Kinetic theory of gases
Section: Ideal Gases | Syllabus: Cambridge AS Level Physics 9702
Assumptions of the Kinetic Theory To model an ideal gas, we make several simplifying assumptions: Huge number: The gas consists of a huge number of molecules in random motion. Negligible volume: Total volume of molecules is negligible compared to the container volume.
Perfectly elastic: Collisions are perfectly elastic (KE is conserved). No forces: No intermolecular forces exist except during collisions (PE = 0). Negligible collision time: Time spent in collisions is tiny compared to time between them.
Hard spheres: Molecules behave as hard spheres obeying Newton's laws. Deriving pV = 1/3Nm c^2 1. Momentum Change Consider a molecule of mass m hitting a wall with velocity +c_x. It bounces back with -c_x.
Change in momentum Δ p = -mc_x - (mc_x) = -2mc_x. 2. Time Between Collisions In a cube of side L, the distance for a round trip is 2L. Time taken Δ t = 2L / c_x. 3. Force and Pressure Force: F = Δ p/Δ t = 2mc_x/2L/c_x = mc_x^2/L For N molecules, total force: F = Nm c_x^2 /L Pressure: p = F/L^2 = Nm c_x^2 /L^3 = Nm c_x^2 /V 4.
From 1D to 3D In 3D: c^2 = c_x^2 + c_y^2 + c_z^2 Since motion is random: c_x^2 = c_y^2 = c_z^2 Therefore: c_x^2 = 1/3 c^2 pV = 1/3Nm c^2 Mean-Square and R.M.S. Speed Mean-square speed ( c^2 ): The average of the squares of the speeds of all molecules.
Root-mean-square speed (c_r.m.s.): √ c^2 . It is an approximation of the average speed but not identical. Average Translational Kinetic Energy By equating 1/3Nm c^2 = NkT: Average E_K = 1/2m c^2 = 3/2kT This reveals that thermodynamic temperature is directly proportional to the average translational kinetic energy of the gas molecules.
Worked Examples Worked Example: Oxygen Speeds Question: The density of oxygen at 101 kPa is 1.33 kg m^-3 at 20 ^C. Determine the c_r.m.s. of the oxygen molecules. Solution 1. From pV = 1/3Nm c^2 , we get p = 1/3ρ c^2 .
2. c^2 = 3p / ρ = (3 × 101 × 10^3) / 1.33 = 227819 m^2 s^-2. 3. c_r.m.s. = √227819 = 477 m s^-1. Figure 15.5: Laboratory Kinetic Theory Model Illustration of a vibrating plate at the bottom of a tube containing small plastic balls.
A polystyrene cylinder on top is supported by the collisions of the balls, demonstrating how temperature (vibration speed) relates to pressure/volume. Figure 15.6: Derivation Cube A cube of side L showing a single molecule of mass m moving with velocity c_x towards a shaded wall, illustrating the basis of the pV = rac13Nmlangle c^2 angle derivation.
Figure 15.7: Maxwell-Boltzmann Distribution Graph of number of molecules vs. speed for different temperatures (e.g., 100K vs 1000K). Shows how the peak shifts to the right and flattens as temperature increases.
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