Unformatted text preview: Yes, the beach balls are for you to enjoy. S5ll need a seminar? There’s a talk today at 3:30 in Johnson 337 on a technique to date archaeological ceramics. For sugges5ons of other seminars, see the discussion board. Lecture 10 • Announcements… – Exams will be returned in discussion sec3on on Tuesday. – Lecture 11 is quite technical…if you are o=en confused in class, then read sec3ons 10.2 and 10.13 before the next lecture. • Topics – Reac5on spontaneity (10.1) – Sta5s5cal interpreta5on of entropy and Boltzmann’s formula (10.3) • Ques5ons we’ll answer – Why do chemical reac5ons occur? – What does the term “spontaneous” mean in a chemical context? – How can we understand spontaneity from a microscopic perspec5ve? Spontaneous Process Water FD&C Yellow #5 dye • A spontaneous process occurs in a system leZ to itself; no ac5on from the surroundings is necessary to drive the process. • A non‐spontaneous process requires ac5on from the surroundings to occur. • If a process is spontaneous as wri]en, the reverse process is non‐spontaneous and vice versa. • Some spontaneous chemical reac5ons: – Spontaneous Combus5on of Oily rags: h]p://youtu.be/l0ujMpOOzU4 – … for Co]on: h]p:// – Ba(OH)2.8H2O + NH4NO3: h]p:// • NOTE: Spontaneous does not mean fast! • Spontaneous means thermodynamically favorable. What makes a process “spontaneous?” A gas diﬀusion video: h]p:// A Br2 diﬀusion video: h]p:// thermal barrier The gas spontaneously expands into the evacuated bulb, even though no energy (no heat, no work) is exchanged with the surroundings. Clearly, energy ﬂow is not enough to explain the observa3on… The Concept of Entropy (S) Bad deﬁni5on (because it can be misapplied) but easy to remember: “Entropy is usually associated with an increase in disorder.” More accurately, entropy is a measure of the how likely a par7cular arrangement of energy is. Another way of saying this is that entropy is a measure of the number of states available. The Concept of Entropy (S) First Law of Thermodynamics: Energy is neither created nor destroyed; there is a constant amount of energy in the universe. Second Law of Thermodynamics: The entropy of the universe is always increasing; the energy of the universe is con5nuously evolving towards a more
probable distribu5on. Any process that leads to an increase in the entropy of the universe is spontaneous. En Toss a coin, and there are two possible outcomes (which we’ll call microstates): it could land “heads” side up or “tails” side up. heads tails If you toss two or more coins simultaneously, the number of possible microstates increases exponen5ally. 4 coins 1 coin 2 coins 3 coins heads heads heads heads heads heads tails tails Heads heads heads tails 2 microstates 2 conﬁgura5ons heads Tails heads tails heads tails tails tails heads heads tails tails heads tails heads Tails heads tails Tails tails tails tails 2 x 2 = 22 = 4 microstates 3 conﬁgura5ons If the coins are indis5nguishable, then we can collect similar microstates into groups called “conﬁguraJons.” 2 x 2 x 2 = 23 = 8 microstates 4 conﬁgura5ons heads heads heads heads tails heads heads tails tails heads tails heads tails tails tails tails heads heads heads tails heads heads tails heads heads tails tails tails heads tails tails tails heads heads tails heads heads tails heads heads tails tails heads tails tails heads tails tails heads tails heads heads heads tails tails tails heads heads heads tails tails tails heads tails 24 = 16 microstates 5 conﬁgura5ons A 30‐second note for the persnickety: • In my notes, and in the physical chemistry literature: – ConﬁguraJon is deﬁned as a general arrangement of a system, which can be achieved by one or more microstates (e.g., two heads and two tails in a system of four coins) – Microstate is deﬁned as a speciﬁc arrangement of a system corresponding to a par5cular conﬁgura5on (e.g., heads‐tails‐heads‐tails, or heads‐heads‐tails‐tails, etc.) • Zumdahl’s Chemical Principles uses incorrect terminology. – The term “conﬁgura5on” is used to refer to a speciﬁc microstate. – The term “arrangement” is used to refer to the general arrangement of a system, which can be achieved by one or more microstates. Sta5s5cal Explana5on of Entropy • Imagine we have a collec5on of three dis5nguishable par5cles which has a total energy of 3 energy units, which we’ll call 3ε. quanta
