#### Fundamental Concepts of Rate Laws, Chemistry tutorial

Introduction:

Chemical kinetics is basically the study of rates and procedures of chemical reactions. The rate of a reaction based on many factors like the concentration of the reactants, temperature, catalyst and so on.

Some Fundamental Concepts:

In this concept, the given shall apply:

• The phrase 'rate of reaction' signifies instantaneous reaction rate.
• The sign 'Δ' is to be read as delta. It represents change in property.

=> Convention for expressing reaction rates:

In order to get a single value for the reaction rate, it is essential to divide the rate of consumption of a reactant or the rate of formation of a product via the stoichiometric coefficient of the corresponding species. To describe this, let us taken the reaction:

aA + bB → cC + dD

In the above reaction, A and B are reactants whereas C and D are the products; a, b, c and d are the stoichiometric coefficients. The reaction rate is associated to the rates of consumption of the reactants and the rates of formation of the products as shown:

Reaction rate:

= (1/a)[-d[A]/dt] = (1/b)[-d[B]/dt]

= (1/c)[-d[C]/dt] = (1/d)[-d[D]/dt]

By utilizing this general equation, we observe:

- The number prior the formula of a substance in the balanced equation is its stoichiometric coefficient

Rate of Reaction:

The rate of reaction or the velocity of reaction at a particular time is stated as the decrease in the concentration of a reactant or the increase in the concentration of the product per unit time. The rate of reaction at a particular time is as well termed as instantaneous rate of reaction; it can be usually stated as the rate of change of concentration of a particular species at a specified instant. Whereas specifying the reaction rate, it is suitable to illustrate the component with respect to which it is stated.

Let us taken a simple reaction:

A → B

With respect to the reaction Stoichiometry, one molecule of B is made or prepared for each and every molecule of 'A' consumed. The reaction rate can be specified in the given manners:

We can calculate the concentration of the reactant 'A' at different time intervals. From such values, we can identify the decrease in concentration of 'A' with respect to time at any specified instant.

The reaction rate therefore obtained is the rate of consumption of A.

Rate of consumption of A = Decrease in the concentration of A/Change in time

= Δ[A]/Δt

=> Δ[A] signifies change in the concentration of A and -Δ[A] signifies the decrease  in the concentration of A. As writing a rate expression with respect to a reactant, there is a preceding negative sign (as it is usual to state the rate of a reaction as a positive quantity).

=> We can assess the concentration of the product B at different time intervals. From such values, we can arrive at the rate of formation of B at any specified instant.

Rate of formation of B = Increase in the concentration of B/Change in time

= -Δ[B]/Δt

The rates of consumption of reactants and the rates of formation of products are associated via their stoichiometric coefficients.

For illustration, let's take the decomposition of NO2.

2NO2 (g) → 2NO (g) + O2 (g)

We can represent the relationship between the rates of consumption of NO2 and the rates of formation of NO and O2 as:

1/2 (Rate of consumption of NO2) = 1/2 (Rate of formation of NO)

= Rate of formation of O2

Reaction rate = (1/2) x (-d/dt)[NO2] = (1/2) (d/dt) [NO] = (d/dt)[O2]

We will remove from the above relationship that for each and every two molecules of NO2 consumed; two molecules of NO and one molecule of O2 are formed. In another words, the reaction rate is equivalent to:

• Half the rate of consumption of NO2
• Half the rate of formation of NO
• The rate of formation of O2

Calculation of Reaction Rate:

We might be curious to recognize how the reaction rates are computed. Take for illustration the given reaction:

2NO2 (g) → 2NO (g) + O2 (g)

In the figure shown below, we can observe the graph of concentration (c) against time (t) plots for NO2, NO and O2 from the values given in the table shown below. In such figures, the graphical process of computing the reaction rates for the consumption of NO2 and for the formation of NO and NO2 are described. The reaction rate at any specific instant is obtained via computing the slope of a line tangent to the curve at that point.

