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Problem Set 12 Solutions 1. What is the increase in entropy of one gram of ice at OoC is melted and heated to 500C? The change in entropy is given by dS = dQ T. In this case, the dQ must be calculated in two pieces. First there is the heat needed to melt the ice, and then there is the heat needed to raise the temperature of the system ...
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To show Shannon’s entropy is the only expression that satisfy these three conditions, we design a special compound experiment. Consider an experiment in which we randomly pick 1 object out of Nobjects.
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- 1 Introduction
- S = ¡kBNXi filn fi (5)
- S = 2NkBln2
- 2 N = Sinit 2NkBln ¡ 2 (12)
- 2 N Sinit
- S = Piln Pi
- 4 Information entropy
- 4.3 Uniqueness
- HPQ = Pilog2(Pi) ¡ Xj Qjlog2(Qj) = HP + HQ (35)
- 6 Maxwell's demon
- 7 Quantum mechanical entropy (optional)
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- S = ~c Z 4 G c3 MdM = M2 = A (59) ~c 4~G
- 9 Summary
- H = Pilog2 Pi (61)
In this lecture, we discuss many ways to think about entropy. The most important and most famous property of entropy is that it never decreases
where fi ni = . Since ni = N then = 1 and so has the interpretation of a probability: N are the probabilities of nding a particle P fi fi fi picked at random in the group labeled i. With the factor of kB but without the N, the entropy written in terms of probabilities is called the Gibbs entropy: S = ¡kBXi Piln Pi (Gibbs entropy) (6) If all we do i...
(10) This increase is called the entropy of mixing. The entropy of mixing is a real thing, and can be used to do work. For example, say we had a vessel with xenon on one side and helium on the other, separated by a semi-permeable membrane that lets helium pass through and not xenon. Say the sides start at the same temperature and pressure. As the h...
Thus the entropy has gone down, by exactly the entropy of mixing. If we mix the gases entropy goes, up, if we split them entropy goes down. That entropy could go down by simply sticking a partition in a gas seems very strange, and apparently violates the second law of thermodynamics. This is called the Gibbs paradox. There are two parts to resolvin...
So the entropy is unchanged by adding, or removing the partition. What about the xenon/helium mixture? The gases are independent and do not interact, so each one separately acts just like helium alone. Thus inserting a partition in a helium/xenon mixture has a net e ect of S = 0 on each separately and therefore S = 0 total as well. What about the e...
where Pi ni = are the number ¡kBX of particles with N the properties of group i.
Next, we introduce the concept of information entropy, as proposed by Claude Shannon in 1948. We'll start by discussing information entropy in the context of computation, as it was originally introduced, and then connect it back to physics once we understand what it is. Consider the problem of data compression: we have a certain type of data and wa...
You might like to know that Shannon's formula for information entropy is not as arbitrary as it might seem. This formula is the unique function of the probabilities satisfying three criteria It does not change if something with Pi = 0 is added. It is maximized when Pi are all the same. It is additive on uncorrelated probabilities. This last criteri...
In other words, entropy is extensive. This is the same criterion Gibbs insisted on. So Gibbs entropy is also unique according to these criteria. By the way, there are other measures of information entropy other than Shannon entropy, such as the collision entropy, Renyi entropy and Hartley entropy. These measures do not satisfy the conditions 1-3 ab...
We're now ready to tackle the most famous paradox about entropy, invented by Maxwell in 1867. Suppose we have a gas of helium and xenon, all mixed together in a box. Now say a little demon is sitting by a little shutter between the two sides of the box. When he sees a helium molecule come in from the right, he opens a little door and lets it go lef...
This section requires some advanced appreciation of quantum mechanics. It's not a required part of the course, but some students might nd this discussion interesting. In quantum mechanics, distinguishability takes a more fundamental role, as does measurement. Thus, naturally, there are additional ways to quantify entropy in quantum mechanics. These...
This Hawking temperature is inversely proportional to the mass: very small black holes are very hot, and very large black holes are cold. This unusual behavior is associated with a negative heat capacity. Indeed, the speci c heat of a black hole is
Note that black holes have entropy proportional to their surface area. String theory even provides a way of counting microstates for certain supersymmetric black holes that agrees with this formula. So black holes have entropy, but no hair, and they evaporate in nite time into pure uncorrelated heat. This means that if some data falls into a black ...
We have seen a lot of di erent ways of thinking about entropy this lecture. The Gibbs entropy is
In this equation, Pi is the probability of ¡X certain data showing up. H has the interpretation as the minimal number of bits needed to encode the data, on average. Information entropy quanti es our ignorance of the system. The more entropy, the less information we have. An important result from information theory is Landauer's principle: erasing i...
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In this lecture, we revisit the basic notions of probability and define some quantities like entropy that will form the foundation for the upcoming lectures. 1 Probability. For events A, B, . . . , we will write P(A) to denote the probability of event A happening, and P(A, B) = P(A ∩ B) to denote the probability of both A and B happening.
Answer is (A). 2. Helium is compressed isentropically from 1 atmosphere and 5°C to a pressure of 8 atmospheres. The ratio of specific heats for helium is 5/3. What is the final temperature of the helium? 290°C. 340°C. 370°C. 650°C. Solution. 2 T 1 − k k P 1 . 1 T = P 2.
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EXAMPLE 1. Calculate the free energy change for the complete combustion of one mole of methane, CH4(g), the main component of natural gas. Is this reaction spontaneous? SOLUTION. We begin by writing the equation that represents this reaction.
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How to think about entropy?
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how to measure the entropy of a physical system. We know how to use entropy to solve problems and to place limits on processes. We understand the role of entropy in thermodynamics and in statistical mechanics. We also understand the parallelism between the entropy of physics and chemistry and the entropy of information theory.
