Question

Rotting leaves of mango are composed of 25% of compound P which decomposes exponentially at a rate \( k_P = 0.5 \, {yr}^{-1} \) and 75% of compound Q which decomposes exponentially at a rate \( k_Q = 0.1 \, {yr}^{-1} \). What percentage of the initial amount of compound Q remains when half of the initial amount of compound P has decomposed? Choose the closest numerical value from the options provided.

• A. 87%
• B. 50%
• C. 75%
• D. 37%

Hint:

For exponential decay, remember that the formula \( N(t) = N_0 e^{-kt} \) can be used to calculate the remaining amount of a substance at any time \( t \).

Answer 1

The Correct Option is A

Step 1: Decomposition of compound P.
The amount of compound P remaining after time \( t \) is given by the formula: \[ P(t) = P_0 e^{-k_P t} \] where \( P_0 \) is the initial amount of compound P. We are told that half of the initial amount of compound P has decomposed, so: \[ \frac{1}{2} P_0 = P_0 e^{-k_P t} \] Solving for \( t \): \[ e^{-k_P t} = \frac{1}{2} \quad \Rightarrow \quad -k_P t = \ln \frac{1}{2} \quad \Rightarrow \quad t = \frac{\ln 2}{k_P} \] Substituting \( k_P = 0.5 \): \[ t = \frac{\ln 2}{0.5} \approx 1.386 \, {years}. \] Step 2: Decomposition of compound Q.
Now, we use the same formula for compound Q: \[ Q(t) = Q_0 e^{-k_Q t} \] Substituting the value of \( t \) from the previous step and \( k_Q = 0.1 \): \[ Q(t) = Q_0 e^{-0.1 \times 1.386} \approx Q_0 e^{-0.1386} \approx 0.870 \, Q_0. \] Thus, 87% of compound Q remains after half of compound P has decomposed.