Question

If t\(^{1/2}\) is 46 days, how much of this material will be left from 1 mole of this substance after 184 days?

• A. 0.2625 mole
• B. 0.2652 mole
• C. 0.2662 mole
• D. 0.2623 mole

Hint:

To find the remaining fraction after $n$ half-lives, use powers of 2 in the denominator: 1 half-life = $1/2$, 2 half-lives = $1/4$, 3 half-lives = $1/8$, 4 half-lives = $1/16$. Here, $1/16 = 0.0625$, which is option A with a slight typo.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Radioactive decay or first-order chemical degradation follows a half-life model.
The half-life (\( t_{1/2} \)) is the time required for the concentration of a substance to decrease to half of its initial value.
Key Formula or Approach:
The amount of material remaining after a given time can be calculated using:
\[ N_t = N_0 \left( \frac{1}{2} \right)^n \]
where \( N_t \) is the remaining amount, \( N_0 \