Complete the rate law for this reaction in the box below.

Question

The following initial rate data are for the oxidation of arsenate ion by cerium(IV) ion in aqueous solution:

AsO₃^3- + 2 Ce⁴⁺ + H₂OAsO₄^3- + 2 Ce³⁺ + 2 H⁺

Experiment	[AsO33-]o, M	[Ce4+]o, M	Initial Rate, M s-1
1	3.14×10-2	6.96×10-2	8.09×10-5
2	6.28×10-2	6.96×10-2	1.62×10-4
3	3.14×10-2	0.139	3.23×10-4
4	6.28×10-2	0.139	6.46×10-4

Complete the rate law for this reaction in the box below.

Use the form k[A]^m[B]ⁿ , where '1' is understood for m or n and concentrations taken to the zero power do not appear. Don't enter 1 for m or n.

Rate =

From these data, the rate constant is _____ M^-2s^-1.

Answer 1

First let's review the rules for how zero order, first order, and second order rates affect the concentration of the reactant and the reaction rate.

Zero Order: If the concentration of the reactant is increased, the rate of the reaction does not change.

1. Let's look at some examples:
2. If the concentration of the reactant is doubled, the rate stays the same
3. If the concentration of the reactant is tripled, the rate stays the same

First Order: The concentration of the reactant and the rate of the reaction change in exactly the same way.

1. Let's look at some examples:
2. If the concentration of the reactant is doubled, the rate also doubles
3. If the concentration of the reactant is tripled, the rate also triples

Second Order: The rate of the reaction changes like this: (change in concentration)²

1. Let's look at some examples:
2. If the concentration of the reactant is doubled, the rate quadruples (is multiplied by 4)
3. If the concentration of the reactant is tripled, the rate is increased by 9x (because 3² = 9)

Now let's look at the data you were given. We will focus on AsO3^3- first. In order to focus on AsO3^3-, we must have the concentration of the other reactant (Ce⁴⁺) stay the same. So let's look in the data table for two experiments where the concentration fo AsO3^3- changes, but the concentration of Ce⁴⁺ stays the same. You will see that the concentration of AsO3^3- changes from 3.14 x 10^-2 in Experiment 1 to 6.28 x 10^-2 in Experiment 2. However, the concentration of Ce⁴⁺ stays the same (6.96 x 10^-2) in Experiment 1 and Experiment 2. This means that we can use Experiment 1 and Experiment 2 to examine how changes in the concentration of AsO3^3- affect the rate of the reaction. In other words, we can find the order of the reaction for AsO3^3- by using the data from Experiments 1 and 2.

First let's figure out how the concentration of AsO3^3- changes. We figure this out by doing (the concentration of AsO3^3- in Experiment 2) divided by (the concentration of AsO3^3- in Experiment 1). We know the concentration of AsO3^3- in Experiment 2 is 6.28 x 10^-2, and in Experiment 1 the concentration of AsO3^3- is 3.14 x 10^-2. So we calculate 6.28 x 10^-2 divided by 3.14 x 10^-2. The calculator says

6.28 x 10^-2 / 3.14 x 10^-2 = 2. The number 2 means that the concentration of AsO3^3- in Experiment 2 was 2 times the concentration of AsO3^3- in Experiment 1. In other words, the concentration of AsO3^3- doubled from Experiment 1 to Experiment 2. We will need to remember this info.

Now let's look at how the reaction rate changes when the concentration of AsO3^3- is doubled. We figure this out by doing (the rate for Experiment 2) divided by (the rate for Experiment 1). The rate for Experiment 2 is 1.62 x 10^-4, and the rate for Experiment 1 is 8.09 x 10^-5. So we calculate 1.62 x 10^-4 divided by 8.09 x 10^-5. The calculator says that 1.62 x 10^-4 / 8.09 x 10^-5 = 2. The number 2 means that the rate of the reaction in Experiment 2 was 2 times the rate of the reaction in Experiment 1. In other words, the reaction rate doubled from Experiment 1 to Experiment 2.

Now we can finally figure out the order for AsO3^3-. If we put together what we learned from our calculations, we can say it like this: When the concentration of AsO3^3- doubles, the reaction rate also doubles. If we look at the rules for the zero order, first order, and second orders that we wrote down first, we see that "if the concentration doubles, the reaction rate also doubles" describes first order. Thus, the reaction is first order with regards to AsO3^3-.

Now let's figure out the order of Ce⁴⁺. In order to do this, we need to find two Experiments where the concentration of AsO3^3- does not change, but the concentration of Ce⁴⁺ does change. If we look at the experimental data, we notice that the concentration of AsO3^3- stays the same (3.14 x 10^-2) in Experiments 1 and 3. However, the concentration of Ce⁴⁺ changes from 6.96 x 10^-2 in Experiment 1 to 0.139 in Experiment 3. So, we can use Experiment 1 and Experiment 3 to figure out the order for Ce⁴⁺. (Note that we also could use Experiments 2 and 4 to figure out the order for Ce⁴⁺. Our answer would be exactly the same if we used Experiments 2 and 4.)

Let's figure out how the concentration of Ce⁴⁺ changes from Experiment 1 to Experiment 3. We figure this out by doing (the concentration of Ce⁴⁺ in Experiment 3) divided by (the concentration of Ce⁴⁺ in Experiment 1). We know the concentration of Ce⁴⁺ in Experiment 3 is 0.139, and in Experiment 1 the concentration of Ce₄₊ is 6.96 x 10^-2. So we calculate 0.139 divided by 6.96 x 10^-2. The calculator says

0.139 / 6.96 x 10^-2 = 2. The number 2 means that the concentration of Ce⁴⁺ in Experiment 3 was 2 times the concentration of Ce⁴⁺ in Experiment 1. In other words, the concentration of Ce⁴⁺ doubled from Experiment 1 to Experiment 3. We will need to remember this info.

Now let's figure out how the rate of the reaction changes when the concentration of Ce⁴⁺ is doubled. We figure this out by doing (the rate for Experiment 3) divided by (the rate for Experiment 1). The rate for Experiment 3 is 3.23 x 10^-4, and the rate for Experiment 1 is 8.09 x 10^-5. So we calculate 3.23 x 10^-4 divided by 8.09 x 10^-5. The calculator says that 3.23 x 10^-4 / 8.09 x 10^-5 = 4. The number 4 means that the rate of the reaction in Experiment 3 was 4 times the rate of the reaction in Experiment 1. In other words, the reaction rate quadrupled from Experiment 1 to Experiment 3.

Now we can finally figure out the order for Ce⁴⁺. If we put together what we learned from our calculations, we can say it like this: When the concentration of Ce⁴⁺ doubles, the reaction rate quadruples. If we look at the rules for the zero order, first order, and second orders that we wrote down first, we see that "if the concentration doubles, the reaction rate quadruples" describes second order. Thus, the reaction is second order with regards to Ce⁴⁺.

The reaction is first order with regards to AsO3^3-. The instructions say that we are to put "m" instead of a 1 if the reaction is first order. The instructions say nothing about substituting a letter for second order, so we will be able to write the 2 for Ce⁴⁺.

Rate = k[AsO₃^3- ]^m [Ce4+]²

Now we must find the rate constant (which is k). To do this, we plug information from one of the Experiments into the bolded equation we wrote above. (We will change the m back to a 1 in order to figure out k). Let's use the data from Experiment 1: [AsO3^3-] = 3.14 x 10^-2, [Ce⁴⁺] = 6.96 x 10^-2, and Rate = 8.09 x 10^-5. If we put the data into the bolded equation above and change the m back to a 1, the equation looks like this:

8.09 x 10^-5 = k*(3.14 x 10^-2 )¹ * (6.96 x 10^-2 )²

The calculator says that (3.14 x 10-² )¹ = 3.14 x 10^-2. The calculator says that (6.96 x 10^-2 )² = 0.00484416. If we put these values into the equation, it looks like this:

8.09 x 10^-5 = k*(3.14 x 10^-2 ) * 0.00484416

Now we multiply (3.14 x 10^-2 ) * 0.00484416.

The calculator says that 3.14 x 10^-2 * 0.00484416 = 1.52107 x 10^-4. When we put this value into the equation, it looks like this:

8.09 x 10^-5 = k*1.52107 x 10^-4

Now we divide both sides by 1.52107 x 10^-4. Once we have done that, the equation looks like this:

8.09 x 10^-5 = k

1.52107 x 10^-4

So now we just divide 8.09 x 10^-5 by 1.52107 x 10^-4 .

The calculator says that 8.09 x 10^-5 / 1.52107 x 10^-4 = ^. 0.53186. So:

k = 0.53186

Let's round k to 3 significant figures since all the experimental data has 3 significant figures. Therefore:

k = 0.532 M^-2s^-1