Kinetic derivation of pressure and temperature

Microscopic origin of pressure

Kinetic theory explains gas pressure as the result of molecular collisions with container walls. Consider a gas of $N$ identical molecules, each of mass $m$, in a cubic container of volume $V=L^3$.

A molecule with x-component velocity $v_x$ collides elastically with a wall. Its x-momentum changes from $mv_x$ to $-mv_x$, so the momentum change is

$Delta p_x=-2mv_x$

The wall receives an equal and opposite impulse of magnitude $2mv_x$.

Collision rate

The time between collisions with the same wall is

Delta t=rac{2L}{|v_x|}

The average force from one molecule on the wall is impulse divided by time:

F_1=rac{2mv_x}{2L/v_x}=rac{mv_x^2}{L}

For all molecules,

F=sum_i rac{mv_{xi}^2}{L}

Pressure is force divided by wall area $A=L^2$:

P=rac{F}{A}=rac{m}{L^3}sum_i v_{xi}^2

Since $V=L^3$,

P=rac{m}{V}sum_i v_{xi}^2

Isotropy

For a gas in equilibrium, molecular motion is random with no preferred direction. Therefore the average squared velocity components are equal:

angle=langle v_y^2 angle=langle v_z^2 angle$$ and $$langle v^2 angle=langle v_x^2+v_y^2+v_z^2 angle=3langle v_x^2 angle$$ Thus $$langle v_x^2 angle=rac{1}{3}langle v^2 angle$$ The pressure becomes $$P=rac{Nm}{V}langle v_x^2 angle=rac{1}{3}rac{Nm}{V}langle v^2 angle$$ So $$PV=rac{1}{3}Nmlangle v^2 angle$$ ## Temperature and kinetic energy Compare this with the ideal gas law: $$PV=Nk_BT$$ Then $$Nk_BT=rac{1}{3}Nmlangle v^2 angle$$ so $$rac{1}{2}mlangle v^2 angle=rac{3}{2}k_BT$$ This is the kinetic interpretation of temperature for an ideal monatomic gas: absolute temperature is proportional to average translational kinetic energy per molecule. ## Root-mean-square speed The root-mean-square speed is $$v_{rms}=sqrt{langle v^2 angle}$$ From the temperature relation, $$v_{rms}=sqrt{rac{3k_BT}{m}}$$ or using molar mass $M$, $$v_{rms}=sqrt{rac{3RT}{M}}$$ Lighter molecules move faster than heavier molecules at the same temperature. ## Important assumptions This derivation assumes elastic collisions, negligible molecular size, no long-range interactions, and equilibrium isotropy. It also assumes classical mechanics. At very low temperatures or high densities, quantum and interaction effects can become important. ## The big idea Kinetic theory derives pressure and temperature from molecular motion. Pressure comes from momentum transfer to walls. Temperature measures average translational kinetic energy. The ideal gas law is therefore not just empirical; it emerges from microscopic mechanics under ideal-gas assumptions.