Kinetic theory of gases
Section: Ideal Gases | Syllabus: Cambridge A 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 = (1/3)Nm c^2
1. Momentum ChangeConsider 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.
In a cube of side L, the distance for a round trip is 2L. Time taken Δ t = 2L / c_x.
3. Force and PressureForce: 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)
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 = (1/3)Nm c^2Mean-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 (1/3)Nm c^2 = NkT:
Average E_K = (1/2)m c^2 = (3/2)kTThis reveals that thermodynamic temperature is directly proportional to the average translational kinetic energy of the gas molecules.
Worked Examples
Worked Example: Oxygen SpeedsQuestion: The density of oxygen at 101 kPa is 1.33 kg m^-3 at 20 ^. Determine the c_r.m.s. of the oxygen molecules.
Solution1. From pV = (1/3)Nm c^2 , we get p = (1/3)ρ 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 ModelIllustration 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 CubeA 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 DistributionGraph 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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