For a reaction, N₂O5(g) arrow 2NO2(g) + (1)/(2)O2(g)$\mathrm{N}_2\mathrm{O}_{5(\mathrm{g})} \rightarrow 2\mathrm{NO}_{2(\mathrm{g})} + \frac{1}{2}\mathrm{O}_{2(\mathrm{g})}$ in a constant volume container, no products were present initially. The final pressure of the system when 50%$50\%$ of reaction gets completed is

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.

Related Formula

where n = tt1/2$n = \frac{t}{t_{1/2}}$ (number of half-lives).

Alternatively, using the first-order decay formula:

where λ = 0.693t1/2$\lambda = \frac{0.693}{t_{1/2}}$.

Core Logic

The atmospheric ratio of ¹⁴C/¹²C$^{14}\mathrm{C}/^{12}\mathrm{C}$ acts as the initial activity or amount (N₀$N_0$) when the tree was alive. The current ratio in the wood represents the amount left at time t$t$ (Nₜ$N_t$). Given that Nₜ = (1)/(8) N₀$N_t = \frac{1}{8} N_0$.

Step 1: Calculate Half-lives

So, the number of half-lives passed, n = 3$n = 3$.

Step 2: Calculate Age

Pattern Recognition

Whenever the remaining fraction is a perfect power of 1/2$1/2$ (like 1/2, 1/4, 1/8, 1/16$1/2, 1/4, 1/8, 1/16$), just find the exponent n$n$ and multiply by t1/2$t_{1/2}$. Here, 1/8 = (1/2)³ arrow 3$1/8 = (1/2)^3 \rightarrow 3$ half-lives.

Chapter Mix

Class 12 Chemistry: Chemical Kinetics