# 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?)