Table: Concentration of NO2, NO and O2 at various time interval at 673 K

Time(s)    [NO2](M)     [NO](M)     [O2](M)

0              0.00100           0                0

50            0.0079         0.0021        0.0011

100          0.0065         0.0035        0.0018

150          0.0055         0.0045        0.0023

200          0.0048         0.0052        0.0026

250          0.0043         0.0057        0.0029

300          0.0038         0.0062        0.0031

350          0.0034         0.0066        0.0033 Fig: Concentration of NO2, NO and O2

Note:

a = Concentration against time plot for NO2

b = Concentration against time plot for NO

c = Concentration against time plot for O2

By the slope of the tangent time drawn (that is, corresponding to a specific time) to the concentration (c) against time (t) curve for a component, we can acquire the rate of the reaction.

Rate of reaction:

= - (Slope of tangent to the c against t curve for the reactant)/Stoichiometric coefficient of the reactant

= (Slope of tangent to the c against t curve for the product)/Stoichiometric coefficient of the product

The concentration of components are provided in molarity (M) unit where 1M = 1 mol/dm3

Rate of consumption of NO2 = - Slope of the tangent line at t = 200s at t = 200s

= - (-1.31 × 10-5) Ms-1

= 1.31 × l0-5 Ms-1

Reaction rate = (1/2) (Rate of consumption of NO2) = (1/2) (l.31 × l0-5) Ms-1

= 6.55 x 10-6 Ms-1

=> Concentration against time plot for NO; it will be noted that the rising nature of the curve that is the characteristic of concentration against time plot for a product.

Rate of formation of NO = Slope of the tangent line at t = 200s at t = 200s

= - 1.30 × 10-5 Ms-1

Reaction rate = 1/2 (Rate of formation of NO)

= 1/2 x 1.30 x 10-5 Ms-1

= 6.50 x 10-6 Ms-1

=> Concentration against time plot for O2; it will be noted again the rising curve. Compared to the curve for NO, the curve for O2 increases slowly.

Rate of formation of O2 = Slope of the tangent line at t = 200s at t = 200s

= 6.25 × l0-4 Ms-1

Reaction rate = Rate of formation of O2

= 6.25 × 10-6 Ms-1

As well note that for the curves (a) and (b), the tangents are not pointed.

From the slope values at t = 200 s, we can observe that the given relationship is almost correct.

Reaction rate = 1/2 (Rate of consumption NO2) = 1/2 (Rate of formation of NO) = Rate of formation of O2

In this, we are primarily interested in the concentration against time plots for the reactants. In another words, we are studying the reactions under conditions where the rate of the forward reaction is important however the reverse reaction rate is low. This is made probable, if we study the reaction to a point where the product amounts are not high. For illustration, in the decomposition of NO2, there could be a decrease in the concentration of NO2 to a specific time. Afterwards, adequate nitric oxide and oxygen are made and the reverse reaction as well could occur leading to the formation of NO2. In order to simplify the condition, it is better to study the reaction rates before significant amounts of products are made up. In common, the rates of reactions are complex functions of the concentrations of the reactants and the products at a specified temperature. Though, there are some reactions in which the rates are proportional to simple powers of the concentrations of the reactants. We shall be mainly concerned by this class of reactions.

Example: Decomposition of N2O5

The decomposition of N2O5 in the gas phase was studied at 323 K

2N2O5 (g) → 4NO2 (g) + O2 (g)

The instantaneous rates of this reaction computed from [N2O5] against time plot (identical to the figure above) are provided in the table shown below.

Table: Rates for the Decomposition of N2O5 at 323 K

[N2O5]/M     Rate/Ms-1       Rate S [N2O5]‾

(i)                  (ii)                     (iii)

0.300        2.73 × 10-4            9.1 × 10-4

0.150        1.37 × l0-4             9.1 × 10-4

0.100        9.10 × l0-3             9.1 × 10-4

From columns (i) and (ii), we can observe that the rate for the decomposition of N2O5 reduces with the decrease in the concentration of N2O5.  Moreover, column (iii) gives the ratio of the rate to the concentration of N2O5. In all the three cases, it is a constant. This exhibits that the rate is directly proportional to the concentration of N2O5.

