Simple Nature - Light and Matter

The proportionality of volume to temperature at fixed pressurewas the basis for our definition of temperature.Pressure related to temperature example 8Pressure is proportional to temperature when volume is held constant.An example is the increase in pressure in a car’s tires whenthe car has been driven on the freeway for a while and the tiresand air have become hot.We now connect these empirical facts to the kinetic theory ofa classical ideal gas. For simplicity, we assume that the gas ismonoatomic (i.e., each molecule has only one atom), and that itis confined to a cubical box of volume V , with L being the lengthof each edge and A the area of any wall. An atom whose velocityhas an x component v x will collide regularly with the left-hand wall,traveling a distance 2L parallel to the x axis between collisions withthat wall. The time between collisions is ∆t = 2L/v x , and in eachcollision the x component of the atom’s momentum is reversed from−mv x to mv x . The total force on the wall isF = ∑ ∆p x,i∆t i[monoatomic ideal gas] ,where the index i refers to the individual atoms.∆p x,i = 2mv x,i and ∆t i = 2L/v x,i , we haveF = ∑ mv 2 x,iL[monoatomic ideal gas] .SubstitutingThe quantity mvx,i 2 is twice the contribution to the kinetic energyfrom the part of the atoms’ center of mass motion that is parallel tothe x axis. Since we’re assuming a monoatomic gas, center of massmotion is the only type of motion that gives rise to kinetic energy.(A more complex molecule could rotate and vibrate as well.) If thequantity inside the sum included the y and z components, it wouldbe twice the total kinetic energy of all the molecules. Since weexpect the energy to be equally shared among x, y, and z motion, 1the quantity inside the sum must therefore equal 2/3 of the totalkinetic energy, soF = 2K total3L[monoatomic ideal gas] .Dividing by A and using AL = V , we haveP = 2K total3V[monoatomic ideal gas] .This can be connected to the empirical relation P V ∝ nT if wemultiply by V on both sides and rewrite K total as n ¯K, where ¯K isthe average kinetic energy per molecule:P V = 2 3 n ¯K [monoatomic ideal gas] .1 This equal sharing will be justified more rigorously on page 322.Section 5.2 Microscopic Description of An Ideal Gas 307