Simulation The End Position Of The Molecule example essay topic

Lab Report #1: Diffusion Team 7: Christina DiPaulJames Thomas Nam Nguyen Amanda Velez Introduction: The human body undergoes a variety of processes throughout each and every day in order to sustain life. Tasks such as walking, breathing, and digesting what has been eaten are sometimes considered mundane, even taken for granted. One such process that is necessary to maintain life is diffusion. Diffusion is a key factor in moving ions, fuels, and other molecules into and out of the blood. It is one of the most important components in supplying oxygen to the alveoli and removing carbon dioxide. Without diffusion, substances would find it very difficult to pass through membranes and could cause detrimental effects to the human body.

The paradox scientists have drawn is related to glucose molecules and the directions in which the molecules "know" to move. No single molecule should diffuse in any particular fashion, but should diffuse randomly. This report looks at four simulations attempting to solve the problem presented, how do the molecules know which way to diffuse? In order to understand the obtained research, it is necessary to present and identify the key components of Fick's Law of Diffusion: F = -D A dC / dx F = the flow of material across a real or imaginary planeD = the diffusivity of the diffusing molecules (the ease in which the molecule diffuses in the surrounding medium) A = area of the plane C = concentration of the molecules X = distance dC / dx = the concentration gradient There will be four simulations conducted in order to apply Fick's Law and determine if molecules do in fact know which way to diffuse. The first simulation will look at a single molecule in an open area, the second looks at the movement of several molecules, the third looks at molecules diffusing in a box, and the fourth looks at molecules diffusing through a pipe between two boxes. Understanding the movements of the molecules in each of these simulations will allow the problem to be solved.

Materials / Methods: In order to properly experience each of the simulations it is necessary to have access to Microsoft Excel and Mat Lab, a program designed allowing scientists to view data in a visual format. The first simulation, as mentioned, dealt with a single molecule randomly moving in an open area. It is assumed that the molecule will move from a higher concentration to a lower concentration based on the laws of diffusion and that the molecule will move a set distance in one step before heading randomly into its next direction. The molecule will proceed randomly through several hundred steps and countless collisions with a fixed free path length, or the average distance between each collision.

The second simulation demonstrates several molecules randomly moving in a designated area. This simulation is similar to the first in that it is also a random walk with a fixed free path length and random direction after each collision. Ten thousand molecules start at the origin to show this relationship because we are assuming that the molecules have no size, however this would not be physically possible outside of the simulation. Again the assumption is made that the molecules will move from a higher concentration (near the origin) to a lower concentration (away from the origin.) Another simplification is also made in order to allow the molecules to ignore each other and not bounce off one another like they would do in the actual environment. The third simulation shows the diffusion of molecules in a box. This simulation is unlike the others because there is now a restriction placed on the molecules.

Once again a simplification is made in which the collisions do not cause the molecules to lose their energy. The left half of the box is filled with molecules and a partition is placed so the right half has no molecules. The partition will then be removed and the molecules will once again move randomly. The main assumption for this simulation is that the collisions between molecules are perfectly elastic so researchers do not have to deal with the complicated effects of molecules that may have resorted to the edges of the walls or how rough the walls are to begin with. The final simulation demonstrates the diffusion of molecules between two boxes that are connected by a pipe.

The area is therefore reduced in this simulation yet the concentration remains constant. As before, the molecules will move randomly and they will lose no energy when bounced against another molecule. An assumption included in this simulation is in dealing with the pipe, entrance region effects are therefore ignored to allow simplicity once again. Results: The following graphs are representative data collected from: Simulation #!

: Figure 1-Path of the Particle Figure 2-Distance from Origin vs. Time Figure 1 simulates the path of a single molecule through several hundred time steps and collisions. Figure 2 represents the random motion of a molecule from its origin. Simulation #2: Figure 3-Position of Molecules at the End of Simulation Figure 4-Mean Distance From Origin vs. Time Figure 5- Distance of Molecules of Several Groups vs. Time Figure 3 shows the position of several hundred molecules after the simulation completion of the simulation. Figure 4 represents the average distance of the molecules during the simulation with respect to time. Figure 5 shows the maximum distance of the 5%, 10%, 25%, etc. closet molecules with respect to time.

