how long (in minutes) will it take for the crystal biolet to be 95% gone if the reaction is zero order and the inital concentration is 3.0M? k=2.0M/min
The correct answer and explanation is:
To find the time it takes for 95% of crystal violet to be gone in a zero-order reaction, we use the zero-order integrated rate law: [A]=[A]0−kt[A] = [A]_0 – kt
Where:
- [A][A] is the concentration at time tt
- [A]0=3.0 M[A]_0 = 3.0\ \text{M} is the initial concentration
- k=2.0 M/mink = 2.0\ \text{M/min} is the rate constant
- We want 95% of the substance gone, so 5% remains:
[A]=0.05×3.0=0.15 M[A] = 0.05 \times 3.0 = 0.15\ \text{M}
Step-by-step Calculation:
0.15=3.0−2.0⋅t0.15 = 3.0 – 2.0 \cdot t 2.0t=3.0−0.15=2.852.0t = 3.0 – 0.15 = 2.85 t=2.852.0=1.425 minutest = \frac{2.85}{2.0} = 1.425\ \text{minutes}
✅ Final Answer: 1.43 minutes (rounded to 2 decimal places)
✏️ Explanation (Approx. 300 words)
In chemical kinetics, a zero-order reaction is one in which the rate of reaction is independent of the concentration of the reactant. This means that the reactant is consumed at a constant rate until it is completely used up. Mathematically, the concentration [A][A] of the reactant at time tt is given by the zero-order rate law: [A]=[A]0−kt[A] = [A]_0 – kt
This linear relationship implies that as time increases, the concentration decreases uniformly.
In this problem, the reactant is crystal violet, with an initial concentration [A]0[A]_0 of 3.0 M, and the rate constant kk is 2.0 M/min. We are asked to determine how long it takes for 95% of the crystal violet to be gone, which means only 5% remains. Thus, the final concentration [A][A] is 5% of 3.0 M, which equals 0.15 M.
By substituting these values into the zero-order rate equation and solving for time tt, we find: 0.15=3.0−2.0t⇒t=2.852.0=1.425 minutes0.15 = 3.0 – 2.0t \Rightarrow t = \frac{2.85}{2.0} = 1.425\ \text{minutes}
This result shows that under zero-order kinetics, time to reach a given percentage completion is linearly proportional to the amount of reactant consumed and inversely proportional to the rate constant. Because zero-order reactions proceed at a constant rate, we don’t need to track changes in concentration to determine the rate — only how much has been used and how fast it’s going.