Introduction
The reaction between peroxydisulfate ion (S2O2−8) and iodide ion (I−) in aqueous solution is an example of a clock reaction, where a sudden change in color indicates the completion of a certain reaction. The reaction can be written as:
�2�82−(��)+3�−(��)→2��42−(��)+�3−(��)S2O82−(aq)+3I−(aq)→2SO42−(aq)+I3−(aq)
The rate of this reaction depends on the concentrations of the reactants and the temperature of the solution. In this article, we will explore how the rate of disappearance of S2O2−8 is related to the rate of disappearance of I−, and how to determine the rate law and the activation energy for this reaction.
Rate Law
The rate law for a reaction expresses the relationship between the rate of the reaction and the concentrations of the reactants raised to some powers. For the reaction between S2O2−8 and I−, the general form of the rate law is:
����=�[�2�82−]�[�−]�rate=k[S2O82−]α[I−]β
where k is the rate constant, and α and β are the orders of reaction with respect to S2O2−8 and I−, respectively. The overall order of reaction is the sum of α and β.
To determine the values of α and β, we need to perform experiments with different initial concentrations of S2O2−8 and I−, and measure the time it takes for a certain amount of iodine (I2) to appear in the solution. The iodine is produced by the reaction between S2O2−8 and I−, but it is also consumed by another reaction with thiosulfate ion (S2O32−), which acts as a delaying agent:
�2(��)+2�2�32−(��)→2�−(��)+�4�62−(��)I2(aq)+2S2O32−(aq)→2I−(aq)+S4O62−(aq)
This reaction is much faster than the first one, so it keeps the concentration of iodine low until all the thiosulfate is used up. Then, the iodine accumulates rapidly and reacts with starch indicator to produce a dark blue color. The time for this color change to occur is called the clock time, and it is inversely proportional to the rate of the first reaction.
To find α and β, we can use two methods:
- The method of initial rates: This method involves comparing the initial rates of different experiments at a constant temperature, and finding how they change when one reactant concentration is varied while keeping the other constant. For example, if we vary [S2O2−8] while keeping [I−] constant, we can write:
����1����2=�[�2�82−]�[�−]��[�2�82−]�[�−]�=[�2�82−]�[�2�82−]�rate2rate1=k[S2O82−]α[I−]βk[S2O82−]α[I−]β=[S2O82−]α[S2O82−]α
Taking logarithms on both sides, we get:
log����1����2=�log[�2�82−][�2�82−]lograte2rate1=αlog[S2O82−][S2O82−]
From this equation, we can find α by plotting log(rate) versus log([S2O2−8]) and finding the slope of the line. Similarly, we can find β by varying [I−] while keeping [S2O2−8] constant.
- The integrated rate law: This method involves finding an equation that relates the concentration of a reactant to time, and fitting it to experimental data. For a first-order reaction, such as:
�→��������A→products
the integrated rate law is:
ln[�]=−��+ln[�]0ln[A]=−kt+ln[A]0
where [A] is the concentration of A at time t, [A]0 is the initial concentration of A, and k is the first-order rate constant. This equation can be rearranged as:
ln[�][�]0=−��ln[A]0[A]=−kt
which has a linear form with slope -k and intercept 0. Therefore, we can plot ln([A]/[A]0) versus t and find k from the slope.
If the reaction between S2O2−8 and I− is first-order with respect to both reactants, then we can write:
ln[�2�82−][�2�82−]0=−�1�ln[S2O82−]0[S2O82−]=−k1t
ln[�−][�−]0=−�2�ln[I−]0[I−]=−k2t
where k1 and k2 are the first-order rate constants for S2O2−8 and I−, respectively. However, these equations are not valid for this reaction, because the concentrations of S2O2−8 and I− are not independent of each other. According to the stoichiometry of the reaction, for every mole of S2O2−8 that reacts, three moles of I− also react. Therefore, we can write:
[�2�82−]=[�2�82−]0−13([�−]0−[�−])[S2O82−]=[S2O82−]0−31([I−]0−[I−])
Substituting this into the first equation, we get:
ln[�2�82−]0−13([�−]0−[�−])[�2�82−]0=−�1�ln[S2O82−]0[S2O82−]0−31([I−]0−[I−])=−k1t
This equation is not linear, but it can be linearized by applying a mathematical transformation. One possible transformation is:
1[�2�82−]−13[�−]=�13[�2�82−]0�+1[�2�82−]0−13[�−]0[S2O82−]−31[I−]1=3k1[S2O82−]0t+[S2O82−]0−31[I−]01
This equation has a linear form with slope k1[S2O2−8]0/3 and intercept 1/([S2O2−8]0 − [I−]0/3). Therefore, we can plot 1/([S2O2−8] − [I−]/3) versus t and find k1 from the slope. Similarly, we can find k2 by applying another transformation:
1[�−]−13[�2�82−]=�29[�−]0�+1[�−]0−13[�2�82−]0[I−]−31[S2O82−]1=9k2[I−]0t+[I−]0−31[S2O82−]01
Once we have k1 and k2, we can find the overall rate constant k and the orders of reaction α and β by using the following relations:
�=�1�2k=k1k2
�=��1α=k1k
�=��2β=k2k
Activation Energy
The activation energy for a reaction is the minimum amount of energy required for the reactants to overcome the energy barrier and form products. The activation energy depends on the mechanism of the reaction and the nature of the transition state. The higher the activation energy, the slower the reaction.
The activation energy can be calculated from the temperature dependence of the rate constant, according to the Arrhenius equation:
�=��−��/��k=Ae−Ea/RT
where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature. Taking logarithms on both sides, we get:
ln�=−���1�+ln�lnk=−REaT1+lnA
which has a linear form with slope −Ea/R and intercept ln A. Therefore, we can plot ln k versus 1/T and find Ea from the slope.
To measure the effect of temperature on the rate constant, we need to perform experiments with different temperatures and find the clock times for each one. Then, we can use the method of initial rates or the integrated rate law to find k for each temperature. Finally, we can plot ln k versus 1/T and find Ea from the slope.
Ionic Strength
The ionic strength of a solution is a measure of the concentration of ions in the solution. The ionic strength affects the activity coefficients of the ions, which in turn affect the rate of reaction. The activity coefficient is a correction factor that relates the actual concentration of an ion to its effective concentration in a non-ideal solution. The activity coefficient depends on the charge and size of the ion, as well as the ionic strength of the solution.
The ionic strength of a solution can be calculated by using the following formula:
