TI-Nspire™ Activities: Physics
February 12, 2008 through February 19, 2008
Entropy
| Part 1 – Microstates and macrostates | |
| Step 1: Students should open the file PhysWeek03_Entropy.tns and read the first three pages. Page 1.4 shows a box divided into two sides with a particle on each side. Students should manipulate the particles in the box and then answer questions 1 and 2. Q1. In terms of the two sides of the box, how many different possible arrangements are there for the two particles? What are they? A. There are four possible arrangements: the initial state, with A on the left and B on the right; B on the left, A on the right; A and B both on the right; and A and B both on the left. Q2. Which microstate is most probable? A. Each microstate is equally probable. |
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| Step 2: Next, students should read pages 1.6 and 1.7, which describe the possible macrostates for the two-particle scenario. Then, they should move to page 1.8, which contains a Lists & Spreadsheet application. In the spreadsheet, each row describes a different macrostate. Note: Students may struggle to understand how there can be a different number of microstates than macrostates for this situation. To help them grasp this concept, you may have them delete the labels on the particles on page 1.4. Then, have them describe the arrangement of the particles in each microstate. Guide them to realize that when the particles are interchangeable, the number of possible states decreases. |
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| Step 3: Students should enter the number of microstates that correspond to each of the macrostates in the table. For example, because there is only one possible microstate that corresponds to the macrostate of having both particles on the right, students should enter “1” in row 3, column C. After students have entered the correct number of microstates in column C, they should answer questions 3 and 4. Q3. Which macrostate is most probable? A. The second macrostate, with one gas particle on each side, is the most probable. If students have difficulty recognizing this, remind them that the probability that a given macrostate will occur is equal to the number of microstates corresponding to that macrostate divided by the total possible number of microstates. Q4. How many more times more likely is it to have one particle on each side than to have two particles on the left side of the box? A. two times more likely |
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| Step 4: Next, students should read page 1.10 before moving to page 1.11, which shows a similar two-part box containing four particles. The students should move the particles to produce the most likely arrangement of particles in the box. Encourage student discussion of the most probable arrangement. Also, encourage students to discuss how the interchangeability of the particles makes many of the microstates equivalent. |
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| Step 5: After manipulating the four particles on page 1.11, students should read page 1.12 and then move to page 1.13, which contains a Lists & Spreadsheet application similar to the one on page 1.8. Students should enter the number of microstates corresponding to each macrostate in the spreadsheet. Then, they should answer questions 5 and 6. Q5. Which macrostate is most probable? A. The third macrostate, with two gas particles on each side, is most probable. Q6. How many more times more likely is it to have two particles on each side than to have four particles on the left side of the box? A. six times more likely |
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Step 6: Next, students should read page 1.15. They should then move to page 1.16 and complete the spreadsheet for a situation involving 12 particles. Students will need to use the equation relating the number of microstates to the number of particles on the left- and right-hand sides to calculate the number of microstates for each macrostate. This equation is given below: Students should enter this equation into the formula bar (gray box) in column C. The factorial sign (!) is found in the Symbols menu (/k). After completing the spreadsheet, students should answer questions 7–9. Q7. Which macrostate is most probable? A. The seventh macrostate, with six gas particles on each side, is the most probable. Q8. How many more times more likely is it to have 6 particles on each side than to have all 12 particles on the left side of the box? A. 924 times more likely Q9. Make a general statement about the most likely macrostate for a two-section box containing n particles, where n is an even number. A. The most likely macrostate is one in which there are  particles on each side of the box. |
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| Part 2 – Exponential growth in number of microstates in the most probable macrostate | |
| Step 1: Next, students should read page 1.19 and examine the data on page 1.20. Page 1.20 is a Lists & Spreadsheet application containing data on the number of microstates in the most probable macrostate (wnum) for systems containing different numbers of particles (num). Encourage students to hypothesize about the relationship between wnum and num. |
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| Step 2: Next, students should plot wnum vs. num on the Data & Statistics application on page 1.21. They should attempt to fit various regression lines to the data, using the Regression tool (Menu > Actions > Regression). Encourage students to predict which regression line will produce the best fit, and then test their predictions. After they have found the best-fit line, students should answer questions 10–11. Q10. What type of relationship is there between wnum and num? A. The relationship is closest to an exponential curve. Q11. Write the equation relating the number of particles in the system to the number of microstates in the most probable macrostate. A. wnum = (0.51)1.88)num |
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| Part 3 – Entropy and the second law of thermodynamics | |
| Step 1: Next, students should read page 1.23, which gives a formal definition of entropy. They should then answer questions 12–15. Q12. What is the entropy of the macrostate with all the particles on the left side of the box? A. Zero; if students have trouble grasping this concept, remind them that there is only one microstate corresponding to this macrostate, and that the natural logarithm of 1 is 0. Q13. Is it possible for all the particles to spontaneously move to the same side of the box? A. Yes. Q14. Is it likely that all the particles would spontaneously move to the same side of the box? A. No. Q15. Which macrostate corresponds to the state with the highest entropy? A. The most probable macrostate has the greatest entropy. |