## Linear Graphs

$\frac{d[R]}{dt}=-k[R]^{2}; \frac{1}{[R]}=kt + \frac{1}{[R]_{0}}; t_{\frac{1}{2}}=\frac{1}{k[R]_{0}}$

Glenda Marshall DIS 3M
Posts: 68
Joined: Sat Sep 27, 2014 3:00 am

### Linear Graphs

Why is the graph of the half-life of a second order reaction not linear. Wouldn't the slope be 1/k with the variable being [R]o?

Thanks!

Chem_Mod
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### Re: Linear Graphs

Please specify what do you mean by 'graph of the half-life of a second order reaction' (specify x and y axis).
For a second order reaction integrated rate law is 1/[R] = kt + 1/[R0]
So 1/[R] vs t will give a straight line with slope k
[R] means reactant concentration at any instant t, which is a variable and [R0] is initial concentration which is a constant.

Now think from the equation how you could vary [R0] to get a linear plot with slope 1/k !

Glenda Marshall DIS 3M
Posts: 68
Joined: Sat Sep 27, 2014 3:00 am

### Re: Linear Graphs

My apologies! Homework problem #14.103 says

"which of the following plots would be linear?" and one of the options is "the half-life against [A] for a second order reaction in [A]"

I am confused why this is not linear because I thought it would be, but the solutions manual does not have it listed. Thanks!

Chem_Mod
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### Re: Linear Graphs

Firstly, there was a typo in the question. It would be
g) 'the half-life against [A0] for a zero order reaction in [A]'
h) 'the half-life against [A0] for a second order reaction in [A]
Check the half life of a zero order reaction t1/2 = [A0]/2k so a linear plot with 1/2k slope
Check the half life of a second order reaction t1/2 = 1/k[A0], see t1/2 is inversely proportional with A0 , not linearly
So you will NOT get a linear plot for half life vs [A0] here.

Rather you will get a linear plot for half life vs 1/[A0] with 1/k slope for a second order reaction

Glenda Marshall DIS 3M
Posts: 68
Joined: Sat Sep 27, 2014 3:00 am

### Re: Linear Graphs

Oh that totally makes sense! Thank you so much and sorry for the typo!

Rachel Lipman
Posts: 45
Joined: Fri Sep 25, 2015 3:00 am

### Re: Linear Graphs

There have been no graph questions in the textbook homework, so I was wondering what some examples might be to prepare should it show up on our quiz?