The graphs on the right show the multiplicity Q for each "n out of 36" macrostate, the probability of observing this macrostate and its entropy S = InQ. (The entropy of a macrostate is defined as S = kg InQ, where kg is the Boltzmann constant. Let us measure S in units of kg and use S = InQ.) Which statement below best describes the relationship between probability and entropy for the macrostates we are considering? 1. The most probable macrostates have the lowest entropy. 2. The most probable macrostates have the highest entropy. 3. There is no clear relationship between entropy and probability for these macrostates. If we examine the probabilities for all of the macrostates in the system we find that the probability is 1. spread evenly among all of the macrostates. 2. spread out among a large fraction of the macrostates, but with some macrostates having slightly higher probabilities than others. 3. concentrated within a small fraction of the macrostates centered on a most probable macrostate. 4. entirely concentrated into a single most probable macrostate. How does this behavior scale with the number of coins N? If you have N squares and N coins, the most probable number "head up" coins is N/2. The spread of the distribution increases a N%, so the percentage spread decreases as N'1/2. For 36 square, the spread approximately plus or minus 6, the percentage spreads is (1/6)*100% or 16.6%. That means that if n differs by more than 16.6% from 18, then it is not very likely to occur. If there were N = 1024 squares and coins (just as there might be 1024 molecules in a gas), then the percentage spread is (107'2)*100% = 101%%, that means in n differs from N/2 by more than 1071%% from 0.5*1024, it is not very likely to occur. (e) Consider approximately 1024 gas molecules in a box. A divider which can conduct heat, for example a rubber diaphragm, separates the box into two chambers of equal size. Each chamber contains half of the molecules. Assume that the internal energy of the gas is (approximately) fixed. Describe some possible macrostates of the gas in the two chambers of the box. Describe a low entropy macrostate. Describe a high-entropy macrostate. In which situation described below is the entropy of the gas greatest? 1. When the energy is spread nearly evenly among all the molecules and all the molecules are clustered in small regions of each chamber. 2. When the energy is concentrated in a small number of molecules that are spread nearly evenly throughout the box. 3. When the energy is spread nearly evenly among all the molecules and all the molecules are spread nearly evenly throughout the box. 4. When the energy is concentrated in a small number of molecules which are clustered in a small region of one chamber of the box. Newton's laws work equally well backwards in time as they do forwards. If a sequence of motions and interactions leads to an increase in entropy, then the time- reversed sequence will lead to a decrease in entropy. Both sequences are allowed by Newton's laws. + Why does heat, of itself, not flow from a cold to a hot object, if it is allowed by Newton's laws? What, do you think, determines, the rate of change of the entropy of a system? Entropy can decrease locally (in a subsystem). Heat can be moved from a cold place to a hot place. (We all are familiar with refrigerators and heat pumps.) But it takes ordered energy that comes from outside the subsystem. For the entropy of a subsystem to decrease, the system as a whole must not yet have reached a state of maximum entropy. 10 20 Multiplicity 1E+10 7 9E+09 . BE+09 S 7E+09 B 6E+09 a 5E+09 > > 4E+09 + 3E+09 2E+09 - = g 0 0 10 20 30 n Probability 0.14 2 0,12~ - - 2 F o0 = = 00 . = Q 0.04 = = 0.02 * - - - D 0 10 20 30 n Entropy 2 amn o LuEEE Ty = e & . n 5 u u w10 & 5 u - 5