Kinetic Theory - Answers to Q.

Ideal Gas Law

The equation of state for an ideal gas can be expressed as:
pV = nRT
where:

• p = pressure

• V = volume

• n = number of moles

• R = molar gas constant

This can also be represented as:
pV = NkT
where:

• N = number of gas molecules

• k = Boltzmann constant

Assumptions of Kinetic Theory of Gases

Gas molecules are in constant random motion

Collisions between gas molecules and container walls are perfectly elastic

The volume of the gas molecules is negligible compared to the volume of the container

There are no intermolecular forces between the gas molecules

The average kinetic energy of gas molecules is directly proportional to the temperature of the gas in Kelvin

Molecular Movement and Pressure

Molecular movement contributes to gas pressure as described by:

p = (1/3) ρc²

p = (1/3) (N/V) mc²
where:

• ρ = density

• c = average speed of molecules

• m = mass of a molecule

Avogadro Constant

The Avogadro constant (Nᗩ) is defined as the number of constituent particles (molecules, atoms) in one mole of a substance, approximately equal to 6.022 x 10²³.

It relates to the mole by establishing that one mole of any substance contains Nᗩ particles.

Molar Mass and Moles

The relationship between molar mass (M) and relative molecular mass (Mᵣ) is given by:
M/kg = Mᵣ/1000

This implies molar mass in kilograms per mole given the relative molecular mass.

The number of moles (n) can be calculated from total mass (m) and molar mass (M):
n = m/M.

Combined Kinetic Energy Equations

To demonstrate the total translational kinetic energy of a mole of a monatomic gas:

• Combine:
  pV = (1/3) Nmc²
  with
  pV = nRT

This leads to the conclusion that:

The total kinetic energy is given by:
Total KE = (3/2) nRT

The mean kinetic energy of a molecule is given by:
Mean KE = (3/2) kT

where k = R/Nᗩ is the Boltzmann constant.

Temperature (T) is directly related to the mean kinetic energy of the molecules.