CHEMICAL STATISTICS

Stacy Blaine
(Partner: Kristen Fleming)

Abstract

A die was thrown 30 times for each of 5 sets of polymers for a total of 150 tosses for each individual to represent the statistical synthesis of a polymer. The numbers on the die represented six initiator molecules that might equally react and began polymerization, and the number of times each number was rolled for each of the five polymers represented the number of monomers in that chain. Thus with 150 tosses, 30 polymers were created with an average chain length of 5. This type of polymerization was simulated using the Poisson Distribution Law. The distribution of individual chains making up the polymer, the number average molecular weight (Mn), and the weight average molecular weight (Mw) were determined for both individual results and combined class results. The average molecular weights for the individual results were, Mn=200.0 and Mw=250.7, and those for the class were, Mn=199.6 and Mw=232.9. Plots were made of pk, the probability of finding a chain with k monomers in a total number of N chains, as a function of k for both individual and class results and compared with a theoretical curve. The variance of values for the individual results was 2 = 6.331, and that for the class results was 2 = 4.154. A computer was used to randomly create polymers of varying step lengths with the same number of walks and the same step length in order to determine the root mean square end-to-end distance for a polymer chain as a function of the number of steps as well as the general configuration a polymer takes with increasing numbers of monomers in a chain.





I. Introduction

The purpose of this experiment is to synthesize an ideal, typical polymer statistically, determine the both the number average, Mn, and weight average, Mw, molecular weights, the root mean square end-to-end distance, and the distribution of individual chains. The distributions for each individual as well as the class as a whole will be plotted and compared to a theoretically predicted distribution. A plot of the root mean square (rms) end-to-end distance as a function of individual chain length will be made. Polymerization may be illustrated statistically by a method allowing monomers in a given polymerization reaction to react randomly with a set number of initiator molecules that begin the process of polymer growth. Propagation of the reaction continues as additional monomers add to previously initiated polymer chains and termination of the reaction occurs when all the monomers have been used up. This addition type polymerization may be simulated using the Poisson Distribution Law, with the theoretical probability of finding a chain with k monomers in a polymer containing chains with an average of monomers given by the equation:
p(k; ) = [e- ][ k/k!] (1)
For the purpose of the experimental results the equation for the probability of finding a chain with k monomers is given by, pk = Nk/N (2) where Nk is equal to the number of chains containing k monomers and N is equal to the total number of chains. There are several equations that may be used to find the average molecular weight of a polymer consisting of chains with differing numbers of monomers. The number average molecular weight, Mn, is given:
Mn = Mo [ kpk] (3)
where Mo is the monomer molecular weight, k is the number of monomers in a given chain, and pk is the probability of finding a chain with k monomers in a polymer consisting of a total of N chains. In this case, Mo equals 40. The weight average molecular weight, Mw, is given by:
Mw = WkMk where Wk = [MokNk] / [ MokNk] and Mk = Mok (4)
The variance of individual chains of a polymer can be found using the equation:
var(k) = 2 = pk(k-k)2 (5)
where k is the average value of k. The standard deviation, , is the square root of the variance and will have particular significance in determining polymer configurations. The purpose of the second part of this experiment is to determine the root mean square end-to-end distance for a polymer chain as a function of the respective number of bonds in a certain direction as well as to find the general configuration a polymer takes with increasing numbers of monomers in a chain. A computer simulation was used to randomly create polymers with the same number of walks (25) and the same step length (5) but varying amounts of steps (monomer numbers). The computer was used to calculate the root mean square end-to-end distance (rms) for the experimental runs and the theoretical rms for each number of steps used.

II. Experimental Method

The experimental method was similar to that in the special instructions handout titled "Chemical Statistics". For the first part of the experiment the sides of a die were used to represent six reacting molecules. By each individual in the class, the die was randomly thrown 30 times for each of five sets of polymers labeled A through E. Each time the die was thrown and a number appeared signifying the molecule reacted with that number initiator a mark was placed in the appropriate box (see Table 1). This occurred exactly 30 times for each polymer set for a total of 150 throws creating 30 polymers with an average chain length of 5 monomers ( =5). The Poisson Distribution was used to find the theoretical probability of finding a polymer with k monomers with an average of 5 monomers, assuming each reaction was equally possible (eqn. 1). Equation (2) was used to calculate this probability, pk, for each k value for the experimental results for each individual with N=30 (Table 2). Results were combined for the class, giving N=300, and the experimental pk was determined for each k value for the class using equation (2) (Table 3). The number average molecular weight, Mn, was determined using equation (3) for both the individual and class results (Tables 2&3). The weight average molecular weight, Mw, was calculated using equation (4) for both the individual and class results (Table 2&3). In this experiment, Mo = 40. The variance was calculated using equation (5) for both the individual and class results. A plot of pk as a function of k was made for both individual and class results and compared to the theoretical curve (Graph 1). For the second part a computer program was used to randomly generate polymers and calculate the root mean square end-to-end distance as well as show the general configuration of the polymers as the number of monomers in a specific chain increased. The step length was kept at 5 while the number of steps was increased from 10, 25, 40, 50, 75, 100 with the number of walks kept at 25. The computer also calculated the theoretical rms for each number of steps. The results can be found in Table 4. A plot of the rms as a function of the number of steps was made and compared to the theoretical curve.


