**First
Order Reactions**

Table of Contents:

## 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 _{37}Rb^{87}
decays by beta emission to _{38}Sr^{87}, a stable strontium
nucleus:

_{37}Rb^{87} ---> _{38}Sr^{87} + b.

From the following experimental data, calculate (a) the rate constant and
(b) the half-life of _{37}Rb^{87} . From a 1.00 g sample
of RbCl which is 27.85% _{37}Rb^{87}, 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 C

_{12}H

_{22}O

_{11}+ H

_{2}O ---> 2C

_{6}H

_{12}O

_{6}, 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 | 0.2770 | 0.3726 |

Using the graph below, verify the reaction is first order, and calculate the rate constant.

**Problem
3:** The decomposition reaction SO_{2}Cl_{2}(g)
---> SO_{2}(g) + Cl_{2}(g) is a first order reaction
with rate constant k=2.2 x 10^{-5} sec^{-1} at 320C. What
percent of SO_{2}Cl_{2} 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

_{p}P + n

_{q}Q - n

_{a}A - n

_{b}B (instead of the conventional equation n

_{a}A + n

_{b}B ---> n

_{p}P + n

_{q}Q). This indicates that n

_{a}and n

_{b}moles of reactants A and B, resp., produce n

_{p}and n

_{q}moles of products P and Q.

_{g}) d[G]/dt where n

_{g}is the signed (positive for products, negative for reactants) stoichiometric coefficient of species G in the reaction. Namely, v = (-1/n

_{a}) d[A]/dt = (1/n

_{p}) d[P]/dt, etc.

_{0})/n

_{a}= -([B] - [B]

_{0})/n

_{b}= ([P]-[P]

_{0})/n

_{p}= ([Q]-[Q]

_{0})/n

_{q}. Alternately, each species concentration is a function of the extent of reaction: [A] = [A]

_{0}- n

_{a}x, etc.

*[B]*

^{a}*[P]*

^{b}*[Q]*

^{p}*then*

^{q}*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.

*a*might differ from n

_{a}.

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.