Question

The decomposition of an aqueous solution of ammonium nitrate was studied. The volume of nitrogen gas collected at different intervals of time was follows:

Time (minutes)	$ 10 $	$ 15 $	$ 20 $	∞
Vol. of $ {N_2} $ (mL)	$ 6.25 $	$ 9.00 $	$ 11.40 $	$ 35.05 $

From the above data prove that reaction is of the first order.

Answer 1

Hint: As we all know that order of a reaction is the sum of the power of concentration and pressure terms raised in the rate law expression. It may be 0.1.2,3 and even fractional or negative and rate of reaction is the rate of change of concentration of any one of reactants or product per unit time.

Formula used: $ K = \dfrac{1}{t}In\dfrac{{{V_\infty }}}{{{V_\infty } - {V_t}}} $

Complete answer:
We know that, $ Rate = {[A]^a}{[B]^b} $ where, K is the rate constant and [A] and [B] are the concentrations of the reactants and $ a $ and $ b $ are the power raised to concentrations which we call here the order of reaction.

Rate of reaction of first order is proportional to the first power of the concentration of reactant and in terms of time and volume we can write it as:
 $ K = \dfrac{1}{t}In\dfrac{{{V_\infty }}}{{{V_\infty } - {V_t}}} $
We are given in the question the volume of nitrogen at infinity: $ {V_\infty } = 35.05 $. And we also know that the volume of rate constant should be equal at all intervals of time for the first order reaction to be feasible. So let us calculate the value of rate constant at different intervals of time as follows:
When time is given $ 10 $ minutes then $ {V_\infty } - {V_t} = 35.05 - 6.25 = 28.80 $, so the value of rate constant will be:
 $ \Rightarrow K = \dfrac{1}{{10}}In\dfrac{{35.05}}{{28.80}} = 0.019{\min ^{ - 1}} $
Similarly, at $ 15 $ minutes, $ {V_\infty } - {V_t} = 35.05 - 9.00 = 26.05 $, so the value of rate constant will be:
 $ \Rightarrow K = \dfrac{1}{{15}}In\dfrac{{35.05}}{{26.05}} = 0.019{\min ^{ - 1}} $.
When time is $ 20 $ minutes then $ {V_\infty } - {V_t} = 35.05 - 11.40 = 23.65 $, so the rate constant for this time will be:
 $ \Rightarrow K = \dfrac{1}{{20}}In\dfrac{{35.05}}{{23.65}} = 0.019{\min ^{ - 1}} $
And finally when the time is $ 25 $ minutes then the volume is $ {V_\infty } - {V_t} = 35.05 - 13.65 = 21.40 $ and the rate constant will be:
 $ \Rightarrow K = \dfrac{1}{{25}}In\dfrac{{35.05}}{{21.40}} = 0.019{\min ^{ - 1}} $
As we can see that the value of the rate constant remains the same at all given intervals of time, it proves that order of the reaction is first order.

Note:
Always remember that it is not necessary that the concentration of every reactant changes with passage of time, therefore only that reactant whose concentration has changed with passage of time will be involved in the rate expression. Hence, if any reactant is present in excess then order of reaction with respect to that reactant is zero order. The rate of reaction as proved by Arrhenius equation is inversely proportional to the temperature, it decreases when temperature increases and vice versa.