# Using Graphs to Determine Rate Laws, Rate Constants & Reaction Orders

Instructor: Laura Foist

Laura has a Masters of Science in Food Science and Human Nutrition and has taught college Science.

Graphs are used to visualize how quickly a reaction occurs, but it isn't always a linear relationship. In this lesson we learn how to use data from a graph to determine kinetics of a reaction.

## Rate

If you were at an all-you-can-eat buffet, do you eat your food at the same pace towards the end as you did at the beginning? You probably tend to slow down over time. Some reactions may occur at the same rate over time, but most act like we do at an all-you-can-eat buffet; they slow down over time.

A reaction order tells us how much the initial rate of a reaction will increase when the reactant is doubled. In other words, a reaction order tells us how much faster you will be eating at the start of the buffet if you began twice as hungry.

## Zeroth Order Reaction

Let's start out with a zeroth order reaction. In other words, it doesn't matter how much reactant you start out with, the reaction rate will always be the same. At the all-you-can-eat buffet, it doesn't matter how hungry you are at the beginning, you know how to pace yourself and will always eat at the same rate. In this case the rate stays the same the entire time. Let's look a graph of time vs concentration of product for a zeroth order reaction:

We see a straight, linear line. The rate constant is equal to the slope of the line. We can determine the slope by looking at the equation of the line, which is:

y=-0.04x + 1.5

Recall that these types of equations fit the y= mx + b formula. In this case, 'y' is equal to the concentration of the reactant, 'm' is equal to the rate constant, 'x' is equal to time, and 'b' is equal to the starting concentration of the reactant. Thus, the rate constant of this reaction is 0.04.

The rate law is simply the equation for the line. Typically, the rate law refers to what we need to do to transform the data into a straight line. Since zero order reactions are already a straight line, the rate law is the same as the equation for the line.

## First Order Reaction

In a first order reaction the initial rate of the reaction will be twice as fast when the reactant is doubled. In other words, if you are twice as hungry when you get to the buffet, you will start out eating twice as fast. In this scenario, you slow down over time, so a graph for a first order reaction may look like this:

The slope of this line is a little harder to determine. But we can straighten out the line by integrating it (seeing how quickly the concentration changes over time). Mathematically this is shown using the natural log (ln). Since we are changing how the concentration is shown on the graph, we need to take the natural log of any variable that includes concentration. Recall that 'y' is the concentration and 'b' is the starting concentration. This means that in the equation we need to take the natural log of 'y' and 'b,' making our new rate law:

ln(y) = mx + ln(b)

This changes our data into a straight line, and we can determine the rate constant in the same way as the zero order reaction.

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