Simple Nature - Light and Matter

of heat to be transferred into either system:dS 1dQ 1= dS 2dQ 2.In terms of our previous definition of entropy, this is equivalent to1/T 1 = 1/T 2 , which makes perfect sense since the systems are inthermal equilibrium. According to our new approach, entropy hasalready been defined in a fundamental manner, so we can take thisas a definition of temperature:1T = dSdQwhere dS represents the increase in the system’s entropy from addingheat dQ to it.,Examples with small numbers of atomsLet’s see how this applies to an ideal, monoatomic gas with asmall number of atoms. To start with, consider the phase spaceavailable to one atom. Since we assume the atoms in an ideal gasare noninteracting, their positions relative to each other are really irrelevant.We can therefore enumerate the number of states availableto each atom just by considering the number of momentum vectorsit can have, without considering its possible locations. The relationshipbetween momentum and kinetic energy is E = (p 2 x+p 2 y+p 2 z)/2m,so if for a fixed value of its energy, we arrange all of an atom’s possiblemomentum vectors with their tails at the origin, their tips all lieon the surface of a sphere in phase space with radius |p| = √ 2mE.The number of possible states for that atom is proportional to thesphere’s surface area, which in turn is proportional to the square ofthe sphere’s radius, |p| 2 = 2mE.Now consider two atoms. For any given way of sharing the energybetween the atoms, E = E 1 + E 2 , the number of possiblecombinations of states is proportional to E 1 E 2 . The result is shownin figure h. The greatest number of combinations occurs when wedivide the energy equally, so an equal division gives maximum entropy.By increasing the number of atoms, we get a graph whose peakis narrower, i. With more than one atom in each system, the totalenergy is E = (p 2 x,1 + p2 y,1 + p2 z,1 + p2 x,2 + p2 y,2 + p2 z,2 + ...)/2m. With natoms, a total of 3n momentum coordinates are needed in order tospecify their state, and such a set of numbers is like a single pointin a 3n-dimensional space (which is impossible to visualize). For agiven total energy E, the possible states are like the surface of a3n-dimensional sphere, with a surface area proportional to p 3n−1 ,or E (3n−1)/2 . The graph in figure i, for example, was calculatedaccording to the formula E 29/21 E 29/22 = E 29/21 (E − E 1 ) 29/2 .Since graph i is narrower than graph h, the fluctuations in energysharing are smaller. If we inspect the system at a random moment inh / A two-atom system hasthe highest number of availablestates when the energy is equallydivided. Equal energy division istherefore the most likely possibilityat any given moment in time.i / When two systems of 10atoms each interact, the graph ofthe number of possible states isnarrower than with only one atomin each system.Section 5.4 Entropy As a Microscopic Quantity 319