To investigate how motion of a dye droplet is affected by the temperature of the water medium.

AIM:

PREDICTION:

I predict that there will be a close directly proportional relationship between diffusion rate and temperature. I would analyze the data I get graphically and hopefully the graphs will show a clear relationship and I would try to present it mathematically.

DAY1-3:

The main objective of today is to set up my apparatus to measure the rate of diffusion of dye along a column of water and get some preliminary results for the experiment. The dye I am using is Toluidine blue 1% which is a common biological dye. The set up I have devised is to clamp a circular glass tube vertically and mark the tube in every 10 cm interval starting from 0 to 100cm. This is the set up of my experiment:

First of all I have to find the best way to measure the diffusion rate of the dye. The volume of dye added should be enough to give a conspicuous colour so it can be traced easily. My first attempt is dropping in 1cm3 of the dye into a column of water at 21.5 degrees and measuring the time for it to reach certain distances (10 cm intervals) until it has traveled 100cm. This quantity of dye gives me a visible colour which can be seen easily and I have obtained 10 readings successfully but the results are not satisfactory. Because of the big volume of dye, the dye spreads into droplets and diffuses separately and it is difficult to trace the overall diffusion distance. Moreover when the dye spreads out, it forms droplets of different concentration and each diffusing in its own rate. This will cause error to the measurements and the error is much more significant at higher temperature. I refined the experiment by adding a smaller quantity of dye into the solution to prevent the spreading of the dye; however this shortens the distance the dye can be traced until it becomes invisible to naked eye. I therefore decreased the measuring distance to 30cm with measuring interval of 2cm. The results are more reasonable and the technique is much more practical. I carried out a series of experiments and obtained several sets of data by varying the temperature of water. This is the table of results I have got:

Although this was only a preliminary run, it could give me a brief idea of the range of results I am looking for. All graphs have a steep start and this is because the dye is dropped into the water column at a small distance above the water surface and the drop of dye will reach the water with an initial velocity. It takes some time for the dye to settle down and start diffusing down the water column. Ignoring the first few points of the graphs, we can see there is a general increase in all the graphs. As the temperature increases, the gradient of the graph increases. This is because the diffusion has a direct relationship with the temperature of the molecules and the higher the temperature the faster molecules can diffuse across a medium. However the graphs at 40, 45, 50oC are very close together which contradicts with the relationship among the results of graph at 23, 35, 40oC. This suggests that there is something else which overrides the effect of the diffusion and I believe that is the convection current in the water column which affects the results. Convection current is created when warmer water rising through cooler water creating turbulence and moreover diameter of the tube is far too wide compared to the surface area of the diffusing dye droplet; these two factors will lead to a convection current which will affect the flow of the dye especially at high temperature. Because of this I have decided to use a much thinner glass tube to minimize the current in the water column.

DAY 4:

A series of experiments have been carried out with the improved setup. A tube with a 9mm diameter is now used instead of the 3cm diameter one and more accurate results were obtained as the effect of convection current in the water column is greatly reduced. These are the results I got at 20oC:

The measurements are fairly accurate and fluctuate slightly in the beginning due to the dropping of the dye into the water surface. However if we look at the gradients of the graphs in the later part are very consistent, this suggests that the dye is experiencing the same magnitude of force which is likely to be caused by the same factor as well. The velocity of the dye decreases as it travels further. This can be explained by the decrease in concentration of the dye. As the dye travels along, the dye molecules disperse around and the concentration slowly decreases. As the concentration of dye molecules is lower and the bombardment of dye and water molecules will decrease and the overall diffusion rate will decrease as well. Another factor is the drag force experienced by the dye droplet as it travels in the capillary tube. The drag force increases with the velocity of the dye drop and this will decrease the velocity more when it is traveling faster.

