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# Summary

### Rate Constant

What is a rate constant?

I mentioned earlier that reactions occur at different rates. An explosion from setting hydrogen on fire may last only a couple of seconds. The reactions involved in baking a cake might take a few minutes to a few hours. The chemistry that rusts your bike can take months or years. In our example above, the decay of carbon-14 takes thousands of years.

A rate constant tells you how fast a reaction goes. We might expect our explosion to have a very large rate constant, while carbon-14 might have a very small one. But how do we find out what this rate constant is?

A neat trick that scientists often use is to linearlize data. Remember that exponential decay from before? If we take the natural log of both sides with can get a new equation that looks like this: If you look closely you should be able to see that plotting ln[C-14]t on the y-axis and t on the x-axis should give a straight line with an intercept of ln[C-14] at time 0 and a slope of k. That allows us to find our rate constant! But first let's program this linear data in python. Remember our data file is called carbondecay.csv

1  #We will need to import numpy this time to calculate the natural log

2  import pandas

import matplotlib.pyplot as plt

import numpy

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#If you look at the carbondecay file you will see it has two columns labeled "Time" and "C14_Con" the following line takes the natural log of every entry in the C14_Con column and then prints it. Note that in python the command "log" means natural log, and not log base 10. I will use log to mean natural log for the remainder of this lesson

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11  logC_14=numpy.log(carbon_decay.C14_Con)

12  print(logC_14)

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14  #Now we want to plot time v our new data to get a linear plot

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16  plt.plot(carbon_decay.Time, logC_14)

17  plt.title('Amount of Carbon-14 in Sample') # this is the title of the graph (note it's a string)

18  plt.xlabel('Time (years)') # this is the label for the x-axis

19  plt.ylabel('log of Amount of Carbon-14 (parts per trillion)') # this is the label for the y-axis

20  plt.savefig('foo2.png')

You should be getting a graph that looks like the one below. We will talk about how to calculate the rate constant from this graph in the next section. bottom of page