Question

2. In lecture, we have derived the thermodynamic definition of temperature, OS In this problem, you will examine what happens to the entropy (i.e., the number of microstates) when two systems with different temperatures interact and exchange energy. The result foreshadows what we will later get to know as the second law of thermodynamics (a) *D1* A system at temperature and energy E absorbs a small amount of energy ΔΕ. Show that the change in entropy is approximately equal to AS-ΔΕΤ. Hint: To show this result, write a Taylor series for AS-S(B + ΔΕ)-S(E) around ΔΕ-0 ordered in powers of ΔΕ and apply the formula stated at the beginning of the problem. Since the energy change ΔΕ is small, you may neglect terms in the Taylor series of order ΔΕ2 and higher (i.e., include only the term proportional to ΔΕ). (2 pts) (b) *D1* Now consider two systems initially at temperatures Tı and T2. A small amount of energy E is transferred from system #1 to system #2, and the entropy of both systems changes by the amounts ΔSl and Δ59 Using your result from part (a), write the total change in entropy ASi + ΔS2 in terms of the temperatures T1 and T2, and the amount of exchanged energy ΔΕ. Keep in mind that and ΔΕ2-ΔΕ and ΔΕ, ΔΕ. (2 pts) (c) D1* Our physical intuition says that heat flows from hot systems to cold systems. In other words, if Ti > T2, then energy should spontaneously flow from system #1 to system #2. What does this observation imply about the sign of AS in problem (b)? In other words, does the total entropy increase or decrease when two systems spontaneously exchange energy? (2 pts)

Answer 1

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