Derive the expression for pressure of an ideal gas enclosed in a container. From this, discuss the kinetic interpretation of temperature.

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Accepted Answer

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> [!TLDR]
> Kinetic theory gives gas pressure P = (1/3)ρ⟨v²⟩; comparing with the ideal gas law shows average molecular kinetic energy ⟨KE⟩ = (3/2)kT, giving temperature its kinetic meaning.
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> **Derivation of pressure:**
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> Consider N molecules, each of mass m, inside a cubical container of side L (volume $V=L^3$), moving randomly with velocities having components $(v_x,v_y,v_z)$.
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> Consider one molecule moving with x-component of velocity $v_x$, colliding elastically with the wall perpendicular to the x-axis. Its momentum reverses: change in momentum per collision $= 2mv_x$.
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> Time between two successive collisions with the same wall (travelling distance $2L$ at speed $v_x$): $\Delta t = \dfrac{2L}{v_x}$
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> Force exerted by this molecule on the wall (average, over many collisions):
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> $$f = \dfrac{2mv_x}{2L/v_x} = \dfrac{mv_x^2}{L}$$
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> Summing over all N molecules and using the mean-square value $\langle v_x^2\rangle$:
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> $$F = \dfrac{Nm\langle v_x^2\rangle}{L}$$
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> Pressure:
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> $$P = \dfrac{F}{A} = \dfrac{F}{L^2} = \dfrac{Nm\langle v_x^2\rangle}{L^3} = \dfrac{Nm\langle v_x^2\rangle}{V}$$
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> By symmetry of random motion, $\langle v_x^2\rangle=\langle v_y^2\rangle=\langle v_z^2\rangle=\dfrac{1}{3}\langle v^2\rangle$ (since $v^2=v_x^2+v_y^2+v_z^2$):
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> $$P = \dfrac{1}{3}\dfrac{Nm\langle v^2\rangle}{V} = \dfrac{1}{3}\rho\langle v^2\rangle$$
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> where $\rho = \dfrac{Nm}{V}$ is the density of the gas. Equivalently, $PV = \dfrac{1}{3}Nm\langle v^2\rangle$.
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> **Kinetic interpretation of temperature:**
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> Rewrite: $PV = \dfrac{2}{3}N\left(\dfrac{1}{2}m\langle v^2\rangle\right) = \dfrac{2}{3}N\langle KE\rangle$
> …

Q.Derive the expression for pressure of an ideal gas enclosed in a container. From this, discuss the kinetic interpretation of 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

Q.Derive the expression for pressure of an ideal gas enclosed in a container. From this, discuss the kinetic interpretation of temperature.

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