1. Flippingafaircoin. ConsiderthesequencesofN=4coinflipsHHHH,HHHT,HHTT,HTTT,andTTTT.Thesesequenceshave distinct com- positionswithn_H=0,1,...,4headsandn_T= 0,1,...,4 tails.Make a table for (a)-(c):

(a) What is the probability of each sequence?

(b) How many per-mutationsarethereforeachcomposition?

(c)Whatisthetotalprobability forallsequenceswitheachcomposition?

(Now, let'sagainsuppose thecoinwasflippedfourtimes,butinstead

(d)write all possible results of the four flips and group them by composition (i.e.., numberof heads, H, and tails, T).Let's define a ratio, A:the number of theseconfigurations(ormicrostates)associatedwithaparticularcompositiondividedbythenumberofconfigurationswitha1:1mixtureofH:T.

(d)GivethevalueofAforeachcomposition, and

(e)sketch(approximate)histogramsthatyouwouldwouldexpect forAasthenumberofcoinflipsisincreased. What do you expect after doing 1 mol of flips?

2. Consider an experiment in which a gas expands.Let's use the lattice modelshowninFigure1.InitiallytheNgasparticlesoccupyavolumewithMsites.Assume particles can simultaneously occupy any given site and thatthe particles are distinguishable.

(i) What is the multiplicity of the initial state?Explain your answer.When the wall is removed, what is the multiplicity?

(ii) What is the entropy of the initial state, when the wall is removed,andtheentropychange?Dotheseanswersmatchyourintuition?Justify your answers.