Van der Waals equation

Now that we done this promenade through Flory's original model, we now walk through the gas models and to mean field theory. I recognize that we have already had to slog through a seemingly endless sea of parameters and abstract concepts to arrive at (47). Moreover, at the end, I said this mass of parameters doesn't even help that much and whisked most of them out of sight. The purpose of running through the Van der Waals model is to show that there actually is some meaningful physics happening here even if the Flory model didn't quite arrive at what we wanted. Moreover, I want to show that the direction we are going is physics that is not so far away from things that we can understand without the frustration of an endless trail of parameters.

From statistical mechanics, one can show that the entropy of an ideal gas is expressed as follows

$$S(U,V,N)=Nk_B \ln \left( {\frac{V}{N}} \right)+\frac{3N}{2}k_B \ln \left\{ {\frac{4\pi mU}{3Nh^2}} \right\}+\frac{5}{2}Nk_B$$ (48)

where $U$ is the internal energy, $h$ is Planck's constant and $m$ is the mass. All terms except the first are independent of volume ($V)$. The Helmholtz FE is therefore

$$F=U-TS=-Nk_B T\ln \left( {\frac{V}{N}} \right)+\;\mbox{terms}\;\mbox{independent}\;\mbox{of}\;V$$ (49)

The FE can be used to derive the equation of state for the ideal gas by $\partial F / \partial V$

$$P = - \frac{\partial F}{\partial V} = \frac{ N k_B T}{V}$$ (50)

In general, the virial equation for a real gas or liquid is expressed in a similar form

$$P=-\frac{\partial F}{\partial V}=\frac{Nk_B T}{V}\left\{ {B_... ...p {\left( {\frac{N}{V}} \right)}\nolimits^2 \ldots } \right\}$$ (51)

where one can see that the first term of this expression corresponds to the ideal gas ($i.e.$, $B_1 =1)$. The terms following involve two-body ($B_2 )$ and three-body ($B_3 )$ interactions. The $B_2$ term also corresponds to the excluded volume and is often referred to as such. For a Van der Waals gas, the interaction is weakly attractive on short range, on very short range repulsive and far away neutral

$$V_{ij} (r_{ij} )=4\varepsilon \left[ {-\mathop {\left( {\fra... ...t( {\frac{\sigma }{r_{ij} }} \right)}\nolimits^{12} } \right]$$ (52)

where for an ideal gas, $r_{ij}$ is the distance between particle $i$ and $j$ ( ${\rm {\bf r}}_{ij} ={\rm {\bf r}}_i -{\rm {\bf r}}_j )$, $\sigma$ is on the order of 3-4 Å and $\varepsilon /k_B$ is on the order of 100 - 200 K. This corresponds to a repulsive core and a weakly attractive interaction. Stronger attraction will change this function accordingly.

To evaluate the excluded volume, potentials such as (52) should, in principle, be evaluated in the partition function over all microstates using the full Hamiltonian (both the momentum and potential terms together),

$$Z = \frac{1}{N! h^{3N}} \int{ \exp \left( - \frac{H}{k_B T} \right) d^{N}\mathbf{p} d^{N}\mathbf{r}}$$ (53)

where

$$H = \sum_{i}^{3N}{\frac{p_i^2}{2m}} + \sum_{i<j}^{N}{V(\mathbf{r}_i-\mathbf{r}_j)}$$ (54)

Equation (53) with (54) is difficult to compute exactly because the momentum is directly correlated with the potential energy in any realistic system and the momentum of each particle depends upon its instantaneous interaction with the entire system or at least some local approximation thereof. Therefore, one attempts a work-around by pretending that the momentum of a bound particle is only weakly coupled to the potential energy allowing separation of the momentum ($p)$ and potential energy ($V$) parts of the Hamiltonian in the partition function: $Z = Z(p)Z(V)$, where the momentum part ($Z(p)$) generates the same solution as the ideal gas.

One such work-around is known as the Mayer function. The Mayer function is a clever artifice aimed at encouraging the potential energy terms to converge independent of the momentum terms

$$f(r_{ij}) = \exp \left( - \frac{V(\mathbf{r}_i-\mathbf{r}_j)}{k_B T} \right) - 1$$ (55)

where $i\ne j$. The potential energy part of the partition function becomes

$$Z(V) = \frac{1}{N!} \int{ \exp \left( - \frac{\sum_{i<j}^{N}V(\mathbf{r}_{ij})}{k_B T} \right) d^{N}\mathbf{r}}$$ (56)

whereupon one inserts the Mayer function into (56)

$$Z(V) = \frac{1}{N!} \int{ \prod_{i<j}{(1+f(r_{ij}))} d^{N}\mathbf{r}}$$ (57)

and expanding in a series generates

$$Z(V) = \frac{1}{N!} \int{ \left( 1 + \sum_{i<j}{f(r_{ij})} ... ..._{i<j,k<m}{f(r_{ij})f(r_{km})} \dots \right) d^{N}\mathbf{r}}$$ (58)

After some effort, it is possible to show that the two-body interaction is

$$B_2 =-\frac{1}{2}\smallint f(r)d{\rm {\bf r}}$$ (59)

and likewise, the three-body interaction term () is

$$B_3 =-\frac{1}{3V}\smallint f(r_{ij} )f(r_{ik} )f(r_{jk} )d{\rm {\bf r}}_i d{\rm {\bf r}}_j d{\rm {\bf r}}_k$$ (60)

where $i\ne j\ne k$.

The stratagem of invoking weak coupling to separate the momentum and the potential energy is mitigated by the sheer number of particles involved and the broad distribution of respective momenta. However, it should always be firmly planted in one's mind that real systems as apparently simple as pure water are not so trivial when examined on closer scrutiny.

The term for two-body interaction is considered in the Van der Waals expression. We assume that the weak attraction between particles reduces the internal energy by a small factor proportional to the density of the gas

$$U^\prime =-a\left( {\frac{N}{V}} \right)N$$ (61)

where $a$ is a constant that depends on the specific properties of the molecule involved.

In addition, a small correction accounting for the free volume is introduced into the ideal gas term so that (48) becomes

$$F=-Nk_B T\ln \left( {V-Nb} \right)-a\left( {\frac{N}{V}} \right)N+\;\mbox{terms}\;\mbox{independent}\;\mbox{of}\;V$$ (62)

where all terms independent of $V$ are ignored. Upon differentiation with respect to $V$, equation (62) yields the famous Van der Waals gas equation

$$\left\{ {P+a\mathop {\left( {\frac{N}{V}} \right)}\nolimits^2 } \right\}\left( {V-Nb} \right)=Nk_B T$$ (63)

where by inspection, one can recognize $B_2 = a/k_B T$.

The free volume contribution is meshed into the volume term in (62), however, expanding it in a power series, we obtain $\ln \{ ( V-Nb) \} \sim \ln ( V ) - N b/V + (N b/V)^2/2 \cdots$. Hence, (62) can also be approximated as

$$\left\{ {P+\left( {a-k_B Tb} \right)\mathop {\left( {\frac{N}{V}} \right)}\nolimits^2 } \right\}V=Nk_B T$$ (64)

Term $b$ represents an entropic contribution (the excluded volume) and $a$ reflects the enthalpic contributions (heat of fusion for example). More will be discussed on this point later when we compare this with Flory's model.