From the kinetic theory of gas, prove that $P = \frac{1}{3}\rho \bar{c}^2$, where $\rho$ is the density of the gas and $\bar{c}$ is the root mean square velocity of the gas molecules. Show that $\bar{c} \propto \sqrt{T}$, where T is the absolute temperature.

Question

Accepted Answer

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> [!TLDR]
> Pressure arises from the rate of momentum delivered to the container walls by colliding molecules; working this out for a cubical box gives $P=	frac13ho\bar c^2$, and combining with the ideal gas law shows the rms speed grows as $\sqrt T$.
>
> **Setup:** Consider $N$ identical gas molecules, each of mass $m$, enclosed in a cubical container of side $L$ (volume $V=L^3$). Assume molecules move randomly, collisions with the walls are perfectly elastic, and (for the derivation) resolve each molecule's velocity into components $v_x, v_y, v_z$ along the three mutually perpendicular edges of the cube.
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> **Momentum change per collision:** consider one molecule with x-component of velocity $v_x$, moving toward a wall perpendicular to the x-axis. On elastic collision, its x-velocity reverses ($v_x 	o -v_x$), so its momentum change is:
> $$\Delta p = mv_x - (-mv_x) = 2mv_x$$
>
> **Rate of collisions with that wall:** after rebounding, the molecule must travel a distance $2L$ (to the opposite wall and back) before striking this same wall again, taking time $2L/v_x$. So the number of collisions per unit time (with this one wall) is $v_x/2L$.
>
> **Force from one molecule** (rate of momentum transfer):
> $$f = \Delta p 	imes \left(\frac{v_x}{2L}ight) = 2mv_x 	imes \frac{v_x}{2L} = \frac{mv_x^2}{L}$$
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> **Total force from all N molecules** (summing $v_x^2$ over all molecules, and using $\overline{v_x^2}$, the mean square x-velocity):
> $$F = \frac{Nm\overline{v_x^2}}{L}$$
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> **Pressure** (force per unit area on a wall of area $L^2$):
> $$P = \frac{F}{L^2} = \frac{Nm\overline{v_x^2}}{L^3} = \frac{Nm\overline{v_x^2}}{V}$$
> …

Q.From the kinetic theory of gas, prove that $P = \frac{1}{3}\rho \bar{c}^2$ , where $\rho$ is the density of the gas and $\bar{c}$ is the root mean square velocity of the gas molecules. Show that $\bar{c} \propto \sqrt{T}$ , where T is the absolute temperature.

Concept understanding — Kinetic Theory of Gases

Kinetic Theory of Gases

The Big Idea

The Five Assumptions (The Precise Statement)

How It Explains Pressure

The Key Result: The Kinetic Equation

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