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HW #5 - Chemistry 132 Winter 2008 Solutions to Homework No...

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Chemistry 132, Winter 2008 Solutions to Homework No. 5 Problem 1. Given n possible values x i , i = 1 · · · n , of a discrete random variable X , and given a function f ( X ) of the variable X , the average value of f ( X ) is (Levine eq. 15.40) f ( X ) = n i =1 p i f ( x i ) , p i = N i N , N = n i =1 N i , where p i is the probability that X = x i , N i is the number of observations (or samples) for which X = x i , and N is the total number of observations. In this problem, an observa- tion consists of recording the speed and direction of a car passing below the bridge. The random variable is either the speed or the velocity. The velocity is just the speed with a sign that indicates whether the car was traveling due east (positive) or west (negative). Each part of this problem can be solved by specializing the above formula to a particular random variable and a particular function. (a) Here the variable is the velocity component v x along the x -axis, which is taken to point from west to east, and the function to be averaged is f ( X ) = X , v x = 1 N m i =1 N i v x,i . (b) Here the variable is the speed v , which is the magnitude of v x , and the function to be averaged is again f ( X ) = X , v = 1 N m i =1 N i v i . (c) Here the variable is again the speed v but the function to be averaged is f ( X ) = X 2 . Also, the square root of the average is being asked (Levine eq. 15.21) v 2 = 1 N m i =1 N i v 2 i , v rms = v 2 1 / 2 . The actual calculations using the above formulas and the given data can be done in an Excel spreadsheet, as shown in Figure 1. The results are v x = 2 . 82 km / h , v = 86 . 08 km / h , v rms = 86 . 20 km / h . Figure 1: Excel spreadsheet for problem 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 A B C D E F G H i v i / (km/h) dir. n i v x,i / (km/h) n i ×v x,i / (km/h) n i ×v i / (km/h) n i ×v i 2 / (km/h) 2 1 80E 40 80 3200 3200 256000 2 85E 62 85 5270 5270 447950 3 90E 53 90 4770 4770 429300 4 95E 12 95 1140 1140 108300 5 100E 2 100 200 200 20000 6 80W 38 -80 -3040 3040 243200 7 85W 59 -85 -5015 5015 426275 8 90W 60 -90 -5400 5400 486000 9 100W 2 -100 -200 200 20000 N 328 (a) v x / (km/h) 2.820122 (b) v / (km/h) 86.08232 (c) v 2 ½ / (km/h) 86.19718 cell formula F2 =D2*E2 G2 =D2*B2 H2 =D2*B2^2 F12 =SUM(D2:D10) F13 =SUM(F2:F10)/F12 F14 =SUM(G2:G10)/F12 F15 =SQRT(SUM(H2:H10)/F12) Problem 4. See Levine problem 15.35. (a) True. Swapping b with c everywhere in the formula for Z bc (Levine eq. 15.62) gives Z cb on left-hand side, but leaves the expression on the right-hand side unchanged. Therefore Z cb = Z bc .
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