DAY5-6:

I conducted the experiment in higher temperatures and got a series of data. I first adjusted the temperature I needed by mixing boiling water and bench distilled water and filled the capillary tube with the water of certain temperatures. I dropped the dye into the water column and started the timer at the same time. These are the results I measured:

This is a preview of the whole essay

DAY5-6:

All the traces are quite reasonable except the second trace which at 40oC. When the dye droplet reached 11cm, it became too dispersed that it is untraceable. I believe this is caused by unsettled current in the capillary tube as I started the experiment right after I poured the water into the capillary tube. To improve my next measurement I decided to leave a small period time between the filling of capillary tube and adding of dye, this will give the time for the water to settle down slightly to avoid this fluctuation and inaccuracy in the measurements. These are the graphs plotted with the data points:

All the graphs show a general increase in displacement as time goes on. The gradient of the graphs decreases with time and the higher the temperature, the greater its initial gradient is. The change in gradient in the graph at 50oC is greater than the ones at lower temperatures. The difference in initial gradients is much greater than the final gradient and from the graphs I suspect the final velocity of the dye may be the same in all cases, which means varying the temperature will only affect the initial motion of a dye in a vertical column of water. This hypothesis can not be concluded until more data are collected and show consistency.

DAY 7-9:

More repeats were done at the temperatures where data points show inconsistency or inaccuracy. In the previous experiment, I wasn’t able to get a complete trace of the dye at 40oC because of the turbulence in the capillary tube. These are the results I have got:

The repeats give a better confirmation of the data and I have taken the average of the three sets of data and plotted a graph against the data points. The data are more consistent and the fluctuations in the graph have greatly reduced. It has been observed that there is a consistent fluctuation in the data points when the dye reaches the depth of 10-13cm. This error is present in all experiments at different temperatures and the same capillary tube has been used in all the experiments, so I believe this is a systematic error caused by the uneven inner surface of the capillary tube around that area. From the graphs I have plotted in day 5-6, we can see the velocity of the dye decreases suddenly, which causes the fluctuation in the curves, when the dye is traveling around half way through the 20 cm distance. This shows that at that area, the resistance force opposing the motion of the dye is higher and builds up a pressure which slows down the diffusion rate. The increase in resistance may be cause by the narrowing of the inner surface of the capillary tube and this may be the cause of the error. In order to prove that the error is not due to temperature, I did another trace at 35oC and these are the results I have got:

The fluctuation is present in this trace even though the temperature of the solution is lowered. This shows that the systematic error is due to the capillary tube itself.

DAY 10-14:

More repeats of the experiments have been done in order to improve the accuracy of the results.

300C:

50oC

Analysis:

The colour of the blue dye is able to spread along the capillary tube of water is because of the mechanism of diffusion. Diffusion is a process by which molecules intermingle as a result of their random motions. We can consider a simple case where there are two containers with particle A and B. (1) In the beginning, there is a partition between both particles. The particles are in constant motion and collide with the partition frequently. (2) When the boundary between the two containers is removed, the particles are free to travel to both compartments. They will eventually mix with each other uniformly and this is a completely random process. Diffusion happens because of the high molecular velocities of the particles which are due to their internal energy. Internal energy is the energy associated with random movement of particles and it can be classified into potential and kinetic energies. Even in a cup of still water, all the water molecules are traveling at hundreds of meter per second.

The second law of thermodynamics state that the total entropy in the universe can never decrease and this means system can only get more random with time. Entropy is the measure of how many ways molecules can be arranged in a system without altering the macroscopic properties. A good example of this is when a gas is compressed to half of its volume in a box and then released; it will attain the highest microscopic arrangement when all the gas molecules are spread evenly around the box. Diffusion of dye is a common case of second thermodynamics. When the dye is mixed with water, the dye molecules will be able to attain a lower energy state if they intermingle with the water molecules, thus increase the entropy.

Because of difference in masses, different types of molecules have a different average kinetic energy at the same temperature. Since the diffusion rate is greatly depends on the average velocity of the molecules and the relation among the diffusion rate, temperature and the molecular mass can be defined by the following equation:

