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Chemistry: The Arrhenius Equation

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:11
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 88 MB
  • Posted: 07/15/2009

This lesson is part of the following series:

Chemistry: Full Course (303 lessons, $198.00)
Chemistry: Chemical Kinetics (18 lessons, $25.74)
Chemistry: Collision Model & Arrhenius Equation (3 lessons, $4.95)

This lesson was selected from a broader, comprehensive course, Chemistry, taught by Professor Harman, Professor Yee, and Professor Sammakia. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/chemistry. The full course covers atoms, molecules and ions, stoichiometry, reactions in aqueous solutions, gases, thermochemistry, Modern Atomic Theory, electron configurations, periodicity, chemical bonding, molecular geometry, bonding theory, oxidation-reduction reactions, condensed phases, solution properties, kinetics, acids and bases, organic reactions, thermodynamics, nuclear chemistry, metals, nonmetals, biochemistry, organic chemistry, and more.

Dean Harman is a professor of chemistry at the University of Virginia, where he has been honored with several teaching awards. He heads Harman Research Group, which specializes in the novel organic transformations made possible by electron-rich metal centers such as Os(II), RE(I), AND W(0). He holds a Ph.D. from Stanford University.

Gordon Yee is an associate professor of chemistry at Virginia Tech in Blacksburg, VA. He received his Ph.D. from Stanford University and completed postdoctoral work at DuPont. A widely published author, Professor Yee studies molecule-based magnetism.

Tarek Sammakia is a Professor of Chemistry at the University of Colorado at Boulder where he teaches organic chemistry to undergraduate and graduate students. He received his Ph.D. from Yale University and carried out postdoctoral research at Harvard University. He has received several national awards for his work in synthetic and mechanistic organic chemistry.

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Thinkwell
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Did you ever cook Thanksgiving dinner? You would cook the turkey about 3 hours at 350 degrees. Did you ever wonder if you could cook if for 1 hours at 700 degrees maybe because you forgot to put the turkey in? That doesn't work. There is a difference in how fast the turkey cooks. Remember that cooking is just a series of chemical reactions. But it isn't the case that if you put it in the oven at twice the temperature that it cooks in half the time.
The way rate does depend on temperature is expressed by something called the Arrhenius equation. The idea, first of all, is this, we have already derived from a bimolecular reaction kinetics that the rate constant is equal to A, which is called the frequency factor. It has units of inverse molar inverse seconds, times the exponential of where E[a] is the activation energy for this reaction, R is the gas constant in joules per mole Kelvin, and T is the Kelvin temperature.
Remember that A has included in it Z, which was a reflection of how fast the molecules are moving, and that has a little bit of a temperature dependence, but it is not real strong. Times p, which was the steric factor or how the molecules are oriented and whether they are oriented correctly or how demanding the transition state is for getting the reaction to occur. And this has essentially no temperature dependence. Times f, which is the fraction of molecules that have enough energy in order to surmount the activation barrier and go on to products. And this has a really strong temperature dependence. Things that are exponential in temperature or exponential in one over temperature have really strong temperature dependencies. Whether or not the turkey would cook in twice the time if you cooked it at 700 degrees is embodied in this piece. Again, it wouldn't work to cook your turkey at 700 degrees.
How can we show how this behaves? Let me tell you how it behaves. It's that the rate constant, which we call a constant, but it's not really a constant, it's constant only so long as the temperature is constant. But if the temperature changes, then the rate constant changes. In general, a reaction goes faster when you heat it up and it goes slower when you cool it down.
Here's a great illustration of that idea. We have here three identical light sticks. They are lumenal and hydrogen peroxide, it is exactly the same reaction that goes on inside a firefly. I'm going to initiate these three. These happen to last for about 12 hours so these will go for a long time. To do our demonstration, let me first show you that they are identical. I'm not going to do anything to one of them. I'm going to put one into ice water to cool it down, and I'm going to put one into hot water to warm it up. If our understanding of what is going on here is correct, what ought to happen is that the one that is in the ice water, the reaction is going to slow down. As the reaction slows down, the amount of light that is emitted is going to decrease so it's not going to look as bright. Conversely, the one that is put into the hot water, the reaction rate is going to go up. It's not that the concentrations are changing. It is that the rate constant is changing and so it ought to be getting brighter. If we turn down the lights, I think you are going to see a fairly significant difference between the light emission from the three light sticks.
Let me tell you a story. You may have heard me lecture about how I loved helium balloons when I was little. After playing with the helium balloon all day, I would wake up the next morning and I wanted to play with it again. But of course, it was dead. Similar things happen with light sticks. You take one out to go trick-or-treating and you put it away. You come back the next morning thinking you get to play with it again and of course it is basically dead. There is a way to save a light stick in a way that there isn't a way to save a helium balloon. That is to put the light stick into the freezer. What happens when you put it in a freezer is that you kill the chemical reaction. You slow it down so much that it is not going to proceed. When it doesn't proceed and you come back later that evening and warm it back up again, it will start going again. So, there is a way to save this toy in a way that there isn't a way to save a helium balloon.
Furthermore, this is also the idea for why we refrigerate food. We refrigerate food because when you cool it down it slows down all of the reactions that are bad. By cooling food down or by freezing it, we can make the reactions that are undesirable, such as decomposition reactions and the spoiling reactions very slow.
Going back to this equation. What happens if we take the natural log of both sides? If we take the natural log of both sides, we get to this expression. This is one of those equations like y = mx + b that scientists love so much. It is a linear equation where y is equal to the natural log of the rate constant. The natural log of the frequency factor is b and the activation integer over RT is m times x, y = b mx, where x is and m is . In other words, that much, that's m, and that much right there, that's x. So if we plot the natural log of k versus , we can learn something about the activation energy because that's the slope. Here is a plot of natural log of the rate constant as a function of temperature for some data. You can see that they fall on a straight line where the slope is . This particular equation for this particular set of data is for a certain bimolecular reaction where the following data were obtained. In other words, we would measure the rate constant. You recall that we talked about how you can measure the rate constant based on, for instance, the method of initial rates. So you can determine for these rate constants at a bunch of different temperatures. So up until now we've been doing things all at one temperature, now we're going to do things at a variety of temperatures, keeping concentrations constant now. Now we're not messing with concentration; we're messing with temperature. The way you do science is you try to control some things and vary some other things.
Then take one over the temperature, and you get these values and take the natural log of the rate constants and you get these values, and plot the natural log of the rate constant as a function of and that gives you this plot. Determine the slope of this line by rise over run, just the way you would do the slope of any other straight line, and you get that the slope is equal to -1.88 times 10^3 and this is dimensionless. And then when we multiply by R the gas constant in joules per mole Kelvin, we get that the activation energy is equal to 15.6 kJ/mole, E[a] = 15.6 kJ/mo.
So, this reaction turns out it has a relatively small activation energy. This means that it doesn't take a whole lot of energy to get over that hump.
Let me conclude by saying that we tend to be really interested in reactions and how they behave as a function of temperature. I talked about food already. Part of the reason why you might run a fever is that running a fever destroys the bacteria that might have infected you. What your body is doing is it is trying to make the reactions that destroy the bacteria go faster to help you get rid of whatever it is that is ailing you. Nature uses temperature in ways that take advantage of the fact that reactions go faster at higher temperatures.
Chemical Kinetics
Temperatuer and Rates
The Arrhenius Equation Page [2 of 2]

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