A sample of ideal gas consists of identical atoms of mass contained in a cubical box…

PHYSICS · Multi-part · CBSE · Physics · Kinetic Theory · Kinetic Theory

Problem

A sample of ideal gas consists of

N

identical atoms of mass

m

contained in a cubical box of side length

L

. The atoms move freely between perfectly elastic collisions with the walls. In this problem you will develop an expression that connects the microscopic motion of the atoms to the macroscopic pressure of the gas. (a) Consider a single atom whose velocity component perpendicular to the right wall of the box has magnitude

v_x

. Derive an expression for the average magnitude of the force that this atom exerts on that wall. (b) Generalize your result from part (a) to all

N

atoms, each having (possibly different) velocity component

v_{x,i}

. Derive an expression for the pressure

P

exerted on the wall in terms of

m

N

L

, and the mean of

v_{x,i}^2

. (c) Assuming the atoms move randomly in three dimensions, relate the mean of

v_{x,i}^2

to the mean of the squared speed

v_i^2

of the atoms. Then show that the equation obtained in part (b) can be written in the form

PV = \tfrac{1}{3}Nm\overline{v^2}

, where

V

is the volume of the box and the bar denotes an average over all atoms. (d) Using the empirical relationship between average kinetic energy and absolute temperature, explain how your result leads to the ideal-gas law expressed in terms of Boltzmann’s constant. Identify the assumptions about the nature of atomic collisions that are critical to this derivation.

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