The following data are obtained for the initial rates of...

The following data are obtained for the initial rates of reaction in the reaction $\mathrm{A}+2 \mathrm{B}+\mathrm{C} \longrightarrow 2 \mathrm{D}+\mathrm{E}.$ $$\begin{array}{lllll} \hline & \text { Initial } & \text { Initial } & & \\ \text { Expt } & {[\mathrm{A}], \mathrm{M}} & {[\mathrm{B}], \mathrm{M}} & {[\mathrm{C}], \mathrm{M}} & \text { Initial Rate } \\ \hline 1 & 1.40 & 1.40 & 1.00 & R_{1} \\ 2 & 0.70 & 1.40 & 1.00 & R_{2}=\frac{1}{2} \times R_{1} \\ 3 & 0.70 & 0.70 & 1.00 & R_{3}=\frac{1}{4} \times R_{2} \\ 4 & 1.40 & 1.40 & 0.50 & R_{4}=16 \times R_{3} \\ 5 & 0.70 & 0.70 & 0.50 & R_{5}=? \\ \hline \end{array}$$ (a) What are the reaction orders with respect to $\mathrm{A}, \mathrm{B}$ and C? (b) What is the value of $R_{5}$ in terms of $R_{1} ?$

The following data are obtained for the initial rates of reaction in the reaction $\mathrm{A}+2 \mathrm{B}+\mathrm{C} \longrightarrow 2 \mathrm{D}+\mathrm{E}.$
$$\begin{array}{lllll}
\hline & \text { Initial } & \text { Initial } & & \\
\text { Expt } & {[\mathrm{A}], \mathrm{M}} & {[\mathrm{B}], \mathrm{M}} & {[\mathrm{C}], \mathrm{M}} & \text { Initial Rate } \\
\hline 1 & 1.40 & 1.40 & 1.00 & R_{1} \\
2 & 0.70 & 1.40 & 1.00 & R_{2}=\frac{1}{2} \times R_{1} \\
3 & 0.70 & 0.70 & 1.00 & R_{3}=\frac{1}{4} \times R_{2} \\
4 & 1.40 & 1.40 & 0.50 & R_{4}=16 \times R_{3} \\
5 & 0.70 & 0.70 & 0.50 & R_{5}=? \\
\hline
\end{array}$$
(a) What are the reaction orders with respect to $\mathrm{A}, \mathrm{B}$ and C?
(b) What is the value of $R_{5}$ in terms of $R_{1} ?$

Key Concepts

Reaction Rate Law

The reaction rate law expresses the rate of a chemical reaction as a function of the concentrations of the reactants, each raised to a power corresponding to its reaction order. This law is central in chemical kinetics as it quantitatively links the kinetic behavior of the reaction to the concentrations, typically in the form rate = k[A]^m[B]^n[C]^p, where k is the rate constant and m, n, and p are the respective reaction orders.

Reaction Order

The reaction order with respect to a given reactant indicates how sensitively the reaction rate responds to the changes in that reactant’s concentration. It is determined experimentally and is not necessarily related to the stoichiometric coefficient of the reactant in the overall balanced equation. Reaction orders can be integers or fractions, and their sum gives the overall order of the reaction.

Method of Initial Rates

The method of initial rates involves measuring the reaction rate at the very beginning of the reaction under various reactant concentrations. By systematically varying the concentration of one reactant while holding others constant, one can determine the individual order with respect to each reactant. This method is a fundamental experimental technique in kinetics for deciphering the dependence of reaction rate on reactant concentrations.

Effect of Concentration Changes

Analyzing how changes in reactant concentrations influence the initial reaction rate allows for the determination of the exponent values in the rate law. By comparing the initial rates from multiple experiments in which the concentration of one reactant is doubled, halved, or otherwise altered, one can deduce the power-law relationship between that concentration and the reaction rate, isolating the reaction order for that species.

Transcript

00:01 Hey guys, so in this question, we're given a reaction between a, b, and c, forming d, and e, and we're asked to find the orders with respect to our reactants, and then find the value of rate 5 with respect to rate 1, according to the experiments that they gave us in the chart in the textbook.

00:21 So, as it first gets started, i'm going to find the rate order with respect to a by using experiments 1 and 2.

00:31 So i'm going to use the rate law equation that i have here to solve for m using e1 and e2.

00:41 I'm going to put e2 on the bottom of e1 since e1 has larger values than e2.

00:52 When i do this, i put in my rates.

00:55 So i have r1 for experiment 1.

00:59 And this is going to be equal to my rate constant times the concentration of a that was given to us in molar to the m power, times the concentration of b times our concentration of c.

01:17 I'm using an exponent of p instead of o since p doesn't look like zero like o would.

01:26 For e2, my rate is with respect to r1, so that's r1 over 2.

01:36 And i'm going to write the rest of my equation, so i have my rate constant multiplied by my concentration for a, multiplied by our concentration of b, and finally multiplied by our concentration for c.

01:56 Now i chose these two equations to solve for the order with respect to a, since my concentrations for b and c cancel out, and my rates cancel out, and i'm looking.

02:10 Left with what i'm looking for, which is the constants and numbers that are with respect to my rate constant or with my order for a.

02:25 So simplifying this down, we have r1.

02:29 We'll also cancel out because of multiplication, and we're going to be left with two on the left side, as well as 2 to the m on the right.

02:38 So when i solve for m, i find that m is equal to 1.

02:43 And this is my order with respect to a, showing that this is first order.

02:51 Now that i have that, i can go ahead and do the same things for reactants b and c.

02:57 So for reactant b, i'm going to be using experiments 2 and 3 for the same reasons that i chose experiments 1 and 2 last time.

03:09 And i'm going to put experiment 2 over experiment 3.

03:16 Plugging in my rates, i have r2 for experiment 2 and r2 over 4 for experiment 3.

03:27 And then for the rest of my equation, i'm going to plug in my concentrations for a, b, and c.

03:45 I'm going to do that for both experiments.

03:47 So this is for experiment 2.

03:49 And right here is going to be for experiment 3, my rate constant, times my concentration of a, times the concentration of b, times the concentration of c.

04:07 When i do simplification, my r2s cancel out, k's cancel out, a's concentration cancels out, and c's concentrations cancel out.

04:18 On the left side, i'm going to be ending up with 4, and on the right side i'll have 2 to 3.

04:24 The n power.

04:26 When i solve for n, i find that n is equal to two, making my order two with respect to b.

04:38 Now that i have that, i can go ahead and find my final order for c.

04:45 So for this one, i'm going to be using experiments 1 and 4, and i'm going to be putting experiment 1 over experiment 4.

04:59 When i do that, i'm going to leave this part blank for a second so i can explain how to find the rates.

05:08 So i'm going to go ahead and just plug in my concentrations first for a, b, and c.

05:21 That's experiment 1, and then we have experiment 2 over here.

05:39 Okay, so when i find my rates, i'm going to use the table that they provided us and just look at the relationships between everything.

05:53 So for the first one, we know that r2 is equal to one -half r1.

06:03 So let me just make a little side note over here.

06:12 We know that r2 is equal to one -half r1...

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