Problem 8 In the course of derivi ng the Boltzmann distribution we came upon a v
ID: 551615 • Letter: P
Question
Problem 8 In the course of derivi ng the Boltzmann distribution we came upon a very simple approximation: I would like you to investigate this in Excel. Create a table that has 5 columns and 4 rows. Label the first column p, the second column ni, the third column label with the formula taken from the left side of the. Label the fourth column with the formula taken from the right side of the ~. In the fifth column I want you to calculate the % difference between your calculation of the third and fourth columns. In the second, third, and fourth rows give ni the values 50, 100, and 150. In the second, third, and fourth rows give p the values 5, 5, and 5. Finally, in the third and fourth columns make a formula and calculate the two expressions on either side of the to compare. You will have to use the factorial function, -fact(e5) where "e5" is arbitrary and might contain the value of ni, for instance. You can change the value of p a fair amount but you are much more limited as far as your choice of ni goes since Excel won't calculate a factorial of numbers greater than 170. a. otice the trend. How big do you think ni has to get for the % difference between the left and b. N right sides of the above expression to be within 1% of each other for p-S? Argue as to whether the approximation we used here is va lid. c.Explanation / Answer
when p=5 and ni= 2000 then (p+ni)!/ni! = 3.22 X10^17, nip = 3.2 X10^17
so the % difference goes down below 1%, i.e, 0.62% .
hence approximation here is quite valid.
p ni (p+ni)!/ni! nip % difference 5 50 4.174 X 10 ^8 3.125 X 10^8 25.13 5 100 1.159 X 10^10 1.000 X 10^10 13.72 5 150 8.382 X 10^10 7.594 X 10^10 9.4