University:
Boston University
Course:
CAS CH 214 | Organic Chemistry with Qualitative Analysis
Academic year:

2023
Views:

455

Pages:

7
Author:

Hanusia3lug

Blind chance & dumb luck Slides on particle dispersal, pressure equilibration, and entropy     Enumerating particle dispersal Practice with particle dispersal Maximum particle dispersal = uniform pressure Arrangements → entropy Everything—absolutely everything—that happens, happens solely because of blind chance and dumb luck. To quantify this … 1. Learn to count the ways 2. Search for greatest number of ways Counting distinguishable (unique) arrange Counting distinguishable (unique) arrangements Say we have three girls and four boys. What is the probability of calling them into line in the order ? 1 Counting distinguishable (unique) arrange Counting distinguishable (unique) arrange Say we have three girls and four boys. Say we have three girls and four boys. What is the probability of calling them into line in the order ? If we ignore whether a child is boy or girl, what is the total number of arrangements? 7 6 5… 1 7! 5040 Counting distinguishable (unique) arrange Counting distinguishable (unique) arrange Say we have three girls and four boys. Say we have three girls and four boys. For each particular arrangement, say about? 3! 4! 144 , how many ways can it come This means the total number of arrangements can be expressed as 7! = W 5040 W 3! 4! 144W 5040/144 35 2 Counting distinguishable (unique) arrange More generally, say we have j girls and k boys. The number of unique arrangements of of another kind is , lecture 27 1. 2. 3. 4. 5. 8 10 12 120 None of these object ! ! [TP] How many distinguishable ways can 3 water molecules and 2 ink molecules be arranged? 0% 0% 0% 0% 0% objects of one kind and [TP] How many distinguishable ways can 5 water molecules and 2 ink molecules be arranged? 0% 0% 0% 0% 0% 1. 2. 3. 4. 5. 14 21 240 5040 None of these 3 [TP] How many distinguishable ways can 2 ink molecules be arranged among 12 water molecules? 0% 0% 0% 0% 0% 1. 2. 3. 4. 5. 36 240 455 720 None of these The more water, the more ways ink disperses 2 ink particles in water Pressure in a gas is unequal Maximum particle dispersal = uniform pressure permeable barrier 4 Pressure in a gas is uniform Pressure in a gas becomes uniform permeable barrier Why? Lattice gas model of pressure 1/(R T) P = n/V = gas density n = particles V = lattice positions Pleft > Pright Left side: n/V = 2/4, Wleft = … 6 Right side: n/V = 1/8, Wright = … 8 Wtotal = Wleft × Wright = 6 × 8 = 48 5 Pleft = Pright Pleft < Pright Left side: n/V = 1/4, Wleft = … 4 Right side: n/V = 2/8, Wright = … Left side: n/V = 0/4, Wleft … 1 Right side: n/V = 3/8, Wright = … 28 56 Wtotal = Wleft × Wright = 4 × 28 = 112 Wtotal = Wleft × Wright = 1 x 56 = 56 Pressure in a gas becomes uniform Arrangements  Entropy Why? Pleft > Pright has Wtotal = 48 Pleft = Pright has Wtotal = 112 Pleft < Pright has Wtotal = 56 Uniform pressure maximizes W! 6 S = kB ln(W) S = kB ln(W) Why natural log? Doubling size of system: W  W × W = W2 Doubling size of system: S  2 S, so … Boltzmann’s definition makes S scale with size of system (extensive). kB = R/NA = 1.4 × 10-23 J/K Spontaneous? Calculate the entropy change. Wi = 1  Wf = (6 + 3)!/(6! 3!) = 84 ΔS = Sf – Si = kB ln(Wf/Wi) = kB ln(84/1) > 0 7