The kinetic theory of gases is based on a number postulates from which the equation P= 1/3× N/V×m×µ² is derived (P is the pressure of the gas, N the number of molecules in the container, m the mass of each molecule and µ² is the mean square speed). State all the postulates and derive the conclusion that the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature from the given equation.

This may help but I don't think anyone at this site will do this for you. Most freshman texts have it.

http://en.wikipedia.org/wiki/Kinetic_theory

The kinetic theory of gases is based on a set of postulates that describe the behavior of gas molecules. These postulates serve as assumptions on which the theory is built. The postulates are as follows:

1. Gas consists of a large number of tiny, individual particles, called molecules or atoms.
2. The molecules in a gas are in constant, random motion, colliding with each other and the walls of the container.
3. The size of the molecules is negligibly small compared to the space between them. Essentially, the molecules occupy an insignificant fraction of the total volume.
4. The collisions between gas molecules or between molecules and the walls of the container are perfectly elastic, meaning there is no loss of kinetic energy during collisions.
5. The intermolecular forces between gas molecules are negligible.

Using these postulates, we can derive the conclusion that the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature.

First, let's rewrite the given equation in terms of kinetic energy (KE):

P = (1/3) * (N/V) * m * µ²

Multiply both sides of the equation by V/2 (volume divided by 2) to obtain:

P * (V/2) = (1/3) * (N/V) * m * µ² * (V/2)

The left side of the equation represents the total kinetic energy of the gas molecules, as pressure (P) multiplied by volume (V) gives us the force exerted by the gas on the walls of the container. And force multiplied by distance (V/2) gives us the work done, which is equivalent to the total kinetic energy.

Now, substitute (N/V) with n, which represents the number density of the gas (the number of molecules per unit volume):

P * (V/2) = (1/3) * n * m * µ² * (V/2)

Rearranging the equation, we get:

P * V = (2/3) * n * m * µ² * V

Cancel out V on both sides:

P = (2/3) * n * m * µ²

From this equation, we can observe that the pressure (P) is directly proportional to the product of the number density (n), the mass of each molecule (m), and the mean square speed (µ²).

Now, let's consider the ideal gas equation:

PV = nRT

Where R is the gas constant and T is the absolute temperature.

Rearranging this equation, we get:

P = (n/V) * RT

Comparing this equation with the previously derived equation for pressure (P), we can see that (n/V) corresponds to (2/3) * n * m * µ².

Now, assuming both equations represent the pressure of the gas, we can equate them:

(2/3) * n * m * µ² = (n/V) * RT

Cancelling out n and rearranging the equation, we get:

(m/2) * µ² = (3/2) * (RT/N)

The term (3/2) * (RT/N) is a constant, as it depends only on the properties of the gas and is independent of individual molecules.

Since (m/2) * µ² represents the average translational kinetic energy (KE) of a gas molecule, we can conclude that the average translational kinetic energy is directly proportional to the absolute temperature (T).