A.The monatomic ideal gas 80 isotropic The entropy of a pure classical monatomic ideal gas,as a function of energy E,volume V,and particle number N,is nematic given by the Sackur-Tetrode formula 3 4TmEV2B smectic S(E,V,N)=kRN In 2 3h6W丽 (2) 2 Let us examine this result qualitatively to see whether it agrees with our understanding of entropy as proportional to 0 0.10.20.3 0.40.5 the number of microstates that correspond to a given mac- composition rostate.If the volume is increased,then the formula states that the entropy S increases,which certainly seems reason- Fig.1.Phase diagram of a liquid crystal mixture.The variable"composi- able:If the volume goes up,then each particle has more tion''refers to the molecular weight ratio of 60CB to 80CB.Figure modi- places where it can be,so the entropy ought to increase.If fied from Refs.21,22. the energy E is increased,then S increases,which again seems reasonable:If there is more energy around,then there homogeneous bowl of meltwater,yet it is the ice slivers that will be more different ways to split it up and share it among the particles,so we expect the entropy to increase.But what have the lower entropy.The moral here is that the huge if the mass m of each particle increases?(Experimentally, number of microscopic degrees of freedom in the meltwater one could compare the absolute entropy of,say,argon and completely overshadow the minute number of macroscopic krypton under identical conditions.18)Our formula shows degrees of freedom in the jumbled ice slivers.But the analo- that entropy increases with mass,but is there any way to gies of entropy to "disorder"'or "smoothness''invite us to understand this qualitatively? ignore this moral and concentrate on the system's gross ap- pearance and nearly irrelevant macroscopic features. In fact,I can produce not just one but two qualitative arguments concemning the dependence of S on m.Unfortu- nately the two arguments give opposite results!The first re- C.Reentrant phases lies upon the fact that When the temperature falls at constant pressure,most pure materials pass from gas to liquid to solid.But the unusual (3) materials called "liquid crystals,"which consist of rodlike molecules,display a larger number of phases.For typical so for a given energy E,any individual particle may have a liquid crystals,the high-temperature liquid phase is isotropic. momentum ranging from 0 to v2mE.A larger mass implies meaning that the positions and the orientations of the mol- a wider range of possible momenta,which suggests more ecules are scattered about nearly at random.At lower tem- microstates and a greater entropy.The second argument re- peratures,the substance undergoes a transition into the so- lies upon the fact that called "nematic"phase,in which the molecules tend to orient in the same direction but in which positions are still (4) scattered.At still lower temperatures it passes into the "smectic''phase,in which the molecules orient in the same direction and their positions tend to fall into planes.Finally, so for a given energy E,any individual particle may have a at even lower temperatures,the molecules freeze into a con- speed ranging from 0 to v2E/m.A larger mass implies a ventional solid.The story told so far reinforces the picture of narrowed range of possible speeds,which suggests fewer "entropy as disorder,with lower-temperature(hence lower microstates and a smaller entropy.The moral is simple: entropy)phases showing more and more qualitative order. Qualitative arguments can backfire! But not all liquid crystals behave in exactly this fashion. One material called "hexyloxycyanobiphenyl'or 60CB' B.Freezing water passes from isotropic liquid to nematic to smectic and then back to nematic again as the temperature is lowered.2122 The It is common to hear entropy associated with"disorder," first transition suggests that the nematic phase is "less or- s“smoothness,.”ors“homogeneity.”How do these associa- derly''than the smectic phase,while the second transition tions stand up to the simple situation of a bowl of liquid suggests the opposite! water placed into a freezer?Initially the water is smooth and One might argue that the lower-temperature nematic homogeneous.As its temperature falls,the sample remains phase-the so-called "reentrant nematic"-is somehow homogeneous until the freezing point is reached.At the qualitatively different in character from the higher- freezing temperature the sample is an inhomogeneous mix- temperature nematic,but the experiments summarized in Fig. ture of ice and liquid water until all the liquid freezes.Then 1 demonstrate that this is not the case.These experiments the sample is homogeneous again as the temperature contin- involve a similar liquid crystal material called "octyloxy- ues to fall.Thus the sample has passed from homogeneous to cyanobiphenyl'or 8OCB'which has no smectic phase at inhomogeneous to homogeneous,yet all the while its entropy all.Adding a bit of 8OCB into a sample of 6OCB reduces has decreased. the temperature range over which the smectic phase exists. Suppose the ice is then cracked out of its bowl to make Adding a bit more reduces that range further.Finally,addi- slivers,which are placed back into the bowl and allowed to tion of enough 80CB makes the smectic phase disappear rest at room temperature until they melt.The jumble of altogether.The implication of Fig.I is clear:there is no irregular ice slivers certainly seems disordered relative to the qualitative difference between the usual nematic and the re- 1091 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1091A. The monatomic ideal gas The entropy of a pure