PX154 - C3 - a real gas equation of state

ideal gas equation does not predict liquification
- as the conditions approach $g a s \to l i q u i d / s o l i d$ phase boundary, it fails to predict the observed properties

things to consider

molecules have finite volume

eg: $1 m^{3} N_{2}$ gas at $S T P$
- $S T P =$ "standard temperature and pressure" $= 1.013 \times 10^{5} P a / 1 a t m$ and $0 ° C (o r 20 ° C)$
- the volumes of the molecules is $0.00075 m^{3}$
  - $1 m^{3} = 1000 l ⟹ V = 0.75 l$
  - effect of $V$ in these conditions on observed gas properties can be assumed to be negligible
  - as we reduce $V (d e n s i t y = n / V i n c r e a s e s)$ , and $V$ becomes significant
- in the ideal gas equation, we replace $V$ with $(V - n b)$
  - where,
    $n =$ number of moles
    $b =$ "co-volume" (represents the volume occupied by a mole of molecules)

intermolecular forces

in an ideal gas, molecules collide elastically and don't interact otherwise
now, we allow for interactions between the molecules (eg: lennard-jones potential)
we replace $p$ with $(p + a \frac{n^{2}}{V^{2}})$
- where,
  $a$ is a constant
  $\frac{n^{2}}{V^{2}} = d e n s i t y^{2}$
  - further reading [grant+phillips : elements of physics]

van der waal's equation of state

(p + \frac{a n^{2}}{V^{2}}) (V - n b) = n R T

at high volumes/low densities, it is well approximate by the ideal gas law