1 2 3 3
2
1 • Each par5cle can accept only whole numbers of energy quanta…0ε, 1ε, 2ε, or 3ε. 1 2 Particle Number Let’s ask the ques5on, “What is the most likely distribu3on of this energy over the par3cles?” 3 First Conﬁgura5on: 3, 0, 0 The ﬁrst possible conﬁgura5on we consider is one in which all energy resides on one par5cle 3 3 3 2 2 2 1 1 1 1 2
3
Particle Number 1 2
3
Particle Number 1 2 Particle Number There are three ways (microstates) to do this 3 Second Conﬁgura5on: 2, 1, 0 Next conﬁgura5on: 2ε on one, 1ε on another, and 0ε on the third. 3 3 3 2 2 2 1 1 1 1 2 1 3 Particle Number 2 3 Particle Number 3 3 2 2 2 1 1 1 2 3 Particle Number 1 2 3 Particle Number 2 1 2 3 Particle Number 3 1 1 Particle Number Six ways to do this…six microstates 3 Third Conﬁgura5on: 1, 1, 1 The ﬁnal possible conﬁgura5on is 1ε on each par5cle. 3
2
1
1 2 3 Particle Number Only one way to do this…one microstate Which Conﬁgura5on? 3
2 3 ways • Which conﬁgura5on is most probable? 6 ways • Answer: The conﬁgura5on with the greatest number of microstates 1 way • In this case: “2, 1, 0” 1
1 2 3 3
2
1
1 2 3 3
2
1
1 2 3 The Dominant Conﬁgura5on • ConﬁguraJon: general distribu5on of total energy in the system. e.g., (3, 0, 0), (2, 1, 0), and (1, 1, 1) from the previous example • Microstate: a speciﬁc distribu5on of energy corresponding to a conﬁgura5on. 3
e.g., 2 from the (3, 0, 0) conﬁgura5on 1
1 2 3 Dominant Conﬁgura5on: (2,1,0) 3
• Which conﬁgura5on will you see? 2
6 ways
The one with the largest # of microstates. 1
This is called the dominant conﬁguraJon. 1
2
3 • A system in a non‐dominant conﬁgura5on will spontaneously evolve un5l it achieves the dominant conﬁgura5on. Sta5s5cal Entropy Weight (Ω, “omega”): the number of microstates associated with a given conﬁgura5on. The more microstates, the higher the entropy (S): S = kB ln Ω Boltzmann’s constant: kB = R/NAvogadro = 1.38 x 10‐23 J/K Example: Ideal Gas Expansion What is ΔS for the constant‐
Initial # microstates = Ω
temperature expansion of a monatomic ideal gas from V1 to 2V1? Final # microstates = 2Ω
Then ΔS = S2 – S1 ΔS = kB ln ( 2Ω) − kB ln ( Ω)
Start by considering only one atom. AZer expansion, the atom has twice the number of posi5ons available. ⎛ 2Ω ⎞
= kB ln ⎜
⎟
⎝ Ω ⎠ = kB ln ( 2 )
Change in entropy per atom in the monatomic ideal gas. Example: Ideal Gas Expansion What is ΔS for expanding the volume available to a single atom in a monatomic ideal gas from V1 to 2V1 (at constant temperature)? For N of par5cles, the volume available doubles for each of the NA par5cles: ΔS = N ⋅ the previous answer () = N ⋅ k B ln 2 If the number of par5cles is NA, then: Change in entropy ΔS
per par3cle () = kB ln 2 () ΔS = N A ⋅ k B ln 2 ΔS = R ln ( 2 ) Entropy change associated with doubling volume for NA par3cles. General Expression for Entropy • Note that in the previous example, the number of microstates was directly propor5onal to volume. • Generalizing: ΔS = kB ln Ω f − kB ln ( Ωi ) ⎛ Ω f ⎞
= kB ln ⎜
…for one par3cle ⎟
⎝ Ωi ⎠
⎛ Ω f ⎞
= N ⋅ kB ln ⎜
⎟ …for N par3cles ⎝ Ωi ⎠ ( ) ⎛ V f ⎞
ΔS = N ⋅ kB ln ⎜ ⎟
⎝ Vi ⎠ …since number of available posi3ons scales linearly with volume. Explaining the Spontaneity Demos Gas Expansion into Vacuum SoluJon FormaJon Food Coloring Volume of gas increases SYSTEM ENTROPY INCREASES Water Volume of both liquids increases SYSTEM ENTROPY INCREASES ...
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