classical monatomic ideal gas, as a function of energy E, volume V, and particle number N, is given by the Sackur–Tetrode formula S~E,V,N!5kBNF 3 2 lnS 4pmEV2/3 3h0 2 N5/3 D 1 5 2G . ~2! Let us examine this result qualitatively to see whether it agrees with our understanding of entropy as proportional to the number of microstates that correspond to a given mac￾rostate. If the volume V is increased, then the formula states that the entropy S increases, which certainly seems reason￾able: If the volume goes up, then each particle has more places where it can be, so the entropy ought to increase. If the energy E is increased, then S increases, which again seems reasonable: If there is more energy around, then there will be more different ways to split it up and share it among the particles, so we expect the entropy to increase. But what if the mass m of each particle increases? ~Experimentally, one could compare the absolute entropy of, say, argon and krypton under identical conditions.18! Our formula shows that entropy increases with mass, but is there any way to understand this qualitatively? In fact, I can produce not just one but two qualitative arguments concerning the dependence of S on m. Unfortu￾nately the two arguments give opposite results! The first re￾lies upon the fact that E5 1 2m ( i pi 2 , ~3! so for a given energy E, any individual particle may have a momentum ranging from 0 to A2mE. A larger mass implies a wider range of possible momenta, which suggests more microstates and a greater entropy. The second argument re￾lies upon the fact that E5 m 2 ( i vi 2 , ~4! so for a given energy E, any individual particle may have a speed ranging from 0 to A2E/m. A larger mass implies a narrowed range of possible speeds, which suggests fewer microstates and a smaller entropy. The moral19 is simple: Qualitative arguments can backfire! B. Freezing water It is common to hear entropy associated with ‘‘disorder,’’ ‘‘smoothness,’’ or ‘‘homogeneity.’’ How do these associa￾tions stand up to the simple situation of a bowl of liquid water placed into a freezer? Initially the water is smooth and homogeneous. As its temperature falls, the sample remains homogeneous until the freezing point is reached. At the freezing temperature the sample is an inhomogeneous mix￾ture of ice and liquid water until all the liquid freezes. Then the sample is homogeneous again as the temperature contin￾ues to fall. Thus the sample has passed from homogeneous to inhomogeneous to homogeneous, yet all the while its entropy has decreased. Suppose the ice is then cracked out of its bowl to make slivers, which are placed back into the bowl and allowed to rest at room temperature until they melt.9 The jumble of irregular ice slivers certainly seems disordered relative to the homogeneous bowl of meltwater, yet it is the ice slivers that have the lower entropy. The moral here is that the huge number of microscopic degrees of freedom in the meltwater completely overshadow the minute number of macroscopic degrees of freedom in the jumbled ice slivers. But the analo￾gies of entropy to ‘‘disorder’’ or ‘‘smoothness’’ invite us to ignore this moral and concentrate on the system’s gross ap￾pearance and nearly irrelevant macroscopic features. C. Reentrant phases When the temperature falls at constant pressure, most pure materials pass from gas to liquid to solid. But the unusual materials called ‘‘liquid crystals,’’ which consist of rodlike molecules, display a larger number of phases.20 For typical liquid crystals, the high-temperature liquid phase is isotropic, meaning that the positions and the orientations of the mol￾ecules are scattered about nearly at random. At lower tem￾peratures, the substance undergoes a transition into the so￾called ‘‘nematic’’ phase, in which the molecules tend to orient in the same direction but in which positions are still scattered. At still lower temperatures it passes into the ‘‘smectic’’ phase, in which the molecules orient in the same direction and their positions tend to fall into planes. Finally, at even lower temperatures, the molecules freeze into a con￾ventional solid. The story told so far reinforces the picture of ‘‘entropy as disorder,’’ with lower-temperature ~hence lower entropy! phases showing more and more qualitative order. But not all liquid crystals behave in exactly this fashion. One material called ‘‘hexyloxycyanobiphenyl’’ or ‘‘6OCB’’ passes from isotropic liquid to nematic to smectic and then back to nematic again as the temperature is lowered.21,22 The first transition suggests that the nematic phase is ‘‘less or￾derly’’ than the smectic phase, while the second transition suggests the opposite! One might argue that the lower-temperature nematic phase—the so-called ‘‘reentrant nematic’’—is somehow qualitatively different in character from the higher￾temperature nematic, but the experiments summarized in Fig. 1 demonstrate that this is not the case. These experiments involve a similar liquid crystal material called ‘‘octyloxy￾cyanobiphenyl’’ or ‘‘8OCB’’ which has no smectic phase at all. Adding a bit of 8OCB into a sample of 6OCB reduces the temperature range over which the smectic phase exists. Adding a bit more reduces that range further. Finally, addi￾tion of enough 8OCB makes the smectic phase disappear altogether. The implication of Fig. 1 is clear: there is no qualitative difference between the usual nematic and the re￾Fig. 1. Phase diagram of a liquid crystal mixture. The variable ‘‘composi￾tion’’ refers to the molecular weight ratio of 6OCB to 8OCB. Figure modi- fied from Refs. 21, 22. 1091 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1091