Physics

Abstract: This two part experiment is designed to determine the rate law of the following reaction, 2I-(aq) + H2O2(aq) + 2H+I2(aq) + 2H2O(L), and to then determine if a change in temperature has an effect on that rate of this reaction. It was found that the reaction rate=k[I-]^1[H2O2+]^1, and the experimental activation energy is 60. 62 KJ/mol.

Introduction

The rate of a chemical reaction often depends on reactant concentrations, temperature, and if there’s presence of a catalyst.

The rate of reaction for this experiment can be determined by analyzing the amount of iodine (I2) formed. Two chemical reactions are useful to determining the amount of iodine is produced.

1. I2(aq) + 2S2O32-(aq) 2I-(aq)+S4O62-(aq)
2. I2(aq) + starch

Reaction 2 is used only to determine when the production of iodine is occurring by turning a clear colorless solution to a blue color. Without this reaction it would be very difficult to determine how much iodine is being produced, due to how quickly thiosulfate and iodine react.

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However this reaction does not determine the amount of iodine produced, it only determines when/if iodine is present in solution. Reaction 1 is used to determine how much iodine is produced.

To understand how the rate constant (k) is temperature dependent, another set of data is recorded in week two’s experiment using six trials and three different temperatures(two trials per temperature change). Using the graph of this data we determine the energy required to bend of stretch the reactant molecules to the point where bonds can break or form, and then assemble products (Activation Energy, Ea).

Methods To perform the experiment for week 1, we first prepare two solutions, A and B, as shown in the data. After preparing the mixtures, we mix them together in a flask and carefully observe the solution, while timing, to see how long it takes for the solution to change from clear to blue. We use this method for all 5 trials, and record the time it takes to change color, indicating the reaction has taken place fully. This data is used to find p (trials1-3) and q (trials3-5), to use in our rate law. This experiment concluded that both p and q are first order.

The rate constant average of all five trials is used as just one point on the Arrhenius Plot. In week 2, we perform the experiment to test the relation of temperature to the rate of reaction. We start by again, preparing six solutions.

We prepared two trials/solutions at 0 degrees Celsius, two and 40 degrees Celsius, and two at 30 degrees Celsius. Again, for each trial we mixed solution A with B, and carefully timed the reaction to look for a color change that indicates the reaction is complete. The interpretation of this data indicated out results of whether temperature has an effect on the rate of this reaction.

Results- It is determined that the rate of reaction is dependent on the temperature in which the reaction occurs. The solutions observed at 40 degrees Celsius reacted at a quicker rate, than those at lesser temperatures, in a linear manor.

Calculations Week 1

1. Find the moles of S2O3-2 Take the value from NaS2O3 *(0. 2)/1000 (5)*(0. 2)/1000= 0. 001 mol of S2O32-
2. Find moles of I2 Take S2O32- /2 (0. 001)/2=0. 0005mol
3. Find I2 Mol I2*1000/vol mL (0. 0005)*1000/40)= 0. 000799885 mol
4. Find the rate of change Take (I2)/ (seconds) (0. 000799885)/(585)= 1. 36732×10-6 M/s
5. Find [I-]0 (0. 300 M KI)*(2. 00mL)/( the final volume)=0. 015 M 6.

Find the Ln of [I-]0 Ln(0. 015)=-4. 19970508 7. Find [H2O2]0 Take (0. 10 M H2O2)*(6. 00mL)/ ( final volume)=0. 015 M 8. Ln of [H2O2]0 Ln(0. 015)= -4. 19970508 9. Find the Ln of rate: Ln(2. 13675×10-5)=-10. 753638 10.

The last step for week one calculations is to calculate the average value of k. Rate= k [I-]1[H2O2]. (2. 13675*10-5 ) = k [0. 015] [0. 015] then solve for k. For this trial, k=0. 09497. This is then done for all trials. Then, once all five values of k are found, the average is taken by adding all five values of k and dividing by 5. The experimental k average is 0. 05894M/s.

Calculations Week 2

1. Find amount of I2 moles produced in the main reaction using Volume of Na2SO4 used, stock concentration of Na2SO4 solution, and the Stoichiometry (2mol Na2SO4 to 1 mol I2) for all six trials. Trial 1: (. 005 L Na2SO4)(. 02 moles Na2SO4/1. 0L)(1 mol I2/2 mol Na2SO4)= . 00005 mol I2 Use this method for all six trials
2. Find the reaction rate using moles of I2 produced, measured time in seconds, and Volume of total solution for all six trials Trial 1: (. 00005 mol I2/. 0403L)=(. 00124906 mol/L) /(692seconds)= . 00000181mol/L(s) Use this method for all six trials
3. Find the rate constant using the reaction rate, measured volumes used, stock concentrations, and the rate law of the main reaction. Trial 1: K=(. 00000181MOL/L(s))/((. 01 L H2O2)(. 1 M H2O2)/. 0403L total))((. 3MKI)(. 006LKI)/. 0403L total)=. 00107 Use this method for all six trials
4. To graph, we must calculate Ln(k) and 1/Temp(K) for each individual trial.