Diffusion rate = K (T/m)1/2

Where K is a constant depending on the geometric factors in the diffusion which includes the cross section of the diffusion. T is the absolute temperature of the substance and m is the molecular weight of the particles. We can see that given that my dye and apparatus setup are constant in every experiment, but only the temperature is varying, the resulting diffusion rate should all satisfy the equation and give the same value for constant K. Because diffusion rate which depends on the average velocity is directly proportional to the √T, so if the temperature is increased by 4 times, the diffusion rate will double. Diffusion is dependent on temperature because an increase of temperature can lead to an increase in the internal energy as well. In this particular system when the water is heated by 10oC, the water molecules will gain both potential and kinetic energies which allow them to bombard each other in an even greater velocity. Potential energy is associated with the intermolecular forces which hold the molecules. There is hydrogen bond in water and if this bonding is overcome by an input of energy, the molecules can move much more freely. This will lead to a faster random mixing process of the dye and water molecules, thus increases the diffusion rate. Another factor which will affect the diffusion rate is the concentration gradient. In any system without a barrier, substances will diffuse from a region of higher concentration to the region of lower concentration. Let’s say there are two compartments with 12 molecules one on side and 2 on the other. Each particle has the same kinetic energy and is traveling at the same velocity. Assuming the particles are traveling in a random order, initially there will be a higher probability for the molecules to move to the right than to the left because the particles are concentrated on the left. However if this process continues the rate of which the particles move to the right will gradually decreases as the number of particles in the left is still higher but the difference is smaller; the probability that a particle traveling to the left is getting closer to the probability of particles traveling to the right. In the end the particles will spread evenly and there isn’t a concentration gradient in the system. Although the particles will still be constantly traveling to the right or left, the system is in a dynamic equilibrium which means in macroscopic view it is in a stable state. Because diffusion is a random process, the greater the concentration gradient is, the more likely that a particle will travel from the region of high concentration to low concentration. From the results I have got, I could obtain the velocity of the dye at certain displacement through simple calculations. These are the graph plotted with the velocity and displacement of the dye at different temperatures.

The best fit line was drawn by the Microsoft excel and the equations are written next to the graph. All the graphs show an exponential decay of velocity of the dye as the displacement and extreme outliers were removed to give the best trend line. The trend lines are in the form of an exponential decay equation.

v = v0e-us

Where vo is the initial velocity of the dye when it first reaches the water surface. u is the decay constant which shows the rate of which the velocity is decreasing. s is the displacement of the dye after it has reached the water surface. v is the instantaneous velocity when the dye has traveled s cm.

The general trend we can see from the graph is the velocity of the dye decreases as it travels further and this decrease is exponential. From my observation during the experiment, I discovered that when a dye travels across the water column, it will form a comet-like droplet with the colour concentrated at the tip of the dye. It loses concentration to the track it passes as it travels along the column and the tip of the droplet gradually loses its colour concentration. The colour concentration loss is much faster in the case of the 50oC runs. This may be due to the higher density of the dye compared to the water so the dye molecules tend to concentrate at the bottom of the droplet as it travels along. The table below shows the relation between temperature and the decay constant of the velocity.

We can see that the velocity of the dye droplet decreases slower as the dye travels further and this means it will reach a slow speed most quickly at 50oC, however the velocity of the dye at 20oC decreases in a much slower pace. If we conduct the experiment in an ideal condition where there isn’t any effect of eddy current and all the variables except the temperature is kept constant, the dye will start with exactly the same velocity. If my results are correct, the dye droplet at 20oC will travel faster across the distance of 20 cm than the 50 cm one first. This may slightly contradict with the law of diffusion as diffusion increases as the temperature increases and we should expect the dye at 50oC should be traveling the fastest. What I suggest is that the concentration of dye at the tip of the droplet may be another factor affecting the velocity the dye travels. As we have mentioned before, concentration gradient also has an effect on the diffusion rate so the greater the concentration gradient, the faster the diffusion rate it will be. The tip of the droplet is the point where the concentration gradient is the greatest as it is where pure water molecules meet pure dye molecules. They interact readily around that region through bombardments and this leads to diffusion. It is due to this local diffusion, the dye droplet moves forward.

