# IDEA: Internet Differential Equations Activities

First Order Rate Law

# First Order Reactions

## Introduction

These are characterized by the property that their rate is proportional to the amount of reactant. It follows that the differential rate law contains the amount (or concentration) of reactant and a proportionality constant (the rate constant):

### Differential Rate Law: d[A]/dt = -k [A]

Mathematicians call equations that contain the first derivative but no higher derivatives first order differential equations. Chemists call the equation d[A]/dt = -k[A] a first order rate law because the rate is proportional to the first power of [A].  Integration of this ordinary differential equation is elementary, giving:

### Integrated Rate Law: [A] = [A]0 exp(-k t)

A common way for a chemist to discover that a reaction follows first order kinetics is to plot the measured concentration versus the time on a semi-log plot. Namely, the concentration versus time data are fit to the following equation:

### Data Analysis: ln([A]) = ln([A]0) - k t.

A plot of ln([A]) versus t is a straight line with slope -k. Alternatively, a plot of rate versus [A] is a straight line with slope -k. From experimental data the rate constant can be found from the slope of the appropriate plot.

### Software tools for first order reactions

Computer software tools can be used to solve chemical kinetics problems. In first 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 first order kinetics data analysis.

Another tool to solve chemical kinetics models is dynasys. Here is a dynasys file for this method: first order file. In this we apply dynasys numerical integration engine to solving the elementary first order kinetics problem. We show that the semilog plot of concentration versus time is linear.

## Exercises

Problem 1: Unstable atomic nuclei may decay by emitting particles that are detected with special counters. Alpha, beta, and gamma emission are common types of radioactivity. In beta decay the emitted particles are electrons; in alpha decay they are helium nuclei, and in gamma decay they are high energy photons. Counters can be sensitive to either a-, b-, or g-ray particles. The rubidium isotope 37Rb87 decays by beta emission to 38Sr87, a stable strontium nucleus:
37Rb87 ---> 38Sr87 + b.
From the following experimental data, calculate (a) the rate constant and (b) the half-life of 37Rb87 . From a 1.00 g sample of RbCl which is 27.85% 37Rb87, an activity of 478 beta counts per second was found. The molecular weight  of RbCl is 120.9 g mole-1.

• Problem 2: The inversion of sucrose according to the reaction C12H22O11 + H2O ---> 2C6H12O6, was observed at 25C and the experimental times and concentrations are given below. The initial concentration of sucrose was 1.0023 moles per liter.
•  time, min 0 30 60 90 130 sucrose inverted, moles per liter 0 0.1001 0.1946 0.277 0.3726

Using the graph below, verify the reaction is first order, and calculate the rate constant. Problem 3: The decomposition reaction SO2Cl2(g) ---> SO2(g) + Cl2(g) is a first order reaction with rate constant k=2.2 x 10-5 sec-1 at 320C. What percent of SO2Cl2 is decomposed at 320C after 90 minutes?

Problem 4: Fales and Morrell [J. Am. Chem. Soc. 44, 2071 (1922)] measured the inversion of sucrose in the presence of hydrochloric acid. Their approach was to measure the angle of rotation of polarized light passed through the sucrose solution.    They obtained the following data. Plot the logarithm of (
a(t)-a(inf))/(a(0)-a(inf)) versus t to find the rate constant of this first order reaction.

 time/sec a, angle of inversion, degrees 0 11.20 1035 10.35 3113 8.87 4857 7.64 9231 5.19 12834 3.61 18520 1.60 26320 -0.16 32640 -1.10 76969 -3.26 inf -3.37

## 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.