September, 00 Readng: Chapter Four Homework: None Entropy and The Degree of Dsorder: Consder a sold crystallne materal: At zero K: All atoms frozen at fxed postons on a perodc lattce. Add heat to a fnte temperature, S ncreases: all atoms vbrate about ther lattce postons; so nstantaneously atoms are randomly dsplaced from ther lattce postons --- dsorder. Add more heat to the meltng temperature, S ncreases further: atoms movng away completely from ther lattce postons; so nstantaneously atoms are randomly dstrbuted n the space --- more dsorder. In the lqud phase, a perodc lattce s destroyed, however, there stll remans certan degree of local order. On average, the number of nearest neghbors and the nearest-neghbor dstances of any gven atom are about the same. Add more heat to the evaporatng temperature, S ncreases even further: atoms movng completely randomly n the space --- even more dsorder. In the gas phase, even the local order doesn t exst; the moton of ndvdual atoms s completely uncorrelated. Therefore, there must be a correlaton (relatonshp) between the entropy and the degree of dsorder. The dsorder s related to the varous ways atoms can share the total amount of energy. For example, consder we add a fxed amount of heat (thermal energy) to a lattce to ncrease ts lattce vbratons (knetc energy). Ths amount of heat can be dstrbuted nto the lattce n many dfferent ways,.e, nstantaneously each ndvdual atom may ncrease ts velocty of a dfferent amount and movng n dfferent drectons, as long as the sum of total ncrease of knetc energy from all atoms corresponds to the heat nput. Dsorder s also related to varous ways atoms (partcles) can be dstrbuted n the space. For example, consder the mxng of two gases, or lquds, or solds. As an rreversble process, the entropy s ncreased. However, ths may happen wthout any energy transfer (heat nput or work done) but only dsorder s ncreased,.e., the spatal arrangement of partcles become more messy. adabatc Gas A Gas B Gas (A+B) daphragm
Example of lqud: alcohol + water: random soluton Examples of sold: slcon + germanum: random soluton Gold + slver: random soluton Dsorder --- A measure of the number of ways a system can be realzed under a gve set of state varables. The concept of Mcrostates: Consder three partcles localzed n space, each of them may occupy one of the many energy levels that are equally spaced (0, u, u, 3u, ). [It may be easer to vsualze an equvalent case: three men of the same weght clmbng an equally spaced starcase, each one of them may stand on (occupy) one step level wth potental energes of 0, (gh), (gh), or 3(gh),, where h s the step heght; u=gh] If the total energy s zero U = 0, All three men (partcles) must be dstrbuted on the ground level of zero energy. Snce nterchange of them s ndstngushable as they have the same weght (dentcal partcle), there s only one way to realze ths dstrbuton. If the total energy s U = u, One of the partcles must be dstrbuted onto the frst level, the other two remans at the ground level. Snce any one of them can be placed onto the frst level, there are three possble arrangements to realze ths dstrbuton. If the total energy s U = u. We can dstrbute partcles n two dfferent ways: n the frst dstrbuton, one partcle s put nto the second level, the other two remans at the ground level. Smlar to the case of U=u, there are three possble arrangements to realze ths dstrbuton:
In the second dstrbuton, two partcles are put nto the frst level and the thrd one remans on the ground level. Snce any one of them can be left on the frst level, three are also three possble arrangements to realze ths dstrbuton. The total number of arrangements for U=u s 3+3 = 6. If the total energy s U = 3u. We can dstrbute partcles n three dfferent ways: n the frst dstrbuton, all three partcles are put on the frst level. Smlar to the case of U=0, there s only one possble arrangement to realze ths dstrbuton: In the second dstrbuton, one partcle s put nto the thrd level, the other two remans at the ground level. Smlar to the case of U=u, there are three possble arrangements to realze ths dstrbuton:
In the thrd dstrbuton, one partcle each s put nto the ground, frst, and second level, respectvely. Snce any one of three can be put frst on the second level and then any one of the remanng two can be put on the frst level, the total number of ways to arrange them s 3xx=6. and The total number of arrangements of all dstrbutons for U=3u s +3+6 = 0. So, gven a total energy, there are number of dstngushable ways to arrange partcles nto dfferent levels, amountng to the same total energy. These ndvdual arrangements are called mcrostates. So, there can be many mcrostates corresponds to a gven dstrbuton and all the mcrostates correspond to a sngle macrostate of energy U. Calculatng number of mcrostates for a gven dstrbuton: Consder a system contanng number of N partcles of total energy U. They are dstrbuted nto an nfnte number of dfferent energy levels such that n 0 n ε 0, n n ε, n n ε,, subect to n = N and n ε = U, for a gven dstrbuton. The possble dstngushable arrangements,.e., the number of mcrostates for ths dstrbuton can be calculated as followng: The dentcal partcle only become dstngushable when they are on dfferent levels. The total number of ways to dstrbute N dentcal partcles nto all dfferent levels so that all the resultng arrangements are dstngushable s N!. Ths s because there s N ways to pck the frst partcle, N- ways to pck the second partcle,, one way to pck the last partcle; so the total number of ways s Nx(N-)x(N-)x x = N!.
When dentcal partcles are dstrbuted on the same energy level, they are ndstngushable. So, when there are n! ways to put number of n partcles nto the same energy level ε, they resulted nto n dstngushable arrangements;.e. only one dstngushable arrangement. = dstngushable arrangemetns total number of possble arrangements = number of ndstngushable arrangements = n 0 N! 3! = No. of dstngushable ways that system s realzed n ths dstrbuton,.e., ways to arrange partcles nto ths dstrbuton of partcles n dfferent levels. N! = No. of ways to to put N partcles nto ndvdual levels; n! = No. of ways to put n partcles nto level. Consder the example of three partcles we dscussed before, For U=0, = = ; 0!0!0! For U=u, = = 3;!!0!0! For U=u, = = 3 ; = = 3;!0!!0!!!0!0! For U=3u, = = ; = = 3; 3 = = 6 0!0!0!!0!0!!!!!0! So, s the total number of mcrostates of a gven dstrbuton, then = total denotes the total number of mcrostates corresponds to a macrostate.