ESCI 341 Atmospheric Thermodynamics Lesson 10 The Physical Meaning of Entropy

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1 ESCI 341 Atmospherc Thermodynamcs Lesson 10 The Physcal Meanng of Entropy References: An Introducton to Statstcal Thermodynamcs, T.L. Hll An Introducton to Thermodynamcs and Thermostatstcs, H.B. Callen An Introducton to Informaton Theory: Symbols, Sgnals and ose, J.R. Perce STATISTICAL MECHAICS Unlke other thermodynamc varables (e.g. U, T, H, G, F), entropy seemngly lacks a physcal meanng. Entropy does have a concrete physcal meanng, but ts meanng s found va the feld of statstcal mechancs. Statstcal mechancs s also referred to at tmes as statstcal physcs, thermostatstcs, or statstcal thermodynamcs. Thermodynamcs apples to macroscopc (large) systems consstng of on the order of at least 10 0 molecules. Thermodynamcs looks at the large scale propertes. Statstcal mechancs looks at the molecular level. Statstcal mechancs and thermodynamcs are ntmately related. Thermodynamc concepts such as pressure and temperature are ted to processes occurrng on the molecular level, but averaged over a large number of molecules. Ar parcels, cloud droplets, etc. are all macroscopc systems, and so as meteorologsts we get along just fne usng thermodynamcs. However, n order to really understand entropy we have to resort to statstcal mechancs. Ths lesson s just a very, very bref overvew of the feld of statstcal mechancs, and s meant only to gve a flavor of how t relates to thermodynamcs. In partcular, the man objectve of ths lesson s to gve a better understandng of entropy. QUATUM STATES A system of matter exsts n dscrete quantum states. Each quantum state has a certan energy level assocated wth t. A certan energy level may be assocated wth two or more quantum states, n whch case we say that partcular energy level s degenerate.

2 If n quantum states have the same energy level, the degeneracy for that energy level s = n. The possble energy levels for the system are denoted as E where s a postve nteger (0, 1,, 3, ). refers to the avalable energy levels, not the quantum states. The degeneracy of energy level E s denoted as. For macroscopc objects the dfference between adjacent energy levels s nfntesmally small. For macroscopc systems we don t need to bother wth quantum states, as t appears that there s a contnuous spectrum of energy. For mcroscopc systems the energy spectrum s not contnuous, but s dscrete. QUATUM FLUCTUATIOS For smplcty, assume our mcroscopc system s composed of number of molecules of a sngle substance contaned n a volume of V. The system s mmersed n a macroscopc heat bath that has a constant temperature of T. Our mcroscopc system s a closed but not solated system. Our mcroscopc system can exst n an nfnte number of quantum states, each wth ts own energy level. Rather than remanng n a sngle quantum state, the system actually fluctuates through all avalable quantum states. Over a long perod of tme the system wll actually spend some fnte amount of tme n every avalable quantum state. The energy of our mcroscopc system wll therefore fluctuate as t moves from one quantum state to another. The probablty of fndng the system n a specfc quantum state s gven by P E e kt, (1) Q where the functon Q s called the partton functon and s gven by (k s the Boltzmann constant). Q E e kt ()

3 The probablty of the system beng n a certan energy level, E, s gven by E, V kt e P (, V, T), (3) Q(, V, T) Recall that s the degeneracy of energy level, and s the number of ndvdual quantum states that have an energy value of E. PHYSICAL MEAIG OF ETROPY We ve very brefly attempted to descrbe the relatonshp between thermodynamcs and statstcal mechancs. The man reason we ve done ths s so that we can try to gve entropy a physcal meanng. In statstcal mechancs entropy s gven by S k P ln P (4) where P s the probablty of the system beng n quantum state, gven by Eqn. (1). Some examples may help. Imagne that a hypothetcal system has three dfferent possble quantum states, = 1 to 3, wth probabltes gven by P 11/16 1 P 3 /16 P 1/ 8 then from equaton (1) the entropy equal to 3 S k ln /16 ln 3/16 (1/ 8)ln(1/ 8) 0.831k. If nstead all three quantum states had the same energy level, and were therefore all equally probable, the entropy would be S k 1 3 ln 1 3 1/ 3 ln 1/ 3 (1/ 3)ln(1/ 3) 1.099k. The second system has hgher entropy because there s less certanty as to whch quantum state the system s n. Entropy s a measure of the uncertanty of the quantum state of the system. 3

4 ETROPY, TEMPERATURE, AD ABSOLUTE ZERO At hgher temperatures the probabltes of the system occupyng a hgherenergy quantum state become larger. The probabltes are spread over more possble quantum states. Ths leads to more uncertanty as to the quantum state of the system, and s why hgher temperatures generally have hgher entropy. As a system s cooled, the lower-energy quantum states have the hghest probablty, and the hgher-energy quantum states become less probable. There s more certanty of the quantum states at low temperature, and s why lower temperatures are assocated wth lower entropy. As absolute zero s approached, the only quantum states avalable are those that have the mnmum amount of energy. The hgher-energy quantum states are so mprobable that t s nearly certan whch states the system are n, and the entropy approaches zero. If the lowest energy state s nondegenerate, then at absolute zero the system s certanly n ths sngle quantum state. From equaton (4) the entropy would be zero at absolute zero for a system havng a nondegenerate, lowest energy quantum state. If the system has a degenerate lowest-energy level (multple quantum states havng the lowest possble energy), then the entropy of the system wll be nonzero even at absolute zero. ETROPY, UCERTAITY, AD RADOMESS Because entropy s assocated wth the amount of uncertanty of the quantum states of a system, hgher entropy s often assocated wth randomness (hgh uncertanty) whle lower entropy s assocated wth order (low uncertanty). It s mportant to keep n mnd that entropy was defned n terms of quantum states n the context of statstcal mechancs and thermodynamcs. Many tmes we see entropy appled (or msappled) n other contexts. For example, magne a bn contanng both red beads and whte beads. If the red and whte beads are separated, they presumably have more order. If they are mxed together, then presumably they are more dsordered. Some 4

5 would say that the mxed beads have a hgher entropy than the separated beads. However, t s not approprate to apply the thermodynamc concept of entropy to ths stuaton. There s a type of entropy that s applcable to the bead example, but t s not the thermodynamc entropy. Instead, t s a mathematcal entropy that appears n the branch of appled mathematcs known as nformaton theory (see the Perce reference for a good, layman s descrpton of nformaton theory). Thermodynamc entropy and nformaton theory entropy are analogous, and behave n smlar ways, but are two dstnct concepts. When people speak of randomness n nature as havng hgher entropy than order n nature, t s the entropy from nformaton theory that they speak of, not thermodynamc entropy. The second law of thermodynamcs apples to thermodynamc entropy, not to the entropy from nformaton theory. However, there are scentsts and mathematcans who are explorng possble connectons between the two entropes. A FIAL THOUGHT O THE PARTITIO FUCTIO The partton functon s of fundamental mportance when relatng statstcal mechancs to thermodynamcs, because all macroscopc state varables can be found from the partton functon. For example: ln Q p kt V U T, ln Q kt T V, ln Q S kt k ln Q T V, 5