In a reaction A + B arrow C$\mathrm{A} + \mathrm{B} \rightarrow \mathrm{C}$, initial concentrations of A$\mathrm{A}$ and B$\mathrm{B}$ are related as [A]₀ = 8[B]₀$[\mathrm{A}]_0 = 8[\mathrm{B}]_0$. The half lives of A$\mathrm{A}$ and B$\mathrm{B}$ are 10~min$10\mathrm{~min}$ and 40~min$40\mathrm{~min}$ respectively. If they start to disappear at the same time, both following first order kinetics, after how much time will the concentration of both the reactants be same?

Related Formula

From the Arrhenius equation, rate constants vary exponentially with temperature:

When rate constants combine multiplicatively or via roots, their corresponding activation energies combine linearly.

Core Logic

Given the overall rate constant expression:

Substitute the Arrhenius expression (kᵢ = Aᵢ · e^-Eₐᵢ/RT$k_i = A_i \cdot e^{-E_{ai}/RT}$) for each rate constant:

Equating the exponential terms yields the linear relationship for the overall activation energy (Eₐ$E_a$):

Substitute the given activation energy values (Eₐ₁ = 60, Eₐ₂ = 30, Eₐ₃ = 10 kJ/mol$E_{a1} = 60, E_{a2} = 30, E_{a3} = 10\text{ kJ/mol}$):

The overall activation energy is 20 kJ/mol$20\text{ kJ/mol}$.

Pattern Recognition

Shortcut: Convert the rate constant algebraic expression directly into an activation energy formula by swapping k$k$ for Eₐ$E_a$, turning multiplications into additions, divisions into subtractions, and powers into multipliers. Here, k = (k₁ k₃ / k₂)1/2$k = (k_1 k_3 / k_2)^{1/2}$ translates directly to Eₐ = (1)/(2)(Eₐ₁ + Eₐ₃ - Eₐ₂)$E_a = \frac{1}{2}(E_{a1} + E_{a3} - E_{a2})$.

Chapter Mix

Class 12 Chemistry: Chemical Kinetics

Core Logic

Let the initial pressure of the reactant N₂O₅$\mathrm{N}_2\mathrm{O}_{5}$ be P₀$P_0$.

Setting up the stoichiometric reaction table:

arraylcccc & N₂O5(g) & arrow & 2NO2(g) & + & (1)/(2)O2(g) Initially (t = 0): & P₀ & & 0 & & 0 At time t: & P₀ - x & & 2x & & (x)/(2) array$$\begin{array}{lcccc} & \mathrm{N}_2\mathrm{O}_{5(\mathrm{g})} & \rightarrow & 2\mathrm{NO}_{2(\mathrm{g})} & + & \frac{1}{2}\mathrm{O}_{2(\mathrm{g})} \\ \text{Initially } (t = 0): & P_0 & & 0 & & 0 \\ \text{At time } t: & P_0 - x & & 2x & & \frac{x}{2} \end{array}$$

The total pressure of the gaseous mixture at any time t$t$ is given by:

When 50%$50\%$ of the reaction is completed, the change in the reactant's pressure is:

Substituting the value of x$x$ into the total pressure expression:

Pattern Recognition

Track the change in the total pressure carefully using stoichiometric coefficients. For a 50%$50\%$ completion step, substitute the fractional equivalent (x = 0.5 P₀$x = 0.5 P_0$) directly into your total pressure expression.

Related Formula

For an elementary reaction, the rate law corresponds directly to its stoichiometry:

R = K[A]¹[B]¹$R = K[A]^1[B]^1$

Concentration (C$C$) is inversely proportional to volume (V$V$):

Core Logic

Initial rate expression:

When volume is reduced to (1)/(3)V$\frac{1}{3}V$, the new concentration becomes 3$3$ times the initial concentration:

Step 1: Calculating the Value of x

Comparing the two rates:

R₂ = 9R₁$R_2 = 9R_1$

Therefore, the value of x$x$ is 9$9$.

Pattern Recognition

For a second-order overall elementary reaction (1+1=2$1+1=2$), reducing the volume by a factor of n$n$ increases the rate by a factor of n²$n^2$. Here n=3$n=3$, so the rate increases by 3² = 9$3^2 = 9$ times.

Chapter Mix

Class 12 Chemistry: Chemical Kinetics

Related Formula

Exponential growth equation model:

N = N₀ eKt$N = N_0 e^{Kt}$

Normalized configuration formula:

Core Logic

Radioactive decay follows a decreasing exponential path (N = N₀ e-λ t$N = N_0 e^{-\lambda t}$).

Conversely, cell culture growth functions via an increasing exponential pattern because the rate of growth is directly proportional to the current population size (dN/dt = KN$dN/dt = KN$). This results in an exponential curve that starts at (N)/(N₀) = 1$\frac{N}{N_0} = 1$ when t = 0$t = 0$ and curves sharply upward over time.

Step 1: Finding the Matching Curve

Plotting (N)/(N₀)$\frac{N}{N_0}$ against time shows an upward-clinging exponential profile starting from 1$1$, which perfectly matches the curve in option (4).

Pattern Recognition

The expression eKt$e^{Kt}$ dictates an exponential increase. Ensure the curve starts from a non-zero value (1$1$) at t=0$t=0$, as (N₀)/(N₀) = 1$\frac{N_0}{N_0} = 1$, rather than starting from the origin (0$0$).

Chapter Mix

Class 12 Chemistry: Chemical Kinetics

Related Formula

Core Logic

The slowest elementary step controls the net kinetic pathway rate law :

Since [A]$[\mathrm{A}]$ behaves as a transient intermediate species, replace it using the prior fast equilibrium step :

Substitute [A]$[\mathrm{A}]$ back into Equation (1) :

Sum of powers determining overall order:

Hence, Option (1) is correct.

Pattern Recognition

Whenever a fast initial step dissociates a molecule into matching independent halves, it always injects a fractional order component of 0.5$0.5$ relative to that parent species.