meaning of linear lines in graphs (order of reactions)

[Mackenzie Fernandez 3G]

hi!

what is the significance of linearity in graphs? I still don't understand the concept. Also, is the general rule: for a reaction to be "x" order, the graph must be linear??

thanks in advance!

[IsaacLaw1E]

The general rule is more like: for a reaction to be "x" order, the graph between time and "y" must be linear. And depending on what order x is, y will be a different value.
For instance, if x = 1 (1st order), then y = ln[A], and the graph between time and ln[A] is linear. If x = 2, y = 1/[A]; if x = 0, y = [A].
Furthermore, the slope of the line will be equal to k (or negative k, depending on the order), so we can find rate constants with this too, but only if we match the order and the "y" so that it's linear.

[Juliet Cushing_2H]

I've always taken it as a thing to memorize rather than to understand conceptually (but the conceptual understanding of it comes from producing an integrated rate law from the normal differential rate law). If you want to understand these conceptually I'd go back to Dr. Lavelle's week 9 lecture 1 where he explains the integrated rate law. Otherwise, here are the things to memorize:
- For a 0 order reaction, graphing [X] vs time will produce a linear graph
- For a 1st order reaction, graphing ln[X] vs time will produce a linear graph
- For a 2nd order reaction, graphing 1/[X] vs time will produce a linear graph
You use these facts to determine the order of a reaction if you're given experimental data. Hope this helps.

[Marc Lubman 3B]

As a general rule, the appearance of a linear graph means that the value you have chosen to represent on the Y axis corresponds to the correct order of your reaction (the specificities of which can be found in the posts above and by using the integrated rate law). This is because the way you manipulate that value (through certain amounts of integration) to get y takes you down from its original order to a simple linear graph. Basically, the type of value on the Y axis tells you how many times you had to integrate the reaction to get from its original order to a straight line, and each integration takes the equation one order down, so you can then work backwards to figure out its original order.