Trial 1: Ln(. 00107)=-6. 8401 and 1/T = 1/692sec=-. 00365k^-1 Use calculation method 1-4 for all six trials

To Find the Activation Energy we multiply by the rate constant of 8. 314J/mol(K), which equals -60617. 4 J/mol.

We then convert this value to kilojoules by dividing by 1000, equaling 60. 62 kJ/mol. Analysis uncertainty- Due to the limit of significant figures in stock solutions used, the resulting data is limited in correctness. Also, temperature fluctuations during the experiment by even a half degree would obscure the data of the exact rate constant, k.

One of our R^2 coefficients for the experiment was in fact greater than 0. , and the other slightly less than 0. 9 meaning the one lesser is not considered a good fit. The deviation in goodness of fit may have been due to our data recording. Discussion- Determination of the rate law and activation energy of a chemical reaction requires a few steps. By varying the concentrations of reactants it was determined that the reaction is first order with respect to both [I-] and [H2O2+].

Measuring the reaction rate at multiple temperatures allows calculation of the activation energy of the process, in this case the activation energy of the reaction is found to be 60. 2 kJ/mol. As you have seen through all the previous data, charts and graphs, this exothermic rate of a reaction is dependent on solution concentrations, a catalyst, and temperature.

References

1. Determination of a Rate Law lab document, pages 1-6, Mesa Community College CHM152LL website, www. physci. mc. maricopa. edu/Chemistry/CHM152, accessed 10/9/2012.
2. Temperature Dependence of a Rate Constant lab document, pages
3. Mesa Community College CHM152LL website, www. physci. mc. maricopa. edu/Chemistry/CHM152, accessed 10/9/2012.

Impact Of a Jet

Introduction: Over the years, engineers have found many ways to utilize the force that can be imparted by a jet of fluid on a surface diverting the flow. For example, the pelt on wheel has been used to make flour. Furthermore, the impulse turbine is still used in the first and sometimes in the second stage of the steam turbine. Firemen make use of the kinetic energy stored in a jet to deliver water above the level in the nozzle to extinguish fires in high-rise buildings. Fluid jets are also used in industry for cutting metals and debarring. Many other applications of fluid jets can be cited which reveals their technological importance. This experiment aims at assessing the different forces exerted by the same water jet on a variety of geometrical different plates. The results obtained experimentally are to be compared with the ones inferred from theory by utilizing the applicable versions of the Bernoulli and momentum equations. Objectives: i. To measure the force produced by a jet on flat and curved surfaces. ii. To compare the experimental results with the theoretically calculated values

Procedure:

1. Stand the apparatus on the hydraulic bench, with the drainpipe immediately above the hole leading to the weighing tank, see figure 4. Connect the bench supply hose to the inlet pipe on the apparatus, using a hose-clip to secure the connection.
2. Fit the flat plate to the apparatus. If the cup is fitted, remove it by undoing the retaining screw and lifting it out, complete with the loose cover plate. Take care not to drop the cup in the plastic cylinder.
3. Fit the cover plate over the stem of the flat plate and hold it in the position below the beam. Screw in the retaining screw and tighten it.
4.  Set the weigh-beam to its datum position. First set the jockey weight on the beam so that the datum groove is at zero on the scale, figure.
5. Turn the adjusting nut, above the spring, until the grooves on the tally are in line with the top plate as shown in figure
6. This indicates the datum position to which the beam must be returned, during the experiment, to measure the force produced by the jet.
7. Switch on the bench pump and open the bench supply valve to admit water to the apparatus. Check that the drainpipe is over the hole leading to the weighing tank.
8. Fully open the supply valve and slide the jockey weight along the beam until the tally returns to record the reading on the scale corresponding to the groove on the jockey weight. Measure the flow rate by limiting the collection of 8Kg of water in the bench-weighing bank.
9. Move the jockey weight inwards by 10 to 15cm and reduce the flow rate until the beam is approximately level. Set the beam to exactly the correct position (as indicated by the tally) by moving the jockey weight, and record the scale reading. Measure the flow rate.
10. Repeat step 6 until you have about 6 sets of readings over the range flow. For the last set, the jockey should be set at about 10cm from the zero position. At the lower flow rates, you can reduce the mass of water collected in the weighing tank to 8Kg.
11.  Switch off the bench pump and fit the hemispherical cup to the apparatus using the method in steps 2 and 3. Repeat step 4 to check the datum setting.
12. Repeat steps 5 to 9, but this time move the jockey in steps of about 25cm and take the last set of readings at about 20cm.
13. Switch off the bench pump and record the mass m of the jockey weight, the diameter d of the nozzle, and the distance s of the vanes from the outlet of the nozzle.

