### The Ideal Gas

For a low-density/high-temperature gas, we neglect the long-range intermolecular attraction and treat the short-range repulsion as perfect elastic. This leads to:

#### The Ideal Gas Model

1. The gas consists of a large number of identical molecules (for air, density ≈ 1025 molecules/m3).
2. The volume occupied by molecules themselves is negligible compared to the volume of the container (molecular volume ≈ 10-29 m3).
3. Molecules move at constant speed in random directions between collisions (typical speed 550 m/s).
4. Molecules experience forces only during collisions that are elastic and instantaneous (typical time between collisions 10-11 sec).

An easy-to-remember number:   ρair ≈ ρwater/1000.

Ideal Gas Law:  P V = n R T

where P is pressure, V is volume, n is the number of moles, T is the absolute temperature, and R = 8.31 J/(mol K) is the universal gas constant.

This can be written in terms of N, the number of molecules, instead. Using N = nNA, where NA is Avogadro's number, we obtain:
PV = nNA
 R NA
T = N k T

where the Boltzmann constant k = 1.38 x 10-23 J/K is the universal gas constant divided by Avogadro's number.

Intuition for the Ideal Gas Law:

• The pressure of an ideal gas is proportional to 1/V ---squeezing a gas leads to higher pressure.
• Pressure is caused by molecules colliding with the container walls. This leads to P ∝ T . As T → 0, the thermal energy → 0. Then there are no collisions with container walls and therefore no pressure.