How many half-lives are required for the concentration of reactant to decrease to 25% of its original value?

Question

BlackTom AI · Accepted Answer

Each half-life reduces the reactant concentration to one-half of its value, so the remaining fraction after n half-lives is (1/2)^n.

1. Relevant knowledge points and concepts
- Half-life (t1/2) definition: time required for a quantity to decrease to half its initial value.
- Exponential decay law for discrete half-lives: fraction remaining = (1/2)^n, where n is the number of half-lives.
- Solving for n can use exponent rules or logarithms.

2. Step-by-step reasoning
We are given the target remaining fraction is 25% = 0.25 of the original. Set up the decay equation with n half-lives:
(1/2)^n = 0.25.

3. Intermediate calculations
Recognize 0.25 as a power of 1/2:
0.25 = 1/4 = (1/2)^2.
Therefore (1/2)^n = (1/2)^2, which implies n = 2.

Alternatively using logarithms:
n · log(1/2) = log(0.25)
n = log(0.25) / log(0.5) =  (-0.60206) / (-0.30103) = 2.

4. Why the correct answer is correct
Because two successive halvings reduce the concentration to one-half and then one-half of that, i.e., (1/2) × (1/2) = 1/4 (25%). The exponential relation directly gives n = 2.

5. Why the other options are incorrect
- 1 half-life gives 50% remaining, not 25%.
- 3 half-lives give (1/2)^3 = 12.5%, too small.
- 7 half-lives give (1/2)^7 ≈ 0.78%, far smaller.
- 3.5 half-lives would give (1/2)^{3.5} ≈ 8.84%, not 25%.
Only n = 2 yields exactly 25%.

6. Conclusion
The number of half-lives required is 2, which matches the provided correct answer.

How many half-lives are required for the concentration of reactant to decrease to 25% of its original value?单项选择题

类似问题

A drug with concentration 0.90 M decomposes via acid-catalysed hydrolysis with a rate constant of 0.031 M−1 h−1. What is the shelf-life of this drug in hours? Report your answer to 2 significant figures.

The rate of hydrolysis of a compound A is measured in mol/(L × h). Express the rate of 0.61 mol/(L × h) in g/(mL × s). The molar mass of compound A is 280 g/mol.

A new drug candidate has been developed and the rate constant for its base-catalysed hydrolysis has been was found to vary with temperature. The rate constant at 25 °C was determined to be 1.053 × 10-3 h-1. Which formulation and storage recommendations, of the ones listed below, are appropriate?

Consider a process where a product is formed from a reactant A via two different mechanistic pathways, characterised by rate constants k1 and k2. Which rate law correctly describes this process?

Question at position 16 Which of the following decreases as temperature increases? Half life (t1/2) Activation energy (Ea) Rate constant (k) Reaction rate k onlyEa onlyt1/2 onlyk and Ea onlyreaction rate onlyreaction rate and t1/2 only

Each hour 6% of Chemical A converts to Chemical B, and 9% of Chemical B converts to Chemical A. If initially there is 12 mg of Chemical A and 13 mg of Chemical B, what is the steady state weight of Chemical A (to 2 decimal places)?

Each hour 7% of Chemical A converts to Chemical B, and 6% of Chemical B converts to Chemical A. If initially there is 16 mg of Chemical A and 13 mg of Chemical B, what is the steady state weight of Chemical A (to 2 decimal places)?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单