Discussion

1. The adjusting nut above the spring until the grooves on the tally are in the line with the top plate as shown in figure 6.
2. Recording the reading on the scale corresponding to the groove on the jockey weight. 3. Starting timer and adding weights when the beam moves to horizontal. Stopping timer when the beam moves to horizontal again.
3. The values of F theoretical (calculated from 4g? x) are close to those found experimentally. So we connect these points with a straight line
4. Also from this graph, we see that the calculated F (4g? ) is equal to the double of mu 2mu
5. It is clear from Fig that the force produced on each of the vanes is proportional to the momentum flow in the jet as it strikes the vane. From the data collected during the experiment, it is found that for different plates of vane used, the force exerted on the plate by the water will be different and it varies from the flat and hemispherical plate. This is supported by the data of the column, distance of jockey from zero position which is the mean of knowing the force needed to balance the force exerted by the water.
6. We were to plot graphs of Force versus delivery of momentum for each plate on the same graph and we found the graphs posses different slope where the values are 2 and 1. 1 for hemispherical and flat plate respectively. We were able to plot the two plates on the same graph and although the relative slope is correct where hemispherical has the greater slope followed by flat plate, the calculation of the slope will not be correct because the value of the x-axis is the same for all two graphs. So in order to obtain the correct value of slopes, the individual plotting of the graph has been plotted and the slope has been calculated.
7. When the water from the nozzle strikes the plate, it has the same initial velocity for the two plates but the velocity changes due to the obstruction by the plate and it will be different for each plate due to the geometrical effect. The geometry of the hemispherical plate minimizes the obstruction of the plate so the water will flow more freely relative to that of the flat plate.

So, for the same flow rate, the hemispherical has a relatively higher final velocity than a flat plate.

Percentage of error of the experiment:

Accuracy = (mum-4g? X /4g? X) *100%

For flat plate: (31. 20-1. 96/1. 96)=10. 2% (2. 10-1. 96/1. 96)=7. 14% (1. 73-1. 57/1. 57)=10. 2% (1. 35-1. 18/1. 18)=14. 4% (0. 9-0. 78/0. 78)=15. 4%

Factor: Parallax error, during adjusting the level gauge to point, the Water valve was not completely close and the Press the stopwatch start button late.

For hemispherical cup: (4. 74-4. 1/4. 71)=0. 64% (4. 08-3. 92/3. 92)=4. 08% (3. 6-3. 14/3. 14)=14. 6% (2. 7-2. 35/2. 35)=14. 9% (1. 90-1. 57/1. 57)=21. 0% (0. 94-0. 78/0. 78)=20. 5%

Factor: Parallax error, during adjusting the level gauge to point, the Water valve was not completely close and Press the stopwatch start button late.

Question: Suggest two ways to improve the accuracy of the results?

1. It is by repeating the experiment a few times which makes the results more reliable.
2. Measuring the use of highly precise digital measurements.
3. If the line didn’t pass through the origin that means that there is an error because if the force is zero ( the jet doesn’t touch the vane) the should be placed at the origin which means? y=0 so F=0
4. F = m (uo = u) u ? uo because we neglect reduction of speed so that u=uo fo = 2muo but the force on the hemispherical cup less than twice that on the flat plate.
5. The effect on the calculated force on the flat plate if the jet was assumed to leave the plate at 1? upward will be a moment in the x-direction which will decrease the moment in the y-direction F=m (1. 9uo) and it won’t affect the results too much.

Conclusion

In a conclusion, the experiment that has been carried out were successful, even though the data collected are a little bit different compared to the theoretical value. The difference between the theoretical value and the actual value may mainly due to human and service factors such as parallax error. This error occurs during the observer captured the value of the water level. Besides that, error may occur during adjusting the level gauge to point at the white line on the side of the weight pan. Other than that, it also may be because of the water valve. This error may occur because the water valve was not completely closed during collecting the water. This may affect the time taken for the water to be collected. There are a lot of possibilities forth experiment will having an error. Therefore, the recommendation to overcome the error is to ensure that the position of the observer’s eye must be 90° perpendicular to the reading or the position. Then, ensure that the apparatus functioning perfectly in order to get an accurate result.

