{238.86 + 25} = 2.10342 \nonumber \]

and the vapor pressure of methanol at 25 C is \(p_{ M, vap}\) = 126.9 mmHg = 16,912 Pa. The result computed using the Clausius-Clapeyron equation was, \(p_{ M, vap}\) = 19,112 Pa, thus the two results differ by 11%. The results of this example clearly indicate that it is misleading to represent calculated values of the vapor pressure to five significant figures.

## Mixtures

the behavior of vapor-liquid systems having more than one component can be quite complex; however, some mixtures can be treated as *ideal*. In an ideal vapor-liquid multi-component system the partial pressure of species \(A\) in the gas phase is given by

Equilibrium relation: \[p_{A} = x_{A} p_{A,vap} \label{23}\]

Here \(p_{A}\) is the partial pressure of species \(A\) in the *gas phase*, \(x_{A}\) is the mole fraction of species \(A\) in the *liquid phase*, and \(p_{A,vap}\) is the vapor pressure of species \(A\) at the temperature under consideration. It is important to remember that Equation \ref{23} is an *equilibrium relation*; however, when the condition of *local thermodynamic equilibrium* is valid Equation \ref{23} can be used to calculate values of \(p_{A}\) for dynamic processes.

The equilibrium relation given by Equation \ref{23} represents a special case of a more general relation that is described in many texts^{4},^{5} and will be studied in a course on thermodynamics. The general equilibrium relation is based on the *partial molar Gibbs free energy*, or the chemical potential, and it takes the form

Equilibrium relation: \[(\mu_{A} )_{gas} = (\mu_{A} )_{liquid} , \quad \text{ at the gas-liquid interface} \label{24}\]

Here we have used \(\mu_{A}\) to represent the *chemical potential* of species \(A\) that depends on the temperature (strongly), the pressure (weakly), and the composition of the phase under consideration. The dependence on the composition is generally represented in terms of mole fractions so that the chemical potential of species \(A\) in the liquid phase is given by

\[(\mu_{A} )_{liquid} = \mathscr{F}(T,p,x_{A} , x_{B} , \text{ etc.)} \label{25}\]

For some situations it is convenient to represent the functional dependence on the composition in terms of the molar concentration, \(c_{A}\), or the mass fraction, \(\omega_{A}\), or the partial pressure, \(p_{A}\). One can see the similarity between Equation \ref{23} and Equation \ref{24} by expressing the former as

Equilibrium relation: \[(p_{A} )_{gas} = p_{A,vap} (x_{A} )_{liquid} \label{26}\]

The matter of extracting Equation \ref{26} from Equation \ref{24} will be taken up in a subsequent course on thermodynamics.

For an ideal gas, use of Equation \((5.2.1)\) along with Equation \((5.2.2)\) indicates that the gas-phase mole fraction can be expressed as

\[y_{A} = {p_{A} / p} \label{27}\]

This result can be used with Equation \ref{23} to obtain a relation between the gas and liquid-phase mole fractions that is given by

Equilibrium relation: \[y_{A} = x_{A} \left({p_{A,vap} / p} \right) \label{28}\]

This equilibrium relation is sometimes referred to as *Raoult’s law*. For a two-component system we can use Equation \ref{28} along with the constraint on the mole fractions

\[x_{A} +x_{B} = 1 , \quad y_{A} +y_{B} = 1 \label{29}\]

to obtain the following expression for the mole fraction of species \(A\) in the gas phase:

Equilibrium relation: \[y_{A} = \frac{\alpha_{AB} x_{A} }{1 + x_{A} (\alpha_{AB} -1)} \label{30}\]

Here \(\alpha_{AB}\) is the *relative volatility* defined by

\[\alpha_{AB} = \frac{p_{A,vap} }{p_{B,vap} } \label{31}\]

For a dilute binary solution of species \(A\), one can express Equation \ref{30} as

\[y_{A} = \alpha_{AB} x_{A} , \quad \text{ for } x_{A} (\alpha_{AB} -1) << 1 \label{32}\]

and this special form of Raoult’s law is often referred to as *Henry’s law*. For an \(N\)-component system, one can express Henry’s law as

Henry’s Law: \[y_{A} = K_{eq,A} x_{A} \label{33}\]

Here \(K_{eq,A}\) is referred to as the Henry’s law *constant* even though it is not a constant since it depends on the temperature and composition of the liquid, i.e.

\[K_{eq,A} = \mathscr{F} \left(T, x_{A} , x_{B} , x_{C} , ....x_{N-1} \right) \label{34}\]

This treatment of gas-liquid systems is extremely brief and devoid of the rigor that will be encountered in a comprehensive discussion of phase equilibrium. However, we now have sufficient information to solve a few simple mass balance problems that involve two-phase systems.