# EoS: The Ideal Gas Law

### 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:

$P=f(T,v)$

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:

$z \equiv \frac{Pv}{RT}$

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.