3.2. Decoding the ideal gas law[[Student version, December 8, 2002]] 73
Figure 3.5: (Schematic.) Origin of gas pressure. (a)Amolecule traveling parallel to an edge with velocityvx
bounces elastically off a wall of its container. The effect of the collision is to reverse the direction of the molecule,
transferring momentum 2mvxto the wall. (b)Amolecule traveling with arbitrary velocityv.Ifits next collision
is with a wall parallel to theyz-plane, the effect of the collision is to reverse thex-component of the molecule’s
momentum, again transferring momentum 2mvxto the wall.
3.2.1 Temperature reflects the average kinetic energy of thermal motion
When faced with a mysterious new formula, our first impulse should be to think about it in the
light of dimensional analysis.
Your Turn 3g
Examine the left side of the ideal gas law (Equation 1.11 on page 23), and show that the product
kBThas the units of energy, consistent with the numerical value given in Equation 1.12.So we have a law of Nature, and it contains a fundamental, universal constant with units of energy.
Westill haven’t interpreted the meaning of that constant, but we will in a moment; knowing its
units will help us.
Let’s think some more about the box of gas introduced in Section 1.5.4 on page 23. If the
density is low enough (an ideal gas), the molecules don’t hit each other very often.^1 But certainly
each one does hit thewallsof the box. We now ask whether that constant hitting of the walls can
explain the phenomenon of pressure. Suppose that a gas molecule is traveling parallel to one edge
of the box, say in thexdirection, with speedvx,and the box is a cube of lengthL,sothat its
volume isV=L^3 (see Figure 3.5a).
Every time the molecule hits the wall, the molecule’s momentum changes frommvxto−mvx;
it delivers 2mvxto the wall. This happens every time the molecule makes a round trip, which takes
atime ∆t=2L/vx.Ifthere areNmolecules, all with this velocity, then the total rate at which
they deliver momentum to the wall is (2mvx)(vx/ 2 L)N.But you learned in first-year physics that
therate of deliveryof momentum is precisely the force on the wall of the box.
Your Turn 3h
Check the dimensions of the formulaf=(2mvx)(vx/ 2 L)Nto make sure they are appropriate
for a force.
Actually every molecule has its own, individual velocityvx. Really, then, what we need is notN
times one molecule’s velocity-squared, but rather the sum over all molecules, or equivalently,N
(^1) T 2 The precise way to say this is that we define an ideal gas to be one for which the time-averaged potential
energy of each molecule in its neighbors’ potential fields is neglibible compared to its kinetic energy.