Unit 3: Gases

Table of Contents

Kinetic Theory of Gases (KTG)

Postulates of Kinetic Theory of Gases

This theory explains the macroscopic properties of ideal gases based on the motion of their molecules. The key postulates are:

  1. Gases consist of many tiny particles (molecules) in continuous, random motion.
  2. The actual volume of the gas molecules is negligible compared to the total volume of the container.
  3. There are no intermolecular forces (attraction or repulsion) between molecules.
  4. Collisions between molecules and with the container walls are perfectly elastic (no net loss of kinetic energy).
  5. The pressure of the gas is due to the collisions of molecules with the walls.
  6. The average kinetic energy of the gas molecules is directly proportional to the absolute temperature (Kelvin).

Derivation of the Kinetic Gas Equation

This equation relates the pressure (P) to the microscopic properties of the molecules.

  1. Consider a cubic container of side l with N molecules, each of mass m.
  2. A single molecule has velocity c1 with components u1, v1, w1.
  3. Momentum change when hitting the x-wall = (mu1) - (-mu1) = 2mu1.
  4. Time between collisions on the same wall = Δ t = (2l) / (u1).
  5. Force from one molecule = (Momentum change) / (Time) = (2mu1) / (2l/u1) = (mu12) / (l).
  6. Pressure from one molecule = (Force) / (Area) = (mu12/l) / (l2) = (mu12) / (l3) = (mu12) / (V).
  7. Total pressure P from all N molecules is P = (m) / (V)(u12 + u22 + ... + uN2) = (mNu2) / (V).
  8. Since motion is random, u2 = v2 = w2 = (1) / (3)c2 (where c2 is the mean square velocity).
  9. Substituting gives: P = (mN) / (V) ( (1) / (3)c2 ).
Kinetic Gas Equation: PV = (1) / (3)mNc2
This can also be written as PV = (2) / (3)KE, where KE is the total kinetic energy.

Real Gases and Deviation from Ideal Behaviour

Real gases only behave ideally at low pressure and high temperature. They deviate under other conditions.

Compressibility Factor (Z)

The deviation is quantified by the compressibility factor, Z.

Formula: Z = (PVreal) / (nRT)
  • For an Ideal Gas, Z = 1.
  • Negative Deviation (Z < 1): Occurs at moderate pressures. Gas is more compressible than ideal. Attractive forces are dominant.
  • Positive Deviation (Z > 1): Occurs at high pressures. Gas is less compressible than ideal. Repulsive forces (molecular volume) are dominant.

Causes of Deviation

Deviations arise because two postulates of the KTG are incorrect for real gases:

  1. Faulty Postulate 1 (Volume): "The volume of molecules is negligible."
    Correction: At high pressure, molecular volume is significant. This leads to positive deviation (Z > 1).
  2. Faulty Postulate 2 (Forces): "There are no intermolecular forces."
    Correction: At low temperature/moderate pressure, attractive forces are significant. This leads to negative deviation (Z < 1).

Vander Waals Equation of State for Real Gases

Van der Waals modified the ideal gas equation (PV = nRT) by introducing two correction terms, 'a' and 'b'.

  • Volume Correction: The "free space" is not V, but (V - nb). The term b is the "excluded volume" per mole.
  • Pressure Correction: The observed pressure P is lower than the ideal pressure due to attractions. The correction term an2/V2 is added. The term a measures the strength of intermolecular attraction.
Van der Waals Equation (n moles): ( P + (an2) / (V2) ) (V - nb) = nRT
(For 1 mole): ( P + (a) / (Vm2) ) (Vm - b) = RT

Molecular Velocities and Collision Properties

Most probable, average and root mean square velocities (no derivation)

Molecules in a gas have a range of speeds (Maxwell-Boltzmann distribution). We use three statistical measures:

Velocity Type Formula (M = Molar Mass) Description
Most Probable (vmp) vmp = √((2RT) / (M)) The speed possessed by the maximum number of molecules.
Average (vavg) vavg = √((8RT) / (π M)) The arithmetic mean of the speeds of all molecules.
Root Mean Square (vrms) vrms = √((3RT) / (M)) The square root of the mean of the squares of the speeds. (Used in KTG).
Key Relationship: vrms > vavg > vmp

Collision number and mean free path of molecules

  • Collision Diameter (σ): The distance of closest approach between the centers of two colliding molecules.
  • Collision Number / Frequency (Z1): The average number of collisions experienced by a single molecule per unit time.
  • Mean Free Path (λ): The average distance a molecule travels between successive collisions.
    Formula: λ = (1) / (√(2)πσ2 n*)
    Where n* is the number density (N/V).

Relationship: λ is inversely proportional to pressure (or density). Higher pressure means more molecules, more collisions, and a shorter mean free path.