Question

For the decomposition reaction: ${N_2}{O_{4(g)}} \to 2N{O_{2(g)}}$ ; the initial pressure of ${N_2}{O_4}$ falls from $0.46atm$ to $0.28atm$ in 30 minute. What is the rate of appearance of $N{O_2}$ ?
A. $12 \times {10^2}atm.{\min ^{ - 1}}$
B. $1.2 \times {10^2}atm.{\min ^{ - 1}}$
C. $1.2 \times {10^{ - 2}}atm.{\min ^{ - 1}}$
D. $1.8 \times {10^{ - 1}}atm.{\min ^{ - 1}}$

Answer 1

Hint:The reaction rate or rate of reaction is the speed at which a chemical reaction takes place. Reaction rate is defined as the speed at which reactants are converted into products. Reaction rates can vary dramatically. The rate of a reaction is always positive. A negative sign is present to indicate that the reactant concentration is decreasing.

Complete step by step answer:
The rate of reaction involves the rate of appearance of reactants as well as the rate of appearance of the products. Suppose we have been provided with a chemical reaction given as follows:
$aA + bB \to cC + dD$
The rate of the reaction for the above reaction can be written as follows:
$ - \dfrac{1}{a}\dfrac{{d[A]}}{{dt}} = - \dfrac{1}{b}\dfrac{{d[B]}}{{dt}} = \dfrac{1}{c}\dfrac{{d[C]}}{{dt}} = \dfrac{1}{d}\dfrac{{d[D]}}{{dt}}$
The negative sign indicates the rate of disappearance of the reactants and the positive sign indicates the rate of appearance of the products. The small letters $a,b,c,d$ are the coefficients of the reactants and products, respectively.
As per the question, the reaction given is as follows:
${N_2}{O_{4(g)}} \to 2N{O_{2(g)}}$
Rate of reaction = $ - \dfrac{{d[{N_2}{O_4}]}}{{dt}} = \dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}}$
We know from the ideal gas equation, $PV = nRT$
$ \Rightarrow P = \left( {\dfrac{n}{V}} \right)RT$
$ \Rightarrow P = CRT$ (where $C = $ concentration)
The rate of appearance of $N{O_2}$ = $\dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}}$ … (i)
The rate of appearance of $N{O_2}$ can also be written as = $ - \dfrac{{(0.28 - 0.46)atm}}{{30\min }}$….(ii)
Thus, equating equations (i) and (ii), we have:
$ \Rightarrow \dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}} = $$ - \dfrac{{(0.28 - 0.46)atm}}{{30\min }}$
Thus, $\dfrac{{d[N{O_2}]}}{{dt}} = 2 \times 6 \times {10^{ - 3}}atm.{\min ^{ - 1}}$
$ \Rightarrow \dfrac{{d[N{O_2}]}}{{dt}} = 1.2 \times {10^{ - 2}}atm.{\min ^{ - 1}}$
Thus, the correct option is C. $1.2 \times {10^{ - 2}}atm.{\min ^{ - 1}}$.

Note:
The rate of reaction differs from the rate of increase of concentration of a product C by a constant factor (the reciprocal of its stoichiometric number) and for a reactant A by minus the reciprocal of the stoichiometric number. The stoichiometric numbers are included so that the defined rate is independent of which reactant or product species is chosen for measurement.