## Kinetic Theory of Gases--State Equations and Specific Heats

**Introduction****:**

The ideal gas equation of state can be derived from the kinetic theory of gases.

**Description:**

• According to kinetic theory, the pressure exerted by an ideal gas on the walls of a container results from the action of a great number of molecules striking and rebounding.

• Let’s first look at one molecule in a box, with mass m and velocity components Vx, Vy, and Vz.

• Consider a molecule striking the wall at x=L. When it rebounds, there is no change in Vy and Vz, but Vx changes to –Vx.

• The corresponding x-component of momentum changes from mVx to –mVx. Thus, the =magnitude of momentum change is 2mVx.

• Also, the molecule travels the distance L in time L/Vx. Thus it could cross the chamber and return in time Δt=2L/Vx. The number of collisions per unit time (collision frequency) is

• Now, the change in x-momentum per unit time is the product of the change per collision and the frequency of collision,

• Thus, the force on the wall is

• The force of all molecules is

or, since m and L are constant,

• Define the average V^{2} of n molecules as

then • The pressure is thus

• If it is assumed that the pressure is the same in all directions, and that all directions of velocity are equally probable,