Module Goal: To demonstrate thermodynamic quantification using modern cubic EOS.
Module Objective: To introduce you to the basis premises of cubic EOS and their behavior.
If we multiply the vdW EOS (expression 7.11a in Module 7) by ${\tilde{v}}^{2}$ and expand the factorized product by applying the distributive law, the result is the vdW EOS expressed in terms of molar volume, as follows:
$${\tilde{v}}^{3}-\left(b+\frac{RT}{P}\right){\tilde{v}}^{2}+\left(\frac{a}{P}\right)\tilde{v}-\frac{ab}{P}=0$$ (9.1)
Note that equation (9.1) is a third order polynomial in $\tilde{v}$ i.e., it is cubic in molar volume. Additionally, we can substitute the definition of compressibility factor Z,
$$Z=\frac{P\tilde{v}}{RT}$$ (9.2)
into equation (9.1) and obtain a different cubic polynomial in Z, as shown:
$${Z}^{3}-\left(1+\frac{bP}{RT}\right){Z}^{2}+\left(\frac{aP}{{R}^{2}{T}^{2}}\right)Z-\frac{ab{P}^{2}}{{\left(RT\right)}^{3}}=0$$ (9.3)
As we see, vdW EOS is referred to as cubic because it is a polynomial of order 3 in molar volume (and hence in compressibility factor Z). In general, any equation of state that is cubic in volume (and Z) and explicit in pressure (equation 7.11b) is regarded as a cubic equation of state. vdW EOS is a cubic EOS, and all the transformations and modifications that it has undergone during the more than one hundred years since its publication are also cubic EOS; or better, they are in-the-van-der-Waals-spirit EOS or of-the-van-der-Waals-family EOS.
Let us see what we have accomplished thus far. First, remember that we are not satisfied with the fact that the ideal gas law is not able to predict the discontinuity of Figure 6.3, which corresponds to the condensation that an all-gas pure substance undergoes during isothermal compression.The isotherm that we get from the ideal model was shown in Figure 6.2. Condensation, as we know, is to be expected at some point when you isothermally compress any pure gas while below its critical conditions (T < T_{c}). This lack of conformance to actual behavior led us to realize that the ideal model is quantitatively and qualitatively wrong at high pressures. To remove the weaknesses of the ideal model, we recall that vdW, in developing his equation of state (equation 7.11), introduced the concepts of co-volume and the attraction term.
Now, we wonder, what did he really accomplish? Are we now able to predict the condensation phenomena? Are these “new” cubic-type EOS capable of showing where such a discontinuity occurs? So far, we have not seen what a cubic isotherm looks like. Let us plot the cubic isotherm for conditions below critical (T < T_{c}), superimpose it on Figure 6.3, and see what we get.
Figure 9.1 shows a typical cubic behavior. That is, around condensation conditions, the equation of state presents an S-shaped curve. This should not come as a surprise; because the equation is cubic in volume, it will provide three roots for volume, and hence the S-shaped cubic feature. For the sake of our discussion, we have shown the gas and liquid branches of the cubic equation lying directly on top of the experimental ones. This kind of matching is the “ultimate goal” of any cubic EOS, but in reality, the matching is not this good — especially for the case of the original vdW EOS, as we will discuss later.
The cubic behavior is bounded by the two extremes of real fluid behavior, given by the zero pressure and infinite pressure limits. On one hand, it is clear from equation (7.11b) that we have a singularity at $\tilde{v}=b$ , where “b” represents the co-volume, or physical space that molecules themselves occupy. This singularity conveniently creates an asymptotic behavior (high pressure asymptote) of the cubic equation liquid branch, by which $\tilde{v}\to b$ as $P\to \infty $ . Recall from our previous discussions that predictions for $\tilde{v}<b$ are meaningless. Accordingly, we need an infinite amount of pressure if we are to compress a fluid to the extent that no free space is available among molecules $(v\u02dc=b)$ . On the other hand, it is clear from vdW EOS (equation 7.11a) that as $v\to \infty \left(P\to 0\right)$ , the cubic EOS collapses to the ideal EOS (equation 7.6). At this low pressure limit, vdW
corrections to the ideal model become inconsequential $\left(a/{\tilde{v}}^{2}\to 0,\text{}\tilde{v}b\right)$ .
Mathematically, this is the low-pressure asymptote of the cubic-equation gas branch by which $\tilde{v}\to \infty $ as $P\to 0$ .
Let us see how “good” the cubic behavior is for the ideal-gas model. Refer to Figure 9.2, where we have superimposed the ideal gas isotherm of Figure 6.2 on the cubic EOS behavior.
A look at Figure 9.2 helps us confirm that the cubic equation of state collapses to the prediction of the ideal gas model at reduced pressure — i.e., they share the same low-pressure asymptote. This is to be expected, since the assumptions underlying the ideal model are satisfied. As we recall, these assumptions are that the attractive forces between molecules are very weak and that the physical volume of the molecules can be disregarded when compared to the total volume of the container. It is worthy to note that the high pressure asymptote is not the same for both models; as $P\to \infty $ , $\tilde{v}\to b$ as for the cubic model while $\tilde{v}\to 0$ for the ideal model. The latter is a direct consequence of neglecting the molecular volume in the ideal model.
A major issue that has kept us banging our heads has been how to mathematically represent the discontinuity of the P-v isotherm during the vapor-liquid transition (Figure 6.3). Such a discontinuity shows up during the isothermal compression of any pure substance at sub-critical conditions (T < T_{c}). What we want here requires fitting a continuous mathematical function to a discontinuous, real-life event. Strictly speaking, it would be contradictory to find a single continuous mathematical function that can capture such a discontinuity in its full nature.
Can we really model the discontinuity? Not really, but we can get around it. van der Waals provided a possible solution in his dissertation on the “continuity of vapor and liquid.” Even though neither cubic equations nor any other continuous mathematical function is able to follow the discontinuity, what they can do is good enough for engineering purposes. The “cubic behavior” can reasonably match the liquid and vapor branches for the real, experimental isotherms.
Since van der Waals’ EOS, we have been able to consider the continuity between gas and liquid phases. Now, we need to learn how to deal with the S-shaped behavior, and to look at it as a minor, inconsequential price that we pay for the modeling of the vapor — liquid discontinuous transition with a continuous mathematical function. Let us zoom in on Figure 9.1, as shown in Figure 9.3.
There are several features of the S-shaped behavior that should be noted.
Answer the following problems, and submit your answers to the drop box in Canvas that has been created for this module.
Please note:
Cubic EOS yield three roots for Z-factor for a pure substance at subcritical conditions. Does that mean that there is a “Z” factor for liquids? Isn’t the “Z” factor concept only applicable to gases? What does a “Z factor” connote for liquids?