Rate/[N2O5] = K

Therefore, rate = k [N2O5]

Here 'k' is the proportionality constant.

Example: Decomposition of Hydrogen Iodide

The decomposition of hydrogen iodide was given at a constant temperature

2HI (g) → H2 (g) + I2 (g)

The instantaneous rates of this reaction were computed by using the [HI] against time plot similar to that of the figure shown above. These values are provided in the table shown below:

Table: Rates for the decomposition of HI

[HI]/M         Rate/Ms-1         Rate [HI]        Rate Ms-1 [HI]2

(i)                          (ii)                   (iii)                   (iv)

3.00 × 10-2    3.60 × 10-5     1.2 × 10-3       4.00 × 10-2

2.00 × 10-2    1.60 × 10-3     8.0 × 10-4       4.00 × 10-2

1.50 × 10-2    9.01 × 10-6     6.0 × 10-4       4.00 × 10-2

From the table shown above, we can observe that the rate of decomposition of HI reduces with decrease in the concentration of HI, as in the case of the decomposition of N2O5. Moreover, it is evident from column (iii) that rate/[HI] is not a constant. However for column (iv), rate/[HI]2 is a constant.

From the table illustrated above, it is obvious that,

Rate/[HI]2 = k

Therefore, rate = k[HI]2

Here, 'k' is the proportionality constant

For numerous chemical reactions, the relationship between the reaction rate and the concentration can be deduced in a simple manner as in rate = k [N2O5] or rate = k[HI]2.

The Rate Law and the Rate Constant:

The relationship deduced as in rate = k [N2O5] or rate = k[HI]2 is termed as the rate law. A rate law is an equation deducing the relationship between the instantaneous reaction rate and the concentrations of the reactants in a particular reaction.

The rate law for a simple reaction having one reactant is provided below:

Reaction rate = k [Reactant]n

Here, 'k' is termed as the rate constant or rate coefficient or the specific rate or the reaction.

Therefore by definition, the rate constant is independent of concentration; however it might base on other factors. In this equation, 'n' refers to the order of the reaction. The order with respect to a component is the power to which the concentration of that component is increased in the rate law.

Comparing the equation Reaction rate = k [Reactant]n with equation rate = k [N2O5] and rate = k[HI]2 we can conclude that:

a) n = 1 in equation rate = k [N2O5];  that is; decomposition of N2O5 is a first order reaction. The importance of this statement is that the reaction rate is proportional to the first power of concentration of N2O5

That is, Rate = k [N2O5]1

Here, 'k' is the first order rate constant

From the equation Reaction rate = k [Reactant]n, it can be observed that if [reactant] = 1 then k = rate. For this reason, 'k' is termed as the specific rate.

b) n = 2 for the decomposition of HI; that is, the decomposition of HI is a second order reaction. Again, this means that the decomposition rate of HI is proportional to the second power or square of the concentration of HI.

That is, Rate = k [HI]2

Here 'k' is the second order rate constant.

Order of Reaction and Stoichiometry:

The rate laws and as well the order of the reaction are generally found out experimentally; these can't be predicted from the Stoichiometry of the reaction. The Stoichiometry of reaction provides the relationship between the amounts of the reactants and the amount of product. The Stoichiometry of a reaction should be differentiated from the order of a reaction. Let us take the given illustrations:

Example:

The gas-phase decomposition of N2O5 results NO2 and O2 at a specific temperature.

2N2O5 (g) → 4 NO2 (g) + O2 (g)

The experimentally examines rate law for the reaction rate = k[N2O5]

Can you remark on the order and the Stoichiometry of the reaction?

Solution:

This can be observing that the stoichiometric coefficient of N2O5 is 2 while the order of reaction is 1.

Example:

The balanced equation for the decomposition of nitrous oxide is represented below:

2N2O5 (g) → 2NO2 (g) + O2 (g)

The rate law is,

Rate = k [N2O5

Remark on the order of the reaction and Stoichiometry.

