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

• Notes
• BarristerStrawOryx9273
• 21
• 100% (1) 1 out of 1 people found this document helpful

This preview shows pages 1–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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
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

This preview has intentionally blurred sections. Sign up to view the full version.

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?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern