# IDEA: Internet Differential Equations Activities

Second Order Rate Law

# Second Order Reactions

### Introduction

Second Order Reactions are characterized by the property that their rate is proportional to the product of two reactant concentrations (or the square of one concentration). Suppose that A ---> products is second order in A, or suppose that  A + B ---> products is first order in A and also first order in B. Then the differential rate laws in these two cases are given by Differential Rate Laws:

### d[A]/dt = -k [A]2  (for 2A ---> products)                                or dx/dt = -k [A][B]  (for A + B ---> products)

In mathematical language, these are first order differential equations because they contain the first derivative and no higher derivatives. A chemist calls them second order rate laws because the rate is proportional to the product of two concentrations. By elementary integration of these differential equations Integrated Rate Laws can be obtained:

### 1/[A] - 1/[A]0 = k t  (for 2A ---> products)                             or (1/(a-b)) [ln((a-x)/(b-x))-ln(a/b)] = k t  (for A + B ---> products)

where a and b are the initial concentrations of A and B (assuming a not equal to b), and x is the extent of reaction at time t. Note that the latter can also be written:

#### (a-x)/(b-x) = (a/b)exp[(a-b)kt].

A common way for a chemist to discover that a reaction follows second order kinetics is to plot 1/[A] versus the time in the former case, or ln(b(a-x)/a(b-x) versus t in the latter case.

### Data Analysis: 1/[A] = 1/[A]0 + k t

A plot of 1/[A] versus t is a straight line with slope k.

### Software tools for second order reactions

Computer software tools can be used to solve chemical kinetics problems. In second order reactions it is often useful to plot and fit a straight line to data. One tool for this is the "slope(x,y)" command in the product MathCad. Here is a mathcad file that can serve as template for second order kinetics data analysis.

### Exercises

• Problem 1: Ammonium cyanate, NH4CNO, in water solution gradually isomerizes to urea, H2NCONH2 according to the reaction: NH4CNO ---> H2NCONH2 . A solution was prepared by dissolving 22.9 g of ammonium cyanate in enough water to make 1.00 liter of solution. After times t had elapsed, portions of the solution were analysed and converted into the mass of urea that had formed in the entire solution. The results are tabulated here.
•  t/min 0 20 50 65 150 m(urea)/g 0 7 12.1 13.8 17.7

Using the graph below verify that this is a second order reaction and calculate the rate constant. Problem 2: A certain chemical reaction follows the stoichiometric equation

A + 2B ---> 2Z.
Measured rates of formation of the product, Z, are shown for several concentrations of reactants, A and B:

 [A]/mole liter-1 [B]/mole liter-1 rate/mole liter-1 sec-1 2.5 x 10-2 3.3 x 10-3 1.0 x 10-2 5.0 x 10-2 6.6 x 10-3 4.0 x 10-2 5.0 x 10-2 1.32 x 10-2 8.0 x 10-2

Assuming a differential rate law of the form

rate = k [A]a [B]b,

what is the value of a (the order of reaction with respect to A), what is b (the order of reaction with respect to B) and what is the value of  k (the rate constant)?

Problem 3: Solutions of A=H3COC6H4CNO in carbon tetrachloride dimerize slowly as shown by the following data

 t/hr 0 3.5 7 10.5 14 17.5 21 24.5 28 31.5 35 [A]/mole/liter 0.995 0.745 0.595 0.494 0.424 0.37 0.33 0.295 0.27 0.247 0.229

Determine the order of the reaction and find the rate constant.

### Glossary of Terms

• Stoichiometry determines the molar ratios of reactants and products in an overall chemical reaction. We express the stoichiometry as a balanced chemical equation. For kinetics it is convient to write this as products minus reactants: npP + nqQ - naA - nbB (instead of the conventional equation naA + nbB ---> npP + nqQ). This indicates that na and nb moles of reactants A and B, resp., produce np and nq moles of products P and Q.
• The rate of a chemical reaction is defined in such a way that it is independent of which reactant or product is monitored. We define the rate, v, of a reaction to be v = (1/ng) d[G]/dt where ng is the signed (positive for products, negative for reactants) stoichiometric coefficient of species G in the reaction. Namely, v = (-1/na) d[A]/dt = (1/np) d[P]/dt, etc.
• It is convenient to refer to the extent of reaction. As the reactants are sonsumed and the products are produced, their concentrations change. If the initial concentrations of A, B, P and Q are [A], [B], [P] and [Q], resp., then the extent of reaction is defined: x = -([A]-[A]0)/na = -([B] - [B]0)/nb = ([P]-[P]0)/np = ([Q]-[Q]0)/nq. Alternately, each species concentration is a function of the extent of reaction: [A] = [A]0 - nax, etc.
• Many reactions follow elementary differential rate laws such as v = k f([A], [B], ...) where f([A], [B], ...) is a function of the concentrations of reactants and products. That is, the rate varies as the concentrations change. A proportionality constant, k, is called the rate constant of the reaction.
• When the rate law has the special form of a product (or quotient) of powers, f([A], [B], ...) = [A]a [B]b [P]p [Q]q then a is the order of the reaction with respect to A, b is the order w.r.t. B, etc. Note that order may be positive, negative, integer, or non-integer. Further, the sum a + b + p + q is the overall order of the reaction rate law.
• NOTE: there is no necessary relation between orders and stoichiometric coefficients. That is, a might differ from na.
• Reaction rate constants are usually temperature dependent; the rate of a reaction usually increases as the temperature rises. The temperature dependence often follows Arrhenius' equation: k(T) = A exp(-Ea/RT) where T is the absolute temperature, R the universal gas constant, Ea is the activation energy (specific to each reaction), and A is the "pre-exponential" or "frequency" or "entropy" factor.
• One objective of chemical kinetics is to solve the differential rate law d[G]/dt = k f([A], [B], ...), and thereby express each species concentration as a function of time: [G](t). Since solution requires integration, we call it the integrated rate law.
• A reaction mechanism is a set of steps at the molecular level. Each step involves combinations or re-arrangements of individual molecular species. The steps in combination describe the path or route that reactant molecules follow to reach the product molecules. The result of all steps is to produce the overall balanced stoichiometric chemical equation for reactants producing products. With the advent of HTML5, Javascript is now ready for prime time for mathematical applications. There are new Javascript demos illustrating how we might use interactive web objects to help students learn Calculus.

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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.