For relatively lower temperature like 20oC, the overall diffusion rate is slower so the dye molecules are not spread out that quickly and they remain together at the bottom of the droplet due to its higher density as the dye travels. This however will retain the concentration gradient at the tip of the droplet where the water molecules interact with the dye, so there will be a fast local diffusion occurring in that region and dye moves forward. For higher temperature like 50oC, the overall diffusion rate is faster so the dye molecules are dispersed much faster and the concentration at the tip of the droplet is much lesser. This will lead to a gentler concentration gradient around that region. Although the molecules have a higher kinetic energy and are traveling faster, the rate of which the dye travels down the water column is slower than the other runs. This may suggest the concentration gradient of the bottom of the droplet may have a stronger effect than the overall temperature in this case. Therefore I believe the result doesn’t contradict with the law of diffusion. What I was measuring in this experiment is how long does it take for the dye droplet to travel down a 20cm water column, instead of the diffusion rate of the dye in the water column and in other words I am measuring the local diffusion rate of the dye at the bottom tip of the droplet. I can think of two major factors controlling this local diffusion rate. The first one temperature as it affects the overall diffusion rate and the second one is the local concentration gradient at the tip of droplet which is also affected by the temperature as well. It is difficult to make an accurate prediction on the trend of decay constant of the velocity of the dye at lower or higher temperatures because there aren’t enough data points and the data points are only in a small range. The initial velocities at different temperatures do not seem to show any correlation or trend. This is the table of the initial value of the velocities at different temperatures.

Initial velocity shows the speed of the dye droplet when it first touches the water surface. It solely depends on the height which the dye droplet was dropped and it is independent to the temperature. However it is difficult to drop the dye at the same height in all the experiment and I think this is how the deviation in the initial velocities is resulted.

ACCURACY:

The accuracy of the ruler is up to 0.1cm. The thermometer readings have an accuracy of 0.1oC and the stop watch can time up to 0.01s even though I was only taking measurements up to 1s accuracy.

SAFETY PRECAUTION:

All glassware which includes the capillary tube and beakers should be handled in care. It is also important not to overfill the water kettle as this may lead to spillage of boiling water. Toluidine blue, the stain I used in the experiment, is a biological stain. It stains organic tissues quite readily and it is quite difficult to clean it off. It is also carcinogenic. Therefore plastic gloves should be worn to avoid direct contact with the stain. Safety spectacles should be worn to avoid the splashing of dye into the eyes.

EVALUATION:

Although the results I have got are quite reasonable, there are still some fluctuations and error in the data points. For example it takes some time for the dye droplet to settle down after its impact with the water surface. There are several improvements we can do to enhance the setup to provide more accurate results. First of all I think I should submerge the whole capillary tube into a water bath so as to ensure the temperature is constant through out the whole experiment. We should inject the dye into the capillary tube with a micro-syringe instead of a dropper because this can make sure we are using exactly the same volume of dye in every case. Although I was using the same dropper in all the experiment, it was observed that the size of dye droplet from the dropper could be varied so the volume of dye is only roughly constant. The height of which the dye is dropped should be kept the same so as to keep the initial velocity the same. It is also a good technique to minimize the height of the dropping because this can minimize the impact force of the dye droplet on the water surface and the droplet will be able to settle down more quickly. An even thinner tube should be used so the eddy current can be further decreased. In my experiment I have used the smallest tube as possible however there was still a certain extent of uneven diffusion happening. Using a capillary tube with a smaller diameter can certainly improve the accuracy of the results. If the investigation is continued, I would make more measurements at a wider range of temperatures and more repeats will be done for each temperature. I would also introduce a data logging system to trace the displacement of the dye with a colorimeter instead of using naked eye to do the measurements. Because when the dye has traveled for a certain distance, its colour will get so diluted that it is difficult to trace by eye so a colorimeter will give us a more accurate trace of the motion of the dye. I would also use a different dye to investigate the effect of density on the motion of the dye droplet.

CONCLUSIONS:

I have successfully completed my aim of investigating how a dye travels along a water column and discovered the effect of temperature on its motion. The results have proved that my prediction that the dye will travel faster if the temperature is increased doesn’t seem to be correct. I have discovered the velocity of the Toluidine blue dye droplet at higher temperature, however, decreases in a faster rate as the dye travels along the column than the lower temperature ones. I have developed a hypothesis trying to explain this phenomenon and I think it is the concentration gradient at the bottom of the droplet and the temperature which are counteracting each other; if the temperature is higher, the more dispersed the dye molecules will be and the less concentrated the bottom of the dye droplet will be. In order to consolidate my hypothesis I believe more experiments have to be conducted to obtain a wider range of data and more repeats. I also suspect that this will only be the case around a certain range of temperature. So if the temperature exceeds this range, it may not necessarily follow this trend as the temperature may completely override the local concentration gradient effect.

BIBLIOGRAPHY:

Books:

Advanced Physics---Steve Adams and Jonathan Allday

Advanced Physics A2

Internet: