## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 4. Describing Liquids: Phenomenological Behavior

### 4.2. Equations of State

For sufficiently low temperature and a not too-high pressure, gas is described by the well-known *perfect gas equation of state* (EoS), given by *P* = *nRT*/*V*, where the pressure *P* is given in terms of the gas constant *R*, the temperature *T*, and the volume *V* for *n* moles of gas. All gases obey this equation in the low pressure limit, as illustrated in __Figure 4.2__, where the behavior of the *compression factor* *Z* = *PV*/*RT* is shown for H_{2}, N_{2}, and CH_{4}. Typically, the accuracy for *P* according to the perfect gas EoS is ∼1% for *P* < 10 bar. However, for somewhat higher pressures, deviations from the perfect gas law occur and the EoS becomes material-specific (__Figure 4.2__; see also __Figure 4.4__).

** Figure 4.2** Approaching the perfect gas EoS for gases as exemplified by H

_{2}, N

_{2}, and CH

_{4}. The limiting value of

*PV*= 2271 Pa m

^{3}mol

^{–1}at the triple point of water,

*T*= 273.16 K. This yields

*R*= 8.314 Pa m

^{3}mol

^{–1}K

^{–1}or 8.314 J mol

^{–1}K

^{–1}.

An initially empirical description for a somewhat higher pressure is the “virial” equation of state. For the perfect gas the compression factor *Z* = 1. For a real gas, *Z* becomes a function of volume, which can be expanded as a power series in *V*^{−1}

(4.1)

This equation is called the *virial equation of state*. The parameter *B _{j}* is denoted as the

*j*th virial coefficient which is typically a function of temperature, except

*B*

_{1}for which obviously holds

*B*

_{1}= 1. Alternatively, one can expand in

*P*

(4.2)

The coefficients *B _{j}*′ can be expressed in terms of the coefficients

*B*by inverting the series. The first three coefficients then read

_{j}If only the second virial coefficient is available, one would be inclined to use *Z* = 1 + *B*_{2}/*V*. However, the expression *Z* = 1 + *B*_{2}*P*/*RT*, obtained from substituting *B*_{2}′ = *B*_{2}/*RT* in *Z* = 1 + *B*_{2}′*P*, is usually much more accurate. On the contrary, if the third virial coefficient is available, the expression *Z* = 1 + (*B*_{2}/*V*) + (*B*_{3}/*V*^{2}) is usually much more accurate than *Z* = 1 + *B*_{2}′*P* + *B*_{3}′*P*^{2}.

For liquids, an expression equivalent to the perfect gas EoS does not exist. A simple EoS for liquids doing remarkably well is the *Tait equation* [1], given by

(4.3)

Here, *V*_{0} represents the molar volume at “zero” pressure, while *P*_{0}, *A* and *B* are (positive) material-specific parameters. Although there is little theoretical justification, the Tait equation in the integrated form represents the behavior of many types of liquid quite well up to pressures of about 1000 bar. Some values for the Tait parameters are given in __Table 4.1__. A simple empirical expression for the compressibility *κ _{T}* reads

*κ*=

_{T}*κ*

_{0}(

*V*/

*V*

_{0})

*. The effect of volume on the Gibbs energy can be estimated by integrating d*

^{γ}*G*=

*V*d

*P*−

*S*d

*T*at constant

*T*, meanwhile realizing that . This leads to

(4.4)

** Table 4.1** Parameters for Tait's isotherm for H

_{2}O and CCl

_{4}.

Similarly, the temperature dependence of *κ _{T}* can be described empirically

(4.5)

with *κ*_{0} and *β* constants. Some data for *κ*_{0}, *β* and *γ* are given in __Table 4.2__. Water behaves anomalously, however, and *κ _{T}* passes through a minimum at ∼45 °C. For the molar volume

*V*

_{sat}of a saturated liquid – that is, a liquid in equilibrium with its vapor – several other empirical equations exist.

** Table 4.2** Compressibility parameters for a few compounds.

It will be clear that the various equations of state describe either the behavior of the gas or the liquid. As early as 1873, *van der Waals* emphasized the continuity between the liquid and the gas state and advocated the EoS named after him [2]. This EoS describes liquids and gases, and for *n* moles it reads

(4.6)

with *ρ* = *N*/*V* the number density. The parameters *a* and *b* are material-specific. The intuitive interpretation for *a* is that there is attraction between the molecules so that part of the pressure expected from the perfect gas law is reduced in overcoming the force of the intermolecular attraction. For *b*, one considers that a molecule is not a point mass but rather has a certain radius, so that there is a certain excluded volume given by *b*. Since for a rigid particle with radius *σ*/2 the volume that cannot be occupied by another particle is 4π*σ*^{3}/3, one finds 2*b* = 4π*σ*^{3}/3. __Figure 4.3__ illustrates the van der Waals (vdW) behavior. For low densities, the vdW expression reduces to the perfect gas law, but for high densities sinuous curves are obtained that represent an unstable state of affairs. The usual interpretation is that the region of the sinuous curves corresponds to a separation of the fluid into liquid and vapor, and that the equation breaks down here. Maxwell argued that a horizontal line should be inserted so as to make the areas below and above that line equal (see __Chapter 16__). In this way, a general resemblance to the actual isotherms of a real fluid is obtained. For mixtures it is often assumed that *a _{ij}* = (

*a*)

_{i}a_{j}^{1/2}and

*b*= (

_{ij}*b*+

_{i}*b*)/2.

_{j}** Figure 4.3** The van der Waals equation of state, as illustrated by isotherms in the

*PV*-plane for

*T*

_{red}= 0.85, 0.90, 0.95, 1.00, 1.05, 1.10, and 1.15. The critical point is indicated by •.