Simulation #3: Figure 6-Box Diffusion: Relative Frequency vs. Position vs. Time Figure 7-Fraction of Molecules in Boxes vs. Time Figure 6 represents a simulation where several hundred molecules are released in the left side of a box, separated from the right side of the box via a divider. Molecules begin to diffuse through the box when the divider is removed. The various colored lines represent the frequency of movement and position with respect to time. Figure 7 shows the fraction of molecules in each cell vs. time. Simulation #4: Figure 8-Pipe Diffusion: Relative Frequency vs. Position vs. Time Figure 9-Pipe Diffusion: Fraction of Molecules in Cell and Pipe vs. Time Figure 10-Pipe Diffusion: Relative Frequency vs. Position vs. Time Figure 11-Pipe Diffusion: Fraction of Molecules in Cell and Pipe vs. Time Figure 12- Pipe Diffusion: Relative Frequency vs. Position vs. Time Figure 13-Pipe Diffusion: Fraction of Molecules in Cell and Pipe vs. Time Figure 14-Pipe Diffusion: Relative Frequency vs. Position vs. Time Figure 15-Pipe Diffusion: Fraction of Molecules in Cell and Pipe vs. Time Figures 8, 10, 12 and 14 are histogram curves of 4 systems similar to the system in simulation 3. The divider is replaced by pipes differing in size, separating the two boxes.

The graphs show the position of the molecules relative to its frequency and time. Figures 9, 11, 13 and 15 are exponential curves showing the fraction of molecules in each box and the pipe with respect to time. As the diameter of the pipe increases we notice the rate of diffusion increases while the amount of time the system takes to stabilize decreases. Discussions: Simulation 1 produces the simulated "drunken path" that a single molecule takes.

In figure 1 titled Path of Particle; we can see that the molecule takes several hundred steps in completely random directions. By running the's simulation repeatedly, we are convinced that that molecule does not have a specified directional path. In each simulation the end position of the molecule varies. If the path it took were not random, then the XY plot on figure 1 would be more ordered and predictable. Figure 2 shows a graphical representation of the varying distances of the molecule from the origin with respect to time. From our analysis of the different simulations, we see there are no visible patterns emerging from the simulation.

In one simulation we " ve run, the molecule returned back to its origin and then farther away. Figure 2 shows a graphical representation of the At the end of simulation 2, shown in figure 3, the cluster of molecules still remained dense in the center. Although the directionality of the "single" molecule is unpredictable, we notice there is a visible trend of the group moving from a higher concentration area to a lower concentration area. In figure 4 we can see a linear plot of the molecules movement away from the origin with respect to time. This suggests that the average movements of the molecules are not random but linear, diffusing outward to a lower concentration. In figure 5 we can see that the molecules do not move much in the beginning but moves faster and farther from the origin afterwards.

We can estimate how fast the boundary of the molecules expands by analyzing figure 4. The radius of the boundary sphere changes with respect to time. Since figure 4 is a linear plot we can calculate this rate of boundary expansion as a function of radius with respect to time. We computed the rise over run value and came up with: (10^0.95 mean distance from origin) / (10^2 time) = 0.089 In simulation 3 we can see a concentrated chamber of molecules on the left side which is released to an empty chamber of same volume on the right side. Figure 6 shows that as time progresses, the molecules move from the higher concentration left chamber to the lower concentration right chamber until equal concentrations are in each chamber. The distribution of molecules can be visually analyzed to be even.

Figure 7 shows 50% of the initial fraction of molecules accumulated space in the second chamber at the end of the simulation. To measure the rapidity of this diffusion towards equilibrium we must calculate the change in concentration with respect to the change in distance. In Simulation 4 we have two chambers connected with a pipe. Like simulation 2, the concentration of molecules across the whole system at equilibrium is equally distributed. If the dimensions of the pipe were small and narrow, only a small percentage of the molecules will be contained in the pipe at equilibrium.

The flow of molecules to the right chamber would decrease. The calculation of the rate of approach to equilibrium in simulation 2 cannot be applied here because we have a pipe of a different volume that alters the flow. As the figure above depicts, the rate of approach to equilibrium increase as the diameter of the pipe increases. However the rate of approach will not increase infinitely but will plateau as some point because the diameter of the pipe can only be so large to connect the boxes.

Accordingly, increasing the length of the pipe will slower the rate of approach to equilibrium because the molecules will have to travel a greater distance.

Bibliography

" Diffusion- Simulation of randomly moving particles" WebCT. Drexel University. April 2005.