Solution:

Again the stoichiometric coefficient of N2O5 is 2 while the order of reaction is 1.

In the above two illustrations, the order of reaction and the Stoichiometry are not similar, however there are cases where the order and stoichiometric coefficient are similar. One of such cases can be seen in the given reaction:

2HI (g) → H2 (g) + I2 (g)

Rate = k [HI]2

In the decomposition of HI, the order of reaction is two. The stoichiometric coefficient of HI is as well 2.

From the above illustrations, it can be expressed that the stoichiometric coefficient and the order of the reaction need not be always similar. We must bear in mind the given points whereas arriving at a rate law.

a) In case of simple reactions, the concentrations of reactants appear in rate law; however the concentrations of the products don't appear in the rate law as the rate measurements are done beneath the conditions where the reverse reaction rate is negligibly low.

b) The order of reaction should be found out experimentally.

c) The order of reaction need not be similar by the stoichiometric coefficient of the reactant.

So far we have taken the reactions comprising only one reactant. In case of reactions comprising numerous reactants, the rate of reaction might base on the concentrations of more than one reactant. In certain cases, we can compute the order of the reaction with respect to the individual reactant and also the overall order. The overall order is the sum of the powers to which the individual concentrations are increased in the rate law.

Usually, for a given reaction;

A + B + C → products

If the rate law is experimentally determined to be,

Rate = k[A]m [B]n [C]p

Then, the total order of the reaction = m + n + p.

For illustration in the given reaction,

BrO3 (aq) + 5 Br- (aq) + 6H+ (aq) → 3 Br2 (aq) + 3 H2O (l)

Rate = k [BrO3] [Br-] [H+]2

The total rate of the reaction is four, being first order in BrO3-, first order in Br- second order in H+

The rate laws illustrated so far are termed as differential rate laws. Such rate laws illustrate the dependence of reaction rate on concentration. From such differential rate laws, we can get the integrated rate laws via integration. Such integrated rate laws assist us in relating the concentration of a substance to time. In other words, by using the integrated rate laws, we can compute the concentration of a substance at any particular time.

For the BrO3- - Br- - H+ reaction let us compare the order of reaction and the stoichiometric coefficient for each and every reactant.

BrO3-    Br-     H+

Order               1          1      1

Stoichiometric

Coefficient       1          5      6

It might be expressed that the stoichiometric coefficients and the respective orders of reaction are not similar all through.

Experimental Methods of Rate Studies:

Most of the chemical and physical methods are available for studying the reaction rates. Some of them are described below:

A) Volume or Pressure Measurement

Whenever one or more of the components are gases, the reaction rate can be obeyed via measuring the volume or pressure change. The partial pressures of species are to be computed by employing the reaction Stoichiometry.

Spectrophotometers comprises of arrangements for the generation of almost monochromatic radiation in visible and ultraviolet regions and as well for the measurement of radiation transmitted via the absorbing substance.

These days, most of the sophisticated instruments like nuclear magnetic resonance spectrometer, mass spectrometer and so on, are employed in reaction kinetics.

ii) Titrimetry:

By using acid-base or oxidation-reduction titrations, the reaction course can be carried out if at least one of the components in the reaction is an acid, a base, an oxidising agent or a reducing agent.

iii) Conductometry or Potentiometry:

Whenever one or more of the ions are present or generated in the reaction, appropriate procedures can be designed based on conductivity or Potentiometric measurements.

iv) Spectrophotometry:

If a component of the reaction consists of a strong absorption band at a specific wavelength region, spectrophotometers could be employed measuring the reaction rate. Photoelectric colorimeters are cheaper instruments and are mostly helpful for reaction rate studies in the visible region.

v) Polarimetry:

If at least one of the components of a reaction is optically active, then the reaction rate can be studied from the measurements of optical rotation.

Based on the reaction under study, the concentration of a reactant or a product is carried out at different time intervals by using any of the methods illustrated above. Such values are then employed for computing the rate constant.

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