KMT is a model that explains the macroscopic properties of a gas (like pressure, temperature) based on the behavior of its microscopic particles (atoms or molecules). This model describes an Ideal Gas.
Postulates of Kinetic Theory of Gases
Gases are made of a large number of tiny particles that are in constant, random, straight-line motion.
The volume of the gas particles themselves is negligible compared to the total volume of the container. (Particles are point masses).
There are no intermolecular forces of attraction or repulsion between gas particles.
Collisions between particles and with the walls of the container are perfectly elastic (no kinetic energy is lost).
The pressure of the gas is due to the collisions of the particles with the walls of the container.
The average kinetic energy of the gas particles is directly proportional to the absolute temperature (in Kelvin).
Derivation of Kinetic Gas Equation and Gas Laws
Kinetic Gas Equation (Derivation not in syllabus, only equation)
Based on the postulates of KMT, the pressure (P) exerted by a gas is given by:
PV = (1/3)mnc²
Where:
P = Pressure
V = Volume of the container
m = mass of one gas particle
n = total number of gas particles in volume V
c² = mean square velocity of the gas particles (the average of the squares of all particle velocities)
We define c as the root mean square (rms) velocity, so c = (c²)¹/².
Deduction of Gas Laws from Kinetic Gas Equation
The average kinetic energy (KE) of *one* particle is (1/2)mc².
The total KE of *n* particles is n · (1/2)mc².
From the kinetic equation, PV = (2/3) · [n · (1/2)mc²]
PV = (2/3) · (Total KE)
Since KMT states that average KE ∝ T, then Total KE = k·T (where k is a constant).
Therefore, PV = (2/3)k·T. Since (2/3)k is a constant, we get PV ∝ T, which is the Ideal Gas Law (PV=nRT).
Boyle's Law (P ∝ 1/V at constant T): From PV = (2/3)k·T, if T is constant, then the entire right side is constant. So, PV = constant.
Charles's Law (V ∝ T at constant P): From PV = (2/3)k·T, V = [(2/3)k/P] · T. If P is constant, then V = (constant) · T.
Molecular Velocities
Gas particles do not all move at the same speed. Their speeds are described by the Maxwell-Boltzmann distribution of molecular velocities. This distribution shows the number of molecules (or probability) versus their speed at a given temperature.
From this distribution, we define three types of molecular velocities:
Root Mean Square Velocity (c_rms): The square root of the mean of the squares of the velocities.
c_rms = √(3RT / M) = √(3PV / M) = √(3P / d)
Average Velocity (c_avg): The arithmetic mean of the speeds of all molecules.
c_avg = √(8RT / πM)
Most Probable Velocity (c_mp): The velocity possessed by the largest number of molecules (the peak of the curve).
c_mp = √(2RT / M)
Where: R = 8.314 J/mol·K, T = temperature in Kelvin, M = Molar mass in kg/mol.
Important: When calculating velocities, Molar Mass (M) must be in kg/mol (e.g., for O₂, M = 0.032 kg/mol) to be consistent with R in J/mol·K.
Ratio: c_mp : c_avg : c_rms = 1 : 1.128 : 1.224
Mean Free Path (no derivation)
Definition: The average distance a gas particle travels between successive collisions.
λ = (1 / √2) · (1 / (πd²N*))
Where: λ = mean free path, d = molecular diameter, N* = number of molecules per unit volume.
Real Gases: Deviation from Ideal Behaviour
An ideal gas (PV=nRT) is hypothetical. Real gases (like H₂, N₂, CO₂) only behave ideally at High Temperatures and Low Pressures.
Causes of Deviation: Real gases deviate from ideal behavior because two postulates of the KMT are false:
Faulty Volume Postulate: KMT assumes gas particles have zero volume. At high pressure, the gas is compressed, and the volume of the particles themselves *is* significant compared to the container volume.
Faulty Force Postulate: KMT assumes no intermolecular forces. At low temperature, particles move slowly and get close, and intermolecular forces (like van der Waals forces) *do* become significant.
Van der Waals Equation
Van der Waals modified the ideal gas equation (PV=nRT) to account for the behavior of real gases. He introduced two correction factors, 'a' and 'b'.
Van der Waals Equation (for n moles):
(P + an²/V²) (V - nb) = nRT
1. Volume Correction (the 'b' term)
Reason: Gas particles have a finite volume ('b'), called the 'excluded volume'.
Correction: The "free" volume available for the gas to move in is not V, but (V - nb).
'b' is a measure of the effective size of the gas particles.
2. Pressure Correction (the 'a' term)
Reason: Gas particles attract each other (intermolecular forces). A particle about to hit the wall is pulled back by its neighbors, so it hits the wall with less force. The observed pressure (P) is *lower* than the ideal pressure.
Correction: The pressure correction term (P_correction) is proportional to the square of the density (n/V)².
P_correction = a(n/V)²
The corrected "ideal" pressure is (P_observed + an²/V²).
'a' is a measure of the strength of intermolecular forces.
Liquefaction of Gases
Real gases can be liquefied by increasing pressure and decreasing temperature. This forces the molecules close together and slows them down, allowing intermolecular forces ('a') to take over and condense the gas into a liquid.
However, for every gas, there is a critical temperature (T_c) above which it cannot be liquefied, no matter how much pressure is applied.
Critical Phenomena
Critical Temperature (T_c): The maximum temperature at which a gas can be liquefied by the application of pressure alone.
T_c = 8a / 27Rb
Critical Pressure (P_c): The minimum pressure required to liquefy a gas *at* its critical temperature.
P_c = a / 27b²
Critical Volume (V_c): The volume occupied by one mole of the gas at T_c and P_c.
V_c = 3b
Significance:
A gas with a high 'a' value (strong intermolecular forces) has a high T_c and is easier to liquefy (e.g., CO₂, NH₃).
A gas with a low 'a' value (weak forces) has a very low T_c and is difficult to liquefy (e.g., H₂, He, so-called "permanent gases").