### Ideal Gas Law

##### DEFINITION

An **equation of state** relates the molar density
(or specific molar volume) of a fluid (i.e., a vapor or a liquid)
to the temperature and pressure of the fluid.

The state postulate claims that any two intensive variables can
fix the state of a system, therefore, we can tabulate all state
variables with respect to any two (measurable) properties (see the
steam tables), or (more conveniently) we can write an explicit
mathematical expression of one property in terms of any two
others:

The most common and simplest equation of state is the ideal gas
law:

$P=\frac{nRT}{V} = \frac{RT}{v}$

##### DEFINITION

A generic expression for an equation of state is to define a
property, **the compressibility factor**, as the ratio
of Pv to RT:

Obviously, $z=1$ for an ideal gas. Real gases often show
$z<1$ at modest pressures and $z>1$ at very high P

##### NOTE

Interestingly, the ideal gas law can be derived from our
molecular understanding of the properties T ($E_{k_{molecular}} =
(1/2)m\bar \vec V^2 = (3/2) kT$) and P ($(1/A)\sum \left
[\frac{d(m\vec V)}{dt} \right ]$). This requires the following
assumptions:

- Ideal gases involve molecules that occupy no volume
- Ideal gases involve molecules that exert no intermolecular
forces

##### IMPORTANT

The ideal gas law is an approximation (!) that has only limited
applicability. It is usually used for diatomic gas when (RT/P)>5
L/mol and for other gases when (RT/P) > 20 L/mol (i.e., at high
specific volumes!).

##### Outcome:

State the molecular assumptions of the ideal gas law

##### Test Yourself:

44 g $CO_2$ are placed in a 0.23 liter container at 26C. The
pressure is recorded to be 6 MPa. Evaluate whether ideal gas
conditions are well-approximated.