Rate Equations and Orders of Reaction - Chemistry

Rate equations and orders of reaction At the end of this topic, students should be able to: explain the terms, rate constant, order of reaction. Perform calculations to determine the rate constant. Rate equations Measuring a rate of reaction There are several simple ways of measuring a...

Rate equations and orders of reaction At the end of this topic, students should be able to: explain the terms, rate constant, order of reaction. Perform calculations to determine the rate constant. Rate equations Measuring a rate of reaction There are several simple ways of measuring a reaction rate. For example, if a gas was being given off during a reaction, you could take some measurements and work out the volume being given off per second at any particular time during the reaction. A rate of 2 cm3 s-1 is obviously twice as fast as one of 1 cm3 s-1. However, for this more formal and mathematical look at rates of reaction, the rate is usually measured by looking at how fast the concentration of one of the reactants is falling at any one time. For example, suppose you had a reaction between two substances A and B. Assume that at least one of them is in a form where it is sensible to measure its concentration - for example, in solution or as a gas.  For this reaction you could measure the rate of the reaction by finding out how fast the concentration of, say, A was falling per second. You might, for example, find that at the beginning of the reaction, its concentration was falling at a rate of 0.0040 mol dm-3 s-1. This means that every second the concentration of A was falling by 0.0040 moles per cubic decimeter. This rate will decrease during the reaction as A gets used up. Summary For the purposes of rate equations and orders of reaction, the rate of a reaction is measured in terms of how fast the concentration of one of the reactants is falling. Its units are mol dm-3 s-1.  Orders of reaction Orders of reaction are always found by doing experiments. You can't deduce anything about the order of a reaction just by looking at the equation for the reaction. So let's suppose that you have done some experiments to find out what happens to the rate of a reaction as the concentration of one of the reactants, A, changes. Some of the simple things that you might find are: One possibility: The rate of reaction is proportional to the concentration of A That means that if you double the concentration of A, the rate doubles as well. If you increase the concentration of A by a factor of 4, the rate goes up 4 times as well. You can express this using symbols as: Writing a formula in square brackets is a standard way of showing a concentration measured in moles per cubic decimeter (litre). Another possibility: The rate of reaction is proportional to the square of the concentration of A This means that if you doubled the concentration of A, the rate would go up 4 times (22). If you tripled the concentration of A, the rate would increase 9 times (32). In symbol terms:  Generalising this By doing experiments involving a reaction between A and B, you would find that the rate of the reaction was related to the concentrations of A and B in this way: This is called the rate equation for the reaction. The concentrations of A and B have to be raised to some power to show how they affect the rate of the reaction. These powers are called the orders of reaction with respect to A and B. The orders of reaction you are likely to meet will be 0, 1 or 2. If the order of reaction with respect to A is 0 (zero), this means that the concentration of A doesn't affect the rate of reaction. Mathematically, any number raised to the power of zero (x0) is equal to 1. That means that that particular term disappears from the rate equation. The overall order of the reaction is found by adding up the individual orders. For example, if the reaction is first order with respect to both A and B (a = 1 and b = 1), the overall order is 2. We call this an overall second order reaction Some examples Each of these examples involves a reaction between A and B, and each rate equation comes from doing some experiments to find out how the concentrations of A and B affect the rate of reaction. Example 1: In this case, the order of reaction with respect to both A and B is 1. The overall order of reaction is 2 - found by adding up the individual orders.  Example 2: This reaction is zero order with respect to A because the concentration of A doesn't affect the rate of the reaction. The order with respect to B is 2 - it's a second order reaction with respect to B. The reaction is also second order overall (because 0 + 2 = 2). Example 3: This reaction is first order with respect to A and zero order with respect to B, because the concentration of B doesn't affect the rate of the reaction. The reaction is first order overall (because 1 + 0 = 1). The rate constant The rate constant is constant for a given reaction only if all you are changing is the concentration of the reactants. We can also deduce the order of reaction with respect to a single reactant by: Plotting a graph of reaction rate against concentration of reactant Plotting a graph of concentration of reactant against time Half life The half-life of a reaction is the time it takes for the concentration of a substance to fall to half of its original value. For a first order reaction (but only for a first order reaction), the half-life is constant. It doesn't matter what concentration you start with, it will take exactly the same time for the concentration to reach half of that value. You can easily discover this from a graph of concentration against time. You simply have to measure how long it takes for the concentration to fall from its original value to half of that value; then measure it from a half to a quarter; then from a quarter to an eighth. Or you could measure it from the original 100% to 50%, and then compare that with the fall from 80% to 40%, and from 60% to 30% - or any other combination of concentrations, as long as you are measuring the time taken to halve the concentration. Looking at this on a graph:.. you can see that the time taken to halve the concentration is always the same. All you need to recognise is that the concentration against time curve for: a zero order reaction will be a straight line; a first order reaction will be a curve with a constant half-life. any other order of reaction will be a curve with a non-constant half-life. Only first order reactions have a constant half-life. The relationship between half-life and rate constant for a first order reaction Half-life is given the symbol t½. For first order reactions there is a simple relationship between this and the rate constant for the reaction, k. This only applies to first order reactions. "ln 2" is the natural logarithm of 2. You will find it gives an answer of 0.69314718. round this off for you to 0.693 Example 1 If the half-life of a first order reaction is 300 seconds, calculate the rate constant for the reaction. Why are the units s-1? 0.693 has no units, and the units of half-life are seconds (s). The unit s was on the bottom of the fraction, and you need to write the units on the same line as the answer. Example 2 If the rate constant of a first order reaction is 7.70 x 10-4 s-1, calculate the half- life of the reaction. First, you will have to rearrange the expression to let you calculate half-life.  Finding the rate equation [A] [B] initial rate Experiment (mol dm-3) (mol dm-3) (mol dm-3 s-1) 1 0.10 0.10 0.0015 2 0.20 0.10 0.0030 3 0.20 0.20 0.0120 In simple cases like this, you can just look at the numbers. When you double the concentration of A between experiments 1 and 2 (keeping [B] constant), you double the rate. The rate is proportional to [A], and so the reaction is first order with respect to A. Between experiments 2 and 3, you keep the concentration of A constant, but double [B]. The rate goes up 4 times. So the rate is proportional to the square of the concentration of B, and the reaction is second order with respect to B. That means that the rate equation is: Rate = k[A][B]2