Physics · Kinetic Theory · NEET
Molecules move randomly in all directions. Speed splits into three components: v² = vx² + vy² + vz². Because motion is random and equal on average, the mean square values are equal: mean(vx²) = mean(vy²) = mean(vz²) = (1/3)v². Only the x-component pushes on a wall facing the x-direction, so only one-third of the total speed-squared contributes to pressure on that wall. That is where the 1/3 comes from.
v² is the average of the squares of the speeds of all molecules, written as v-bar-squared. It is NOT the square of the average velocity. Average velocity of a gas is zero because molecules move equally in every direction and cancel out. But speeds squared are always positive, so their average is not zero. The square root of v² gives the rms speed.
Yes, they are identical. Density ρ = total mass / volume = N·m / V, where N is number of molecules, m is mass of one molecule, and V is volume. Substitute ρ = Nm/V into P = (1/3) ρ v² and you get P = (1/3)(N/V) m v². Same equation, just written using density instead of number density.
When a molecule of mass m hits the wall with x-velocity vx and bounces back, its momentum changes by 2m·vx. Many molecules hit per second. Total momentum given to the wall per second is a force (Newton's second law). Force divided by wall area is pressure. Adding up all molecules gives P = (1/3)(N/V) m v².
Average velocity (a vector) of gas molecules is zero, so it cannot explain a real, non-zero pressure. Pressure depends on how hard molecules hit, which depends on speed squared (through momentum change and collision rate). Speed squared is always positive, so its average survives and correctly predicts pressure.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
P = (1/3) ρ v², where P is pressure, ρ is gas density, and v² is the mean square speed of molecules. It can also be written P = (1/3)(N/V) m v².
Molecules are point-like with negligible volume, collisions are perfectly elastic, there are no forces between molecules except during collisions, molecules move randomly in all directions, and time of collision is negligible compared to time between collisions.
Combining it with the ideal gas law gives the mean kinetic energy per molecule as (3/2) kB T. This shows temperature is a measure of the average kinetic energy of molecules. This is covered next in kinetic interpretation of temperature.
The rms (root mean square) speed is v_rms = square root of v². From P = (1/3) ρ v², we get v_rms = square root of (3P/ρ).
No. Although the derivation often uses a cube for simplicity, the result P = (1/3) ρ v² holds for any container shape because pressure depends only on density and mean square speed, not geometry.
