handout10 - 4 The Second Law of Thermodynamics 4.1...

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4– The Second Law of Thermodynamics 4.1 Introduction In 1867, Rudolf Clausius published "Abhandlungen über die mechanische Wärmetheorie, Zweite Abteilung", where he introduced the term entropy (Greek for ’transformation’). REMINDER: Entropy is an extensive state function. The difference in entropy is a way of measuring the effects of irreversibility of a thermodynamic process. We use the symbol S , units are Energy Temperature . NOTE: There are different definitions and ways of understanding entropy: Thermodynamics (Clausius) If a system moves from state A to B , then the difference in entropy is Δ S = S B - S A = Z B A 1 T ( Q ) d Q . (4.1) We can think of entropy as a ’measure of waste’ in a heat engine: (at least) T Δ S will be given up by the engine to the surroundings as unusable heat. This is also true e.g. for chemical reactions: We will see that the Gibbs free energy change of a thermodynamic process is Δ G = Δ H - T Δ S . (4.2) Δ G is a measure of the actual useful energy that can be 4–1
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extracted from an isothermal, isobaric system. Here, we have to subtract T Δ S from Δ H to account for the ’internal losses’ due to entropy. NOTE: We use the word ’free energy’ because this is energy we can extract; the entropic part is ’not free’ (bound) in this sense. NOTE: The new IUPAC name for this is simply ’Gibbs Energy’. Statistical Mechanics (Boltzmann) The entropy of a (macroscopic) state I is defined as S I = k B log e ( Ω I ) , (4.3) where Ω I is the number of microscopic states in the macrostate. NOTE: We can now see why entropy and the Boltzmann constant have the same units: log e ( Ω I ) is dimensionless. EXAMPLE: We have seen the Ω I before. When we derived the Boltzmann distribution in 2.18, we showed the example of flipping a coin four times and then asked how many times, m 4 ( k H ) , we would observe a game with k H heads: 4–2
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Table 2.B k H m 4 ( k H ) p 4 ( k H ) 0 1 1 16 1 4 4 16 2 6 6 16 3 4 4 16 4 1 1 16 Here, the macroscopic state I is k H and Ω I = m 4 ( k H ) . Then it is easy to calculate the log e ( Ω I ) of each macroscopic state: Table 4.A k H = I m 4 ( k H ) = Ω I log e ( Ω I ) 0 1 0 1 4 1.3863 2 6 1.7918 3 4 1.3863 4 1 0 NOTE: When we derived the Boltzmann distribution, we said that, at equilibrium, the most probable macrostate I max is the one with the most microstates Ω I max . 4–3
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macroscopically: ’S looks blue’ S This also means, that the entropy of a system is maximal when it is at equilibrium . NOTE: This is where the notion of ’entropy as a measure of disorder’ comes from: Macroscopic state with small number of microstates low entropy large number of microstates high entropy QUESTION: So entropy is a measure of the number of microstates in a macroscopic state. Why do we take the logarithm when measuring entropy, instead of just using Ω I directly?
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