Mathematics can sometimes seem scary for me, and I am sure that a lot of other high school students feel the same way. Maybe, it’s because we often see math as merely a series of problems to be solved and rules to master and apply. Calculus is one of the branches of math that some students like me find intimidating to learn.

This paper aims to establish an appreciation and better understanding of calculus by reviewing its historical groundings and giving the practical application of the subject.

The foundation of calculus did not just appear in history, in fact, mathematicians had encountered numerous difficulties and problems that had led to their desire to find ways in which to offer solutions. It is the case that although Isaac Newton and Gottfried Leibniz were the ones to formulate the theorems of Calculus we know today, a fair share of mathematicians began utilizing concepts of calculus as early as the greek period. Calculus was developed from ancient Greek geometry.

It was mainly use to Democritus calculated the volumes of pyramids and cones, probably by regarding them as consisting of infinitely many cross-sections of infinitesimal (infinitely small) thickness, and Eudoxus and Archimedes used the “method of exhaustion”, finding the area of a circle by approximating it arbitrarily closely with inscribed polygons. In fact it was Archimedes who was the first person to find an approximation of the area of the circle using the “method of exhaustion”; it was the first samples of integration and led to the approximated values of ?

(pi). In line with the developments in the field of theoretical mathematics, it can be said that mathematicians encountered their own difficulties with math problems before they were able to actually find the answers through calculus. It was not until the 16th century when mathematicians found the need to further develop the methods that could be used to calculate areas bounded by curves and spheres.

Johannes Kepler for example had to find the area of the sectors of the ellipse in order for him to proceed with his work in planetary motion. He was lucky enough to find the answer in two tries despite the then crude methods of calculus. Imagine if he was unable to compute the area of ellipses during that time, chances are there would have been a delay in the development of astronomical science. It was through Kepler’s exploration of integration that laid groundwork for the further study of Cavalieri, Roberval, and Fermat.

The latter especially contributed a great deal to calculus by generalizing the parabola and hyperbola as y/a = (x/b)2 to (y/a)n = (x/b)m and y/a = b/x to (y/a)n = (b/x)m respectively. It is the case that some mathematicians (like Joseph Louis Langrange) consider Fermat to be the father of calculus, especially with his formulation of the method used in acquiring the maxima and minima by calculating when the derivative of the function was 0; this method is not far from that which we use today in solving such equations.

The formulas we use today to determine motion at variable speeds use calculus. Toricelli and Barrow were the first mathematicians to explore the problem of motion by implicitly applying the inverse of differentiation, integral and derivative as inverses of each other in asserting that the derivative of distance is velocity and vice versa. Newton and Leibniz are considered to be the inventors of calculus because of their discovery of the fundamental theorems of calculus.

However though both shares credit for the latter, Newton was able to apply it further showing its use both in his works in physics and planetary motion which are considered the most significant of all his contributions. The three laws of motion echoed if not are born out of the notion that since the world changes and derivatives are the rates of changes, and then the latter becomes pivotal to any scientific endeavor that attempts to understand the world. Newton was able to use calculus in determine a lot of things during his time.

We must remember though, that in voicing Newton it is good to reminisce his advice that abstractions and concepts don’t stand alone, they’re pieced together with other ideas to find a solution, an answer. This goes with his Newtonian laws, which if we are to really understand we must see how it relates with his law of gravitational force. Calculus bridges the gaps between theoretical math and the applied sciences/mathematics; if we are to look at it exclusively then we would miss the entire point of why we use it as such fail to realize its true value.

Calculus plays a role in the natural, physical as well as the social sciences; it is being employed in solving numerous problems that wishes to determine the maximum and minimum rates of change. It is capable of describing the physical processes that occur around us. It has even been used to solve paradoxes created during the time of Zeno in ancient Greece. It is impossible to imagine how we can be able to understand the world today without the calculus as one of our tools in acquiring knowledge. We may perhaps still be slaves to mystical forces that were claimed to be the cause of change in this world.

Mathematics would remain to us mere abstractions if calculus was not introduced to become the mediator of thought and practice. The development of other disciplines would have not followed without first establishing the existence of the fundamental concepts of calculus. Things which in history were thought to be inconceivable were able to have a figure that man can understand and therefore have the capacity to manipulate though not complete control. Students like me get frustrated when trying to solve a mathematical problem and failing once or twice.

Reading on the history of calculus made me realize that mathematicians would not have come up with the theorems and methods we use today if they too decided to simply get frustrated. In as much as Calculus teaches you at what rate things change and how the infinite can be understood, one could also learn the value of knowing something even if exclusively it seems unimportant. In order for us to appreciate the subject we must look at it as part of the greater system of knowledge, without it all things would not be coherent.