§ 12.2NCERT Class 11 · Physics · Chapter 12
Pressure Exerted by an Ideal Gas & Kinetic Interpretation of Temperature
Fundamental Mechanical Connection: Gas pressure is a macroscopic manifestation of countless microscopic, elastic collisions of gas molecules with the walls of the container.
1. Postulates of Kinetic Theory of Gases
- A gas consists of a very large number of identical, tiny, hard, spherical particles called molecules.
- The actual volume occupied by the molecules is negligibly small compared to the total volume of the container.
- Molecules are in a state of continuous, chaotic, random motion in all directions with all possible speeds.
- Intermolecular forces of attraction or repulsion are negligible, except during momentary collisions.
- All collisions (between molecules and between molecules and container walls) are perfectly elastic (kinetic energy and momentum are conserved).
- The duration of a collision is negligible compared to the time interval between successive collisions.
- Gravity has negligible effect on the motion of gas molecules due to their high speeds and small masses.
2. Derivation of Pressure Exerted by an Ideal Gas
Consider a cubical box of side L containing N identical molecules, each of mass m, moving randomly.
- A molecule moving with velocity (vx, vy, vz) collides elastically with a wall perpendicular to the x-axis.
- Change in momentum per collision along x: Δpx = (−mvx) − (mvx) = −2mvx.
- Momentum imparted to the wall = +2mvx.
- Time between two successive collisions with the same wall: Δt = 2L / vx.
- Rate of momentum transfer (Force on wall by one molecule) = Δp / Δt = (2mvx) / (2L / vx) = m vx2 / L.
- Total force exerted by all N molecules: F = (m / L) · Σ vix2.
- Since motion is isotropic: ⟨vx2⟩ = ⟨vy2⟩ = ⟨vz2⟩ = (1/3) ⟨v2⟩.
- Therefore, Pressure P = F / Area = F / L2:
P = (1/3) · (N · m / V) · ⟨v2⟩ = (1/3) · ρ · vrms2
where ρ = (N · m) / V is the mass density of the gas and vrms = √⟨v2⟩ is the root-mean-square speed.
3. Kinetic Interpretation of Absolute Temperature
Multiplying both sides by volume V:
P · V = (1/3) · N · m · vrms2 = (2/3) · [ N · (1/2 m vrms2) ] = (2/3) · Etotal
where Etotal is the total translational kinetic energy of the gas. Comparing with the ideal gas equation P · V = N · kB · T:
(2/3) · Etotal = N · kB · T ⇒ Etotal = (3/2) · N · kB · T
| Quantity | Formula | Physical Significance |
|---|---|---|
| Average Translational KE per Molecule | ⟨Ek⟩ = (3/2) kB T | Depends only on temperature T; completely independent of the mass or chemical nature of the gas molecule! |
| Average Translational KE per Mole | Emolar = (3/2) R T | For 1 mole of any gas, translational kinetic energy is identical at a given absolute temperature. |
| Total KE for μ Moles | Etotal = (3/2) μ R T | Proportional to the number of moles μ and absolute temperature T. |
Fundamental Insight: Absolute zero (0 K = −273.15 °C) is the temperature at which the translational kinetic energy of all gas molecules becomes zero and molecular motion ceases completely according to classical kinetic theory.
4. Pressure in Terms of Kinetic Energy Density
The total translational kinetic energy per unit volume (Energy Density u = E / V) is:
u = (3/2) · P ⇒ P = (2/3) · u
The pressure exerted by an ideal gas is numerically equal to two-thirds of its translational kinetic energy density.
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