Ideal Gas (chapter 3)

• Ideal Gas: A Gas that obeys PV=nRT at all values of temperature, pressure and volume.

The Assumptions of Ideal Gas

1. It contains the same type of molecules.
2. All collisions with the wall of container are elastic.
3. Kinetic energy of the atoms are directly proportional to the temperature.
4. There are no inter-molecular forces of attraction, subsequently, there is no potential energy among the atoms.
5. At 0 Kelvin the mean kinetic energy becomes 0.

Proofs

• Show that PV=NKT

We know that PV=nRT

we also know that n=N/N_a

so PV=NRT

now since K=R/N_a

PV=NKT

• Pressure of a single atom along the x-axis

Suppose there is a cube shaped container with length L that has a single atom of ideal gas and the atom has speed v:

ΔP = mv - (-mv)

ΔP=2mv

F= ΔP/Δ T =2mv/t

since the collision is elastic T=2L/V

Now putting equation (2) in (1) we get mv² /L³

Now since Pressure = Force/Area

Pressure = mv²/L³

• Total Pressure

Pressure of N atoms on the wall of the container along x-axis:

P_N=mv²_N/l³ 

Total Pressure:

P=P₁+P₂..... P_N

mv²₁/L³+mv²₂/L³+mv²_N/L³

m/L³ (v²₁+v²₂+v²_N)

Now since the mean square speed is denoted by

<v²> = v²₁+v²₂+v²_N / N

Plugging the above mentioned expression in the expression of total Pressure:

P=Nm<v²> /L³

• Total pressure for N atoms

P_x/y/z = Nm <v²_x/y/z> / L³

<c²> = <v²_x>+<v²_y>+ <v²_z>

since it is an ideal gas so <v²_x> = <v²_y> = <v²_z>

<c²> = 3 <v²>

1/3<c²> = <v²>

Plugging this in the expression of Total Pressure

P=1/3 Nm<c²>/L³

• Mean Kinetic Energy of an atom of ideal gas:

PV=NKT

PV=1/3Nm<c²>

3KT=m<c²>

multiplying both sides with 1/2

3/2KT = Kinetic Energy

Brownian Motion

• It is the random motion of microscopic particles in a fluid as a result of countinuous bombardment from molecules of the surrounding medium

The following is observed:

1. Bright tiny Smoke particles are jerking sideways in a random motion
2. Particles randomly moved into and out of focus of the microscope, It indicates random vertical motion of the particles

Explanation for the Brownian motion: The tiny smoke particles are bombarded by millions of small air molecules moving in random directions.