EoS: The principle of corresponding states

The principle of corresponding states

Since the Lennard-Jones potential can be used somewhat generically to describe the interactions between molecules and it depends on only two parameters -- $\sigma$ and $\epsilon$ -- it is intriguing to postulate the following.

DEFINITION

The principle of corresponding states suggests that (for normalized choices of $\epsilon$ and $\sigma$) the dimensionless potential energy is the same for all species.

In other words, rather than having

$\Gamma_{ii} = 4\epsilon_i \left [ \left (\frac{\sigma_i}{r} \right )^{12} - \left ( \frac{\sigma_i}{r}\right )^6 \right ]$

apply solely to interactions between $i$ molecules (i.e., having a new version of this equation for every molecule), we can instead define a function:

$f\left(\frac{\Gamma_{ii}}{\epsilon_i},\frac{r}{\sigma_i}\right ) = 0$

that works for everything.

Adapting this statement to macroscopic quantities, van der Waals suggested that we could instead normalize using the critical properties of a fluid so that

DEFINITION

The principle of corresponding states (take 2) states that all fluids at the same "reduced temperature" ($\frac{T}{T_c}$) and "reduced pressure" ($\frac{P}{P_c}$) have the same compressibility factor.

Mathematically, this means that we can define a function:

$f\left(\frac{T}{T_c},\frac{P}{P_c},\frac{v}{v_c}\right ) = 0$

or

$z = \frac{Pv}{RT} = f'\left(\frac{T}{T_c},\frac{P}{P_c}\right )$

Note

This expression actually works quite well for non-polar fluids. That is, it works for fluids that exhibit only induced dipoles, rather than permanent ones.

In order to extend this idea to asymmetrical (polar) molecules, we define:

DEFINITION

The Pitzer ascentric factor, $\omega$, quantifies the degree of polarity of a molecule (that is, how asymmetrical it is).

$\omega \equiv -1-\log_{10} \left [P^{sat}(T_r=0.7)/P_c\right ]$

Finally, we can mathematically write this generic idea as:

$f\left(\frac{T}{T_c},\frac{P}{P_c},\frac{v}{v_c}, \omega \right ) = 0$

or

$z = \frac{Pv}{RT} = f_o'\left(\frac{T}{T_c},\frac{P}{P_c}\right )+\omega f_1'\left(\frac{T}{T_c},\frac{P}{P_c}\right ) = z_o + \omega z_1$

Note

The generalized compressibility charts plot $z_o$ and $z_1$ as a function of $T_r$ and $P_r$.

Outcome

State and use the principle of corresponding states to develop expressions for the critical property data of a species

Outcome

Describe the purpose of the acentric factor and its role in the construction of compressibility charts

Test Yourself

Let's revisit our friend and calculate the container size when 44 g $CO_2$ is at P=6MPa and 26C (i.e., do we get 0.23 liters?)