Consider the following concentration–time data for the decomposition reaction AB → A + B.

(a) Determine the order of the reaction and the value of the rate constant.

Video transcript

Hello. In this problem we are told the decomposition reaction for C. D. Going to form C. And D. Has the following concentration time data were asked what is the order of the reaction and what is the rate constant for the reaction? So to solve this problem we are going to have to recall plots for determining whether reaction is zero order, first order or second order. So we will make use of the integrated rate law for a zero first and second order reaction for a zero order reaction. Then a plot of concentration. The time T. Will be equal to the negative of the reaction rate constant times time plus the concentration About reacting at time zero. And so we already have those values provided in the table. We know the concentration of our reactant C. D. As a function of time. To determine if it is first order then the integrated rate law for a first order reaction is given by the natural log concentration of the reactant at some time. T. That's equal to the negative of the reaction rate constant times time plus the natural log of the initial concentration of our reactant. And so we're gonna need to add to this table. Then the natural log of our concentration of our reactant. And then to determine if it's second order, the integrated rate law is one over the concentration of our reactant at some time. T. Is equal to negative of the reaction rate constant. Sorry the positive reaction rate constant times time plus one over the concentration of our reactant initially. So we're also going to have to come up with another calm in our table which is one over are concentration of our reactant. So all of this can be done most simply using a software like Excel or other spreadsheet software. So I will write out the answers um or the values for taking the natural log of the concentration and one over the concentration. And then you can verify these using a spreadsheet like Excel, So we take the natural log of the first concentration. This works out to -3.265. The next one works out to -3.399. Next one is 3. And the next is -3.734 and then -3.958. And the last one is -4.241. When we take the natural first I went over the concentration, the first one we get 26.18 And then 0.94 And then 34. And then 41 0.84, 52.36 and 69.44. And then again a plot can most easily be done within a software like Excel. So first thing we'll do is plot, the concentration of our reactant is a function of time. We'll find then when we do this that we get a straight line where the equation for the best fit line is equal to negative y equals negative 0.318 x plus 0.38200. And the r squared value which tells us how good of a fit we have is 1.0. And then if we make a plot of the natural log of the concentration of our reactant as a function of time, we get line that's not quite as straight. So the equation for the best fit line is y equals negative 0.12864, X minus 3.209397. And r square value is 0.979956. And well then take apart, make a part of one over the concentration of our reactant versus time and get a curve. And so a best fit line fit to this data is given by y equals 0. X plus 21.637783. And our r squared value is equal to 0.9 to +1897. So if we compare our r squared values which tells us the goodness of fit best fit then is for this first plot Which represents a zero order reaction. And so the order of the reaction is zero order because a plot of the concentration versus time gives us a linear fit. And the rate constant for the reaction is given by the slope. And so the negative of our reaction rate constant is equal to negative 0.000318. So our rate constant then is equal to 0.000318. And our units are then concentration over time, which is minutes. Right? So we're finding the slope of the line. The slope is the change in Y over the change in X. So that's changing our concentration, which has units of polarity And our change in X is time, which has units of minutes. So our units for our reaction rate constant, our concentration per minute, and the value of our reaction rate constant is 0.00318. Thanks for watching. Hope this helps