To measure the rate of a chemical reaction. we must find out the change in the number of particles of a reactant that is disappearing or that of a product which is being made. We must also record the time taken for the change to take place. Let us discuss a simple example. Suppose we want to measure the rate at which hydrogen peroxide solution decomposes in the presence of a catalyst. The equation for the reaction is

2H2O_{2 }(aq) à 2H_{2}O(l) + O_{2}(g)

There are three different types of molecule in the reaction. and hence,

As we cannot count the actual number of molecules of hydrogen peroxide, water, or oxygen. Therefore we measure the concentrations of substances which are directly related to the number of molecules.

At this stage we should decide which measurements would be the easiest to make. In the present example the measurement of oxygen in terms of volume of oxygen which is collected can be easily done so it is the third equation that we should use. For a gas, the concentration is proportional to its volume at constant pressure and temperature, or its pressure at a constant volume and temperature. Thus we can put,

Rate of appearance of O_{2} = volume of O_{2} produced / time taken

The sample set of apparatus that can be used to measure the volume of oxygen collected every 10 s or so is given in Fig. 20.3

On plotting the volume of oxygen that is given off at regular intervals of time we get a plot as shown in Fig. 20.4.

An alternative way of producing this graph is to connect the pen of a chart recorder to the plunger of the gas syringe. The pen will then produce the graph automatically.

On analyzing the graph it is clear that, the oxygen gas is given off very quickly at the start. This is where the curve is the steepest. As time goes on, the curve becomes less steep, until at about 120 s it levels off. At this stage the reaction has stopped. We can measure the rate at which oxygen is evolved at any time by finding out how steep the curve is. Say, let us measure to rate: at time 60 s. The slope is found by drawing a tangent to the curve at time 60 s. This involves a certain amount of estimation if you do it by hand and eye. (There are more accurate methods, which make use of advanced mathematics and computers.) We have,

slope of line = 0.42 cm^{3} s^{-1}

i.e., rate of appearance of oxygen is 0.42 cm3 s-1 at time 60s.

If we measured the slope at time zero, we would have measured the initial rate of the reaction.

**WRITING EXPRESSIONS FOR REACTION RATES**

** Expression 1.** The expressions for the rates of gaseous reaction NO_{2}(g) + CO(g) à CO_{2}(g) + NO(g) are given as:

= Δ[NO2] / Δt = – Δ[CO] / Δt

= Δ[CO2] / Δt = Δ[NO] / Δt

**Expression 2.** Consider another gaseous reaction between hydrogen and iodine to give hydro-iodic acid.

H_{2}(g) + I_{2}(g) à 2HI(g)

The rate of this reaction can also be given in terms of rate of disappearance of one of the reactants H_{2} or I_{2} or in terms of the rate of formation of HI.

Now, in the present case, for every mole of H_{2} or I_{2 }reacting we get two moles of HI. Thus, the rate of formation of Hl will be twice the rate of disappearance of H_{2} or I_{2 }Now in order to avoid the difference in the two rates, i.e., to get a unique value of the reaction rate irrespective of the species selected, we divide the rate of change of concentration by the number of moles of reactant or product present in the reaction. Thus, we have

Rate of reaction r = – Δ[H2] / Δt = – Δ[I2] / 2 = 1/2 . Δ[HI] / Δt

The rate given as above is average rate of a reaction. It is calculated by dividing the total change in concentration of any one of the reactant or product by the total time taken to do so.

Average rate = change in concentration in given time / time taken

= Δx / Δt or = – ΔR / Δt or = – ΔP /Δt

**UNITS OF RATE OF EACTION**

The units of rate of reaction are concentration time^{-1} As concentration of substance is expressed in mol dm^{-3} and the time is expressed in seconds or minutes or hours the units for reaction rate, therefore, are moles d.m^{-3} sec^{-1} or moles dm^{-3} min^{-1} or moles dm^{-3} hr1.

For gaseous reactions, if the concentrations of reactants and products are given in terms of partial pressures, therefore, the units of rate of reaction will be atm min^{-1} or atm sec^{-1} or atm hr1 (Pressure is expressed in atmosphere).

**INSTANTANEOUS RATE AND ITS DETERMINATION**

The instantaneous rate of a reaction is defined as the decrease in concentration of any one of the reactants or increase in concentration of any one of the product at a particular instant of time for a given temperature. Mathematically, instantaneous rate may be expressed by the expression dx / dt or – d[R] /dt or + d[p] / dt instead of Δx / Δt or – Δ[R] / Δt or + Δ[P] / Δt . Here , dt is the infinitesimally small time interval and dx is the change in concentration of any of the species in time interval dt. Thus, instantaneous rate is given by r _{inst} = – Δ[R] / Δt = Δ[P] / Δt

**CALCULATION OF INSTANTANEOUS RATE**

** **The instantaneous rate can also be calculated from the plot of concentration vs time. Suppose the rate of reaction is to be determined at time interval ‘t’ (marked in the figure). A straight line is then drawn from this point to cut the curve at point P. The slope of line gives the instantaneous rate of the reaction. The calculation of both average as well as instantaneous rate is shown in Figs. 20.6 and 20.7.

(i) Instantaneous rate at time t = r _{inst} = OA / OB= slope

(ii) Average rate at time t r _{av} = Δx / Δt = C2 – C1 / t2 – t1

** Fig.20.7**