III. Results

**data tables not shown

Number Average Molecular Weight

Individual: N = 30
Mo = 40

Mn = Mo x ( kpk)
= 40 x (4.997)
Mn = 199.988 = 200.0

Class: N = 300
Mo = 40

Mn = Mo x ( kpk)
= 40 x (4.989)
Mn = 199.560 = 199.6

Weight Average Molecular Weight

Individual: Mo = 40

Mw = [(MokNk)/( MokNk)] x Mok
Mw = 250.7

Class: Mo = 40

Mw = [(MokNk)/( MokNk)] x Mok
Mw = 232.9

Variance & Standard Deviation

Individual: k= 55/11 = 5
2 = pk(k-k)2
= 0.0333(0-5)2 + 0.1000(2-5)2 + 0.2333(3-5)2 + 0.1000(4-5)2 + 0.0667(6-5)2 + 0.1333(7-5)2
+ 0.1333(9-5)2 + 0.0333(10-5)2
= 0.8325 + 0.9000 + 0.9332 + 0.1000 + 0.0667 + 0.5332 + 2.1328 + 0.8325
2 = 6.331

Variance = ( 2 * 2) = 2.516


Class: k= 55/11 = 5

2 = pk(k-k)2
= 0.0133(0-5)2 + 0.0267(1-5)2 + 0.0700(2-5)2 + 0.1233(3-5)2 + 0.1600(4-5)2 + 0.1733(6-5)2 +
0.1133(7-5)2 + 0.0333(8-5)2 + 0.0533(9-5)2 + 0.0133(10-5)2
= 0.3325 + 0.4272 + 0.6300 + 0.4932 + 0.1600 + 0.1733 + 0.4532 + 0.2997 + 0.8528 + 0.3325
2 = 4.154

Variance= ( 2 * 2) = 2.038


IV. Discussion

The class was successfully able to statistically simulate polymer growth using the method described in the experimental methods section of this paper. The Poisson Distribution Law was used to the theoretical probability, p(k;5), of finding a chain with k monomers in a polymer with chains averaging 5 monomers. Equation (2) was used to find the experimental pk for both individual and combined class results. A plot of pk as a function of k was made for both individual and class results and compared with the theoretical curve (Graph 1). This plot shows that the curve for the class is much closer in distribution to the theoretical curve than the individual curve. The individual curve deviates quite significantly from the theorectical curve at some points and this shows the randomness of the experiment. This was expected and illustrates the randomness of polymer growth. As also expected, the class curve is a closer match to the theoretical curve because the number of chains increased from 30 to 300. In other words the greater the number of chains the closer in distribution the experimental plot will be to the theoretical plot. This plot also showed that the Poisson Distribution is a valid model for polymer growth since the class curve was close in distribution to the theoretical curve. The number average molecular weight, Mn, was slightly higher for my individual results (Mn=200.0) than that for the class results (Mn=199.6). The weight average molecular weight for my individual results (Mw=250.7) was also higher than that for the class results (Mn=232.9). In both cases the weight average molecular weight was greater than the number average molecular weight as expected. The variance and standard deviation for the individual results ( 2=6.331, V=2.516) were greater than those of the class results ( 2=4.154, V=2.038), which is apparent from the plot of pk versus k where the spread of pk values and their deviations from the theoretical curve are much larger for the individual curve. This is not surprising since the data seems to support that as the number of chains evaluated increases the closer the experimental curve comes to the theorectical curve. In part two of this experiment a plot was made of root mean square end-to-end distance (rms) as a function of the number of steps, or monomer subunits. The data points for both the experimental curve and the theorectical curve were obtained from a computer simulation that randomly created several polymers using varying numbers of steps as explained in the experimental method. The plot shows that curve of the randomly generated polymers is very close in distribution to the theoretical curve and deviates slightly in a few areas. This suggests that the statistical method used to randomly create these polymers is a valid for determining the general configuration that a polymer takes as the number of monomers in a specific chain
increases.

V. Error Analysis

The results of this experiment are valid for whatever differences that may occur from individual sample to sample as only a large number of samples will begin to approach the theoretical statistical laws. The only other errors that could have occured are if the die had some flaw in it that would result in some numbers being thrown more often than can be attributed to random chance.. For instance, if the weight of the die was not evenly dispersed, meaning one side was slightly heavier than the other, certain numbers might appear more often than others and the experiment would not be entirely random. Another error might be not hitting the die against that side of the paper cup and actually shaking it up.