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Question: For the decomposition reaction: \({N_2}{O_{4(g)}} \to 2N{O_{2(g)}}\) ; the initial pressure of \({N_...

For the decomposition reaction: N2O4(g)→2NO2(g){N_2}{O_{4(g)}} \to 2N{O_{2(g)}} ; the initial pressure of N2O4{N_2}{O_4} falls from 0.46atm0.46atm to 0.28atm0.28atm in 30 minute. What is the rate of appearance of NO2N{O_2} ?
A. 12×102atm.min⁡−112 \times {10^2}atm.{\min ^{ - 1}}
B. 1.2×102atm.min⁡−11.2 \times {10^2}atm.{\min ^{ - 1}}
C. 1.2×10−2atm.min⁡−11.2 \times {10^{ - 2}}atm.{\min ^{ - 1}}
D. 1.8×10−1atm.min⁡−11.8 \times {10^{ - 1}}atm.{\min ^{ - 1}}

Explanation

Solution

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→cC+dDaA + bB \to cC + dD
The rate of the reaction for the above reaction can be written as follows:
−1ad[A]dt=−1bd[B]dt=1cd[C]dt=1dd[D]dt- \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,da,b,c,d are the coefficients of the reactants and products, respectively.
As per the question, the reaction given is as follows:
N2O4(g)→2NO2(g){N_2}{O_{4(g)}} \to 2N{O_{2(g)}}
Rate of reaction = −d[N2O4]dt=12d[NO2]dt - \dfrac{{d[{N_2}{O_4}]}}{{dt}} = \dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}}
We know from the ideal gas equation, PV=nRTPV = nRT
⇒P=(nV)RT\Rightarrow P = \left( {\dfrac{n}{V}} \right)RT
⇒P=CRT\Rightarrow P = CRT (where C=C = concentration)
The rate of appearance of NO2N{O_2} = 12d[NO2]dt\dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}} … (i)
The rate of appearance of NO2N{O_2} can also be written as = −(0.28−0.46)atm30min⁡ - \dfrac{{(0.28 - 0.46)atm}}{{30\min }}….(ii)
Thus, equating equations (i) and (ii), we have:
⇒12d[NO2]dt=\Rightarrow \dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}} = −(0.28−0.46)atm30min⁡ - \dfrac{{(0.28 - 0.46)atm}}{{30\min }}
Thus, d[NO2]dt=2×6×10−3atm.min⁡−1\dfrac{{d[N{O_2}]}}{{dt}} = 2 \times 6 \times {10^{ - 3}}atm.{\min ^{ - 1}}
⇒d[NO2]dt=1.2×10−2atm.min⁡−1\Rightarrow \dfrac{{d[N{O_2}]}}{{dt}} = 1.2 \times {10^{ - 2}}atm.{\min ^{ - 1}}
Thus, the correct option is C. 1.2×10−2atm.min⁡−11.2 \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.