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.
I hope you will help in derive the equation atleast..
Certainly! The kinetic theory of gases is based on several postulates, which provide a basis for understanding the behavior of gas molecules. Here are the postulates:
1. Gas consists of a large number of tiny particles known as molecules or atoms.
2. The gas molecules are in constant random motion and move in straight lines between collisions.
3. Collisions between gas molecules and the walls of the container are perfectly elastic, meaning there is no loss of kinetic energy during collisions.
4. The volume occupied by the gas molecules themselves is considered negligible compared to the total volume of the gas.
5. There are no attractive or repulsive forces between gas molecules, except during collisions.
Now, let's derive the conclusion that the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature using the given equation.
Starting with the given equation, P = (1/3) * (N/V) * m * µ², where P is the pressure of the gas, N is the number of molecules in the container, V is the volume of the container, m is the mass of each molecule, and µ² is the mean square speed.
We can simplify this equation further by substituting the ideal gas law equation: PV = NRT, where R is the ideal gas constant and T is the absolute temperature.
Rearranging the ideal gas law equation to solve for pressure (P), we get:
P = (N/V) * RT
Now, let's substitute this simplified equation for P into the original equation:
(N/V) * RT = (1/3) * (N/V) * m * µ²
Canceling out (N/V) on both sides, we get:
RT = (1/3) * m * µ²
Now, let's further simplify this equation. The mean square speed (µ²) can be expressed as the average kinetic energy per molecule (KE_avg) divided by the mass of a molecule (m):
KE_avg = (1/2) * m * µ²
Now, let's substitute this expression for KE_avg in terms of m and µ² into the equation:
RT = (1/3) * m * (1/2) * m * µ²
Simplifying further, we get:
RT = (1/6) * m² * µ²
Finally, we can conclude that the average kinetic energy per molecule (KE_avg) is directly proportional to the absolute temperature (T):
KE_avg ∝ T
So, from the given equation P = (1/3) * (N/V) * m * µ², we have derived the conclusion that the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature.