Pressure from Molecular Theory
P = ⅓ ρ⟨v²⟩ — kinetic origin of pressure.
Key Notes
Kinetic-theory derivation: P = (1/3)·n·m·⟨v²⟩, where n = number density, m = molecular mass, ⟨v²⟩ = mean-square speed.
Equivalently: P = (1/3)·ρ·v_rms², where ρ = mass density.
Derived from: molecules in random motion, elastic collisions with walls, momentum transfer 2m·v.
Connects MICROSCOPIC (molecular motion) to MACROSCOPIC (pressure).
Implies pressure depends ONLY on density and speed — independent of detailed molecular interactions (in ideal gas).
At fixed T: ⟨v²⟩ = 3k_BT/m ⇒ P = nk_BT (ideal gas law form).
Total KE per molecule: ⟨KE⟩ = (3/2)k_BT (in 3D, monatomic).
Temperature is a measure of MEAN MOLECULAR KE.
Formulas
Kinetic pressure formula
From molecular dynamics: momentum transfer rate.
Ideal gas connection
Combining with ⟨v²⟩ = 3k_BT/m gives PV = NkT.
Average KE per molecule
Direct from kinetic theory.
RMS speed
Defines the molecular speed scale.
Important Points
P = (1/3)ρv_rms² connects molecular motion to macroscopic pressure.
Doubling number density doubles pressure (same v_rms).
Doubling v_rms ⇒ quadrupling pressure.
Combining with ⟨v²⟩ = 3k_BT/m gives PV = Nk_BT — ideal gas law DERIVED from kinetic theory.
Temperature is microscopically defined: T ∝ ⟨KE⟩.
Real gases deviate from ideal at high P/low T (intermolecular forces, finite molecular size).
Pressure from Molecular Theory notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.