Pressure from Kinetic Model — Theory
The kinetic theory of gases explains macroscopic pressure as the result of many microscopic
collisions of molecules with container walls. In the ideal-gas approximation,
the microscopic and macroscopic descriptions are exactly consistent.
Key formulas
Derivation idea: impulses from wall collisions
Consider a molecule of mass \(m\) moving with velocity component \(v_x\) toward a wall normal to the \(x\)-axis.
For an elastic collision, \(v_x\to -v_x\), so the molecule’s momentum change in the \(x\) direction is
\[
\Delta p_x = m(-v_x)-m(v_x) = -2mv_x.
\]
The wall receives an equal and opposite impulse magnitude \(2mv_x\).
Summing impulses over a time interval \(\Delta t\) gives an average force:
\[
F_x \approx \frac{\sum (2mv_x)}{\Delta t}.
\]
Pressure is force per area \(P=F/A\). Using (i) the number density \(N/V\),
(ii) the typical time between wall hits for particles moving in \(x\),
and (iii) isotropy of velocities, one arrives at:
\[
P = \frac{1}{3}\rho \langle v^2\rangle
= \frac{1}{3}\rho v_{\mathrm{rms}}^2.
\]
Micro ↔ macro consistency
If \(v_{\mathrm{rms}}^2=3kT/m\) and \(\rho = (Nm)/V\), then
\[
P=\frac{1}{3}\left(\frac{Nm}{V}\right)\left(\frac{3kT}{m}\right)
=\frac{NkT}{V}.
\]
And since \(N=nN_A\) and \(k=R/N_A\), we also get \(PV=nRT\).
This is why the calculator shows the same pressure from the three “routes”.
Assumptions and limits
- Ideal gas: negligible intermolecular force
