FIGURE 2.3 Compressibility of nitrogen.
3. Point B at 20 MPa and 300 K exhibits positive deviation from ideal behavior. This is due to the small specific volume of the -gas, which causes the intermolecular repulsive forces to become significant
Abbott and Van Ness (1989) present an extensive discussion of equations of state that reproduce plots such as that shown in Figure 2.3.
We will consider mainly the simplest and oldest of these EOS, namely the one due to the Dutch physicist Van der Waals. This equation of state was developed in 1910, but is still in use for many applications.
Hie Van der Waals equation is most commonly written in the form of the ideal gas law of Equation <2.1) with corrections to the p and v terms:
where a and b are constants for the particular gas.
The radial extent of the repulsive portion of the intermolecular potential function shown in Figure 2.1 is the basis for assigning a hard-sphere radius to the molecule. This quantity determines the constant b, which is four times die 'Volume" of the molecule.*
The term v - ¿in Equation (2.3) corresponds to the free volume available for unimpeded molecular motion after the effective volume occupied by the molecules has been deducted.
a in Equation (2.3) reflects the attractive portion of the intermolecular potential curve in Figure 2.1. If the parameter b is neglected for the moment, the physical meaning of the constant a can be seen by solving Equation (2.3) for the pressure:
* If rHS is the hard-sphere radius, two molecules collide if their center-to-center distance is 2rm (Fig. 2.4). The volume around a molecule from which other molecules are excluded is b = %jt(2rHi)3 - 4 x
FIGURE 2.4 The excluded volume around hard-sphere molecules.
FIGURE 2.4 The excluded volume around hard-sphere molecules.
v v v where pid is the pressure that would be exerted by the gas if it were ideal. Subtraction of the a/v2 term suggests that the actual pressure is less than the ideal-gas pressure. This may be viewed as the result of the attractive intermolecular forces "holding together" the assembly of gas molecules and thereby reducing the intensity of the molecular impacts on the container walls, which is the manifestation of pressure.
Without neglecting the constant b, Equation (2.3) can be written in the form of the compressibility:
Equation (2.3) is the T(p,v) form of the Van der Waals EOS and Equation (2.4) can be converted to the p(v,T) form by multiplying both sides by RT/V. However, being cubic in v, neither of these equations can be converted analytically to the viTjr) form, which is equivalent to the Z(T,p) function that is plotted in Figure 2.3.
The explicit v(T,p) function is useful in many practical problems involving nonideal gases. Fortunately, an accurate approximation provides a satisfactory solution. First, it is assumed that b/v « 1, so that the first term on the right in Equation (2.4) can be approximated by a one-term Taylor series expansion as 1 + b/v. The b/v term so obtained is combined with the second term in Equation (2.4) by factoring the common term 1/v, which is then further approximated by its ideal-gas value p/RT. The result of these mathematical manipulations is the explicit v(p,T) form of the Van der Waals EOS:
or, in terms of the compressibility:
Despite its simplicity, Equation (2.5a) provides a reasonably good approximation to the compressibility curves for nitrogen shown in Figure 2.3 (see Problem 2.6b).
The Van der Waals constants for N2 can be determined by fitting the curves in Figure 2.3 to Equation (2.5a). An exercise in this fitting is given by Problem 2.6a. The results of a detailed analysis yield (for N2):
With these constants, Equation (2.5a) predicts that deviations from ideality should change sign at a temperature given by setting the parenthetical term equal to zero, which yields:
T(Z < 1 <=> Z > 1) = -___________= 432 K
Although the actual switch-over temperature from Figure 2.3 appears to be closer to 300 K, the simplified form of the Van der Waals equation provides a result that is at least not unreasonable.
The Van der Waals EOS can be used to predict the critical temperature and pressure of a nonideal gas (Problem 2.4). Problem 2.10 illustrates how the Van der Waals equation can be used to account for deviations from ideality for steam. In this case, the experimental database is the steam tables.
2.3.3 The Virial Equation of State
In this EOS, the compressibility is expressed as a series in 1/v":
B and C are the second and third virial coefficients, respectively. The chief advantage of this description is that the virial coefficients can be determined by the methods of molecular thermodynamics. The second virial coefficient, for example, is given by:
where NAv and k are Avogadro's number and the Boltzmann constant, respectively. 4> is the intermolecular potential function shown in Figure 2.1 as a function of r, the separation of a pair of molecules. Values of B are shown in Figure 2.2 of Prausnitz et al. (1986) for a number of gases.
If the temperature dependence of Cv (or CP) is known, the interna] enejgy (or the enthalpy) follows by integration:
where Tref is an arbitrary reference temperature where the internal -eneigy or the enthalpy is set equal to u^ or to h^. Either of these could be chosen as zero, but the two must be related (for the ideal gas) by hnf = u^ + RTnf.
The microscopic origins of the specific heat were in Section 1.6.2. The temperature dependence of Cv (or CP) of ideal gases is due to the various ways that the particles store energy. Classical physics accords lh IcT to each mode of energy storage. These energy-storage modes start to contribute at different temperatures, resulting in specific heats that either increase monatonically with temperature or exhibit plateaus if the onset temperatures of the interna) modes are widely separated. Figure 2.5 shows these features in various gases.
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