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Energy and Entropy Contents Energy and Entropy Physics 423 David Roundy Spring 2014 Draft 3/17/2014 Contents Contents i 1 Monday: Lab 1: Heat and Temperature1 Lecture: ...................................... 1 Lecture: ...................................... 1 Lecture: ...................................... 1 2 Tuesday: First and Second Laws3 Lecture: ...................................... 3 Lecture: ...................................... 3 Lecture: ...................................... 3 Lecture: ...................................... 3 Lecture: ...................................... 4 Lecture: ...................................... 4 Lecture: ...................................... 4 Lecture: ...................................... 5 Lecture: ...................................... 5 3 Wednesday: Second Law lab6 Lecture: ...................................... 6 Physics 423 i 4 Thursday: Heat and work 10 5 Friday: 11 Lecture: ...................................... 11 Lecture: ...................................... 11 6 Monday: 13 Lecture: ...................................... 13 Lecture: ...................................... 13 Lecture: ...................................... 14 Lecture: ...................................... 14 7 Tuesday: Lab 2: rubber band 15 8 Wednesday: 16 Lecture: ...................................... 16 Lecture: ...................................... 16 9 Thursday: Thermodynamics practice 18 10 Friday: Black bodies and statistics 19 Lecture: ...................................... 19 Lecture: ...................................... 20 Lecture: ...................................... 20 11 Monday: Statistical approach 22 Lecture: ...................................... 22 Lecture: ...................................... 23 Lecture: ...................................... 24 Lecture: ...................................... 27 12 Tuesday: From statistics to thermodynamics 28 Lecture: ...................................... 28 13 Wednesday: Statistical mechanics of air 31 Lecture: ...................................... 31 Lecture: ...................................... 32 14 Thursday: Statistical mechanics of air 35 Lecture: ...................................... 35 15 Friday: 37 Lecture: ...................................... 37 Physics 423 ii 1 Monday: Lab 1: Heat and Temperature This class is called Energy and Entropy. You've already learned quite a bit about energy, but entropy is probably quite new to you. Thermodynamics is a field that involves making experimental measurements of bulk substances, and using theory to connect those measurements with other measurements. We will begin this course by making measurements of how much energy is needed to heat up water (and ice) by a certain amount. Now that you all have some data being collected, let's talk about how you will be analyzing it. Many thermodynamic measurements are measurements of derivatives. The temperature you can measure directly, and the power of the heater (which is the energy dissipated into the water per unit time) we can work out directly. From the two of those, we will work out the heat capacity Write on board: dQ¯ \C = " p @T p The heat (called Q) is the amount of energy which is thermally transfered, just like work is the amount of energy which is mechanically transfered. I write the derivative funny because Q is not a function of T . The p subscript just means we're keeping the pressure constant. Another quantity we'll be looking at is the entropy. We'll spend much of this course talking about what entropy \really is", but for now, just know that you can measure entropy by measuring heat and integrating: Write on board: dQ¯ ∆S = reversible T Z In 1819, shortly after Dalton had introduced the concept of atomic weight in 1808, Dulong and Petit observed that if they measured the specific heat per unit mass of a variety of solids, and divided by the atomic weights of those solids, the resulting per-atom specific heat was essentially constant. This is the Dulong-Petit law, although we have since given a name to Physics 423 1 Monday 4/8/2014 that constant, which is 3R or 3kB, depending on whether the relative atomic mass (atomic weight) or the absolute atomic mass is used. This law isn't precisely true, and isn't always true, and is never true at low temperatures. But it captures some physics that we will later call the equipartition theorem. We will write Dulong-Petit's law as: Cp = 3NkB (1.1) where N is the total number of atoms. Physics 423 2 Tuesday 4/9/2014 2 Tuesday: First and Second Laws Thermodynamic quantities can be divided according to how they scale with quantity of material present. This division is sort of like dimensional analysis, and allows us to catch mistakes and reason about how properties behave. Extensive properties are those that are proportional to the amount of quantity present. Intensive properties are independent of the quantity present. It helps to imagine dividing your (homogeneous) system into two, and asking whether the quantity you're looking at is divided into two. e.g. imagine only \counting" half of this glass of water. As an example, V is extensive: if you've got twice as much stuff, it's Every time you come up with an answer in this course, you should ask yourself whether the result is extensive or intensive, and whether this is what you expected. It's like checking that the dimensions match up! We distinguish in thermodynamics between a system and its surroundings. The system is the thing we're measuring or predicting the properties of, and the surroundings is everything else. Which is which will depend on what experiment we're doing. The first law of thermodynamics simply states that energy is conserved (or is a substance, to use Aristotle's terminology). But it is useful to look at those two non-state variables work and heat. Both are changes in energy of a system, so we can write the first law as ∆U = Q + W (2.1) where U is the internal energy of the system, Q is the energy added to the system by heating, and W is the work done by the system (or the energy removed from the system by working). It is more convenient mathematically, however, to have a framework for talking about infinitesimal changes in the total energy... Write on board: dU =dQ ¯ +dW ¯ Physics 423 3 Tuesday 4/9/2014 The second law of thermodynamics clarifies this rule, and extends it to cases where there might be other things going on, e.g. in the case of a refrigerator. The second law involves the change in entropy, which I defined for you yesterday: dQ¯ ∆S = reversible (2.2) T Z The Second Law of Thermodynamics simply states that for any possible process, the change in entropy of a system plus its surroundings is either positive or zero. Write on board: ∆S + ∆S 0 system surroundings ≥ In 2011, I ad libbed quite a bit after this. a) Fast vs. slow b) quasistatic vs. reversible vs. irreversible vs. spontaneous c) adiabatic and its various meanings (including isentropic) The internal energy is clearly a state function, and thus its differential must be an exact differential. dU = ? (2.3) =dQ ¯ dW¯ (2.4) − =dQ ¯ pdV only when change is quasistatic (2.5) − I spent some time on work being pdV . What is thisdQ ¯ ? As it turns out, we can define a state function S called entropy and so long as a process is done reversibly dQ¯ = T dS only when change is quasistatic (2.6) so finally we find out that Write on board: dU = T dS pdV − The fact that the T in this equation is actually the physical temperature was originally an experimental observation. At this point, the entropy S is just some weird heat-related state function. Physics 423 4 Tuesday 4/9/2014 As we learned last week, heat capacity is amount of energy required to raise the temper- ature of an object by a small amount. dQ¯ C (2.7) ∼ @T dQ¯ = CdT At constant what? (2.8) If we hole the volume constant, then we can see from the first law that dU =dQ ¯ pdV (2.9) − since dV = 0 for a constant-volume process, @U C = (2.10) V @T V But we didn't measure CV on Monday, since we didn't hold the volume of the water constant. Instead we measured Cp, but what is that? To distinguish between different sorts of heat capacities, we need to specify the sort of path used. So, for instance, we could write dQ¯ = T dS (2.11) dQ¯ = CαdT +?dα (2.12) T dS = CαdT +?dα (2.13) C ? dS = α dT + dα (2.14) T T @S C = T (2.15) α @T α This may look like an overly-tricky derivative, so let's go through the first law and check that we got it right in a few cases. I'll do the CV case. We already know that dU =dQ ¯ pdV (2.16) − @U C = (2.17) V @T V @U @S = (2.18) @S @T V V @S = T (2.19) @T V where the second step just uses the ordinary chain rule. We can find a change in entropy from the heat capacity quite easily: 1 ∆S = dQ¯ (2.20) T QS Z C(T ) = dT (2.21) T Z Physics 423 5 Wednesday 4/10/2014 3 Wednesday: Second Law lab We can view the melting of the ice in the warm water as a two step process, assuming that the ice completely melts. In the first step, the ice completely melts, while the water Ti 0°C Q1 Tm 0°C Q2 Tf Tf cools down. This involves the water heating the ice, transfering an amount of energy which ◦ we will call Q1. After this step, the melted ice water is still at 0 C. In the second step, the cold water (which was formerly ice) is warmed up by the warmer water, until finally they end up at the same temperature. In this case, we will call the heat Q2. Of course, the real ice doesn't undergo these two distinct steps, since it melts gradually, and the melted water doesn't wait to start warming up, but immediately begins step two. What we are doing here is envisioning a simpler (and quasistatic) path that the system could have taken to reach the same final ending point. We have diagrammed three unknowns, Q1, Q2 and Tf . To solve for these quantities, we will need to invoke known properties of water. The entire experiment is done at fixed pressure, rather than fixed volume. If we examine the first law: ∆U = Q + W (3.1) or the thermodynamic identity dU = T dS pdV (3.2) − Physics 423 6 Wednesday 4/10/2014 we can see that the internal energy U isn't easy to relate to the heats Q1 and Q2, since we don't know how much work is done as the ice melts (or as the water changes temperature).
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