Mean Free Path

by

Betty Roombos


This website is hosted by the Illinois Institute of Technology


This deals with finding the size of an object using an indirect method. 

 

    On a sheet of graph paper, 50 units on a side, you will locate between 40 and 100 molecules.   To do this, use the supplied table of random numbers.  Use the first number as the x coordinate and the second number as the y coordinate.  Start anywhere in the table and then continue in a regular sequence.  Once you have your points chosen, shade in the square above and to the right of the point.  See the diagram below.

 

Rules:

  1. The particle can travel only along lines of the graph paper, up or down, left or right.  They start at some point (chosen at random) on the left-hand edge of the paper.  The particle initially moves horizontally from the starting point until it collides with a shaded square or another edge of the paper.

  2. If the particle strikes the upper left-hand corner of a target square, it is diverted upward through a right angle.  If it should strike a lower left-hand corner it is diverted downward, again through ninety degrees.

  3. When the path of the particle meets an edge of the graph paper, the particle is not reflected directly back.  (Such a reversal of path would make the particle retrace its previous paths.)  Rather it moves two spaces to the right along the boundary edge before reversing its direction.

  4. There is an exception to rule 3.  Whenever the particle strikes the edge so near a corner that there isn't room for it to move two spaces to the right without meeting another edge of the paper, it moves two spaces to the left along the edge.

  5. Occasionally two target molecules may occupy adjacent squares and the particle may hit touching corners of the two target molecules at the same time.  The rule is that this counts as two hits and the particle goes straight through without changing direction.

The diagram below illustrates these rules.

Finding the mean free path.

    Using the above rules, trace the path of the particle as it bounces about among the random array of target squares.  Count the number of collisions with targets.  Follow the path of the particle until you get 51 hits with the target squares  (collisions with the edge do not count).  Next, record the 50 lengths of the paths of the particles between collisions.  Distances to and from an edge should be included, but distances along the edge (the two spaces introduced to avoid backtracking) should not be included.  These 50 lengths are the free paths of the particle.  You may measure these lengths in centimeters, or just count the number of squares.  Total them and divide by 50 to obtain the mean free path "L". 

    You may use the mean free path to estimate the width "d" of each square using the following formula.  "A" is the area of the grid and "N" is the number of target squares.

Summing up.

  1. Write down the value of "L" that you obtained from this activity.

  2. Using the equation above, calculate the width "d" of a square.  (This is either measured in centimeters or squares.)

  3. Compare your calculated value of the width with the measured value.

Reference:

The Project Physics Course Handbook   ISBN 0-03-084801-6

 

Table of Random Numbers 0-49

49,30 44,09 34,21 01,15 16,31 46,35 47,14
29,47 31,48 46,39 06,04 08,29 08,17 03,43
17,21 04,31 05,07 35,35 24,48 01,29 37,30
10,35 42,08 26,27 25,07 10,19 43,34 22,22
15,46 28,29 31,16 23,28 42,18 39,27 40,32
             
44,46 02,45 19,29 25,08 37,14 22,38 02,02
18,32 16,29 29,45 45,39 14,09 22,17 37,44
02,27 25,44 35,13 46,47 22,17 11,46 08,42
13,24 08,11 45,13 31,33 37,30 06,33 33,15
31,33 03,29 45,18 06,07 42,05 39,34 15,10
             
37,12 42,17 47,30 11,28 10,18 20,20 36,33
27,10 01,11 46,11 01,20 43,17 18,41 30,31
27,20 40,20 06,28 33,47 37,34 12,39 35,39
01,01 37,33 34,10 16,40 49,20 13,41 49,25
13,33 43,07 14,01 02,33 15,00 38,35 07,38
             
33,40 48,46 12,06 09,47 06,45 21,35 06,32
26,17 23,10 48,40 17,11 16,18 01,01 47,33
16,36 05,11 30,19 15,06 40,07 02,00 33,07
46,30 04,23 23,25 24,45 42,38 01,31 35,48
01,27 15,20 29,29 06,39 04,30 13,21 08,25
             
45,17 35,43 10,39 30,40 24,46 12,10 43,14
25,31 31,42 22,16 27,25 06,35 21,18 06,18
39,03 04,39 37,39 22,43 49,41 19,22 32,24
36,26 25,24 12,16 44,43 43,06 02,11 37,48
11,43 03,17 13,42 32,20 34,36 41,07 14,18
             
37,18 33,15 01,43 35,27 39,06 17,46 18,09
19,10 26,22 02,27 43,34 04,29 08,34 12,44
39,12 46,23 22,20 28,25 20,03 34,07 30,38
25,41 37,21 43,30 13,21 13,40 35,35 43,19
36,45 20,33 04,03 43,24 23,13 23,28 16,21
             
02,15 20,23 48,49 06,27 09,44 21,18 20,28
27,47 20,06 34,22 08,20 08,03 27,18 01,48
42,23 47,26 23,17 31,15 38,06 18,20 45,06
24,46 34,15 09,08 09,06 17,30 05,03 03,34
29,28 18,05 18,41 29,27 24,47 05,31 22,00
             
15,01 32,35 18,40 10,40 47,49 09,48 03,43
41,45 09,44 32,26 26,23 19,48 32,02 16,42
42,04 03,16 25,34 42,30 37,23 35,36 01,20
17,40 03,32 06,06 05,24 01,04 46,43 24,34
15,35 36,09 25,19 18,01 21,27 45,41 11,39
             
38,32 13,49 05,39 00,05 06,10 28,27 10,08
04,24 14,21 33,47 06,26 41,13 08,34 04,15
12,31 42,04 26,37 33,03 34,09 28,04 23,36
14,10 45,07 28,06 03,44 38,16 24,26 14,07
03,21 13,15 18,29 10,17 21,09 43,14 27,02

 

This is a blank 50 x 50 sheet.

 

This is a sample set of data using 40 squares.

I just counted squares, so I expect to get a value of 1 unit for "d".

 

The red dot is the starting point.

The red square indicates a point where the line moved left along the edge to avoid repeating the same path.

This is a table of the 51 path lengths.

# L   # L   # L   # L   # L
1 37   12 21   23 65   34 24   45 12
2 32   13 24   24 73   35 51   46 26
3 21   14 16   25 1   36 19   47 27
4 10   15 5   26 26   37 3   48 9
5 64   16 10   27 33   38 13   49 2
6 13   17 99   28 32   39 2   50 36
7 6   18 10   29 17   40 10   51 7
8 33   19 43   30 47   41 4      
9 2   20 9   31 22   42 10      
10 24   21 30   32 28   43 17    

 

11 77   22 0   33 30   44 17      

The total path length = 1249 units

L = 1249/50

L = 24.98 units

d = 1.25 units

If you want to print this out, it's best to copy this into a word processor and set up the pages to look the same as the web page.

 


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