Kinetic Interpretation Of Pressure

Expressing gas pressure in terms of molecular motion, the kinetic theory gives $P=\frac{1}{3}\rho v_{rms}^2$, where $\rho$ is the mass density and $v_{rms}$ is the root-mean-square speed. This follows from averaging the momentum that molecules transfer to the walls during elastic collisions, with the factor $\frac{1}{3}$ coming from the equal sharing of mean-square speed among the three spatial directions. Substituting $\rho =1.25$ kg/m\textsuperscript{3} and $v_{rms}=500$ m/s gives $P=\frac{1}{3}\times 1.25\times (500{)}^2=\frac{1}{3}\times 1.25\times 2.5\times 10^5\approx 1.04\times 10^5$ Pa. The value $3.13\times 10^5$ Pa omits the factor $1/3$. The value $0.52\times 10^5$ Pa mistakenly uses $\frac{1}{6}$ instead of $\frac{1}{3}$. The value $2.08\times 10^5$ Pa doubles the correct result by misreading the density. This is the NCERT kinetic interpretation of pressure linking the microscopic and macroscopic descriptions, and it shows pressure rises with both density and the square of molecular speed. As a plausibility check, the result is close to one atmosphere ($1.013\times 10^5$ Pa), which is exactly what we expect for a gas at ordinary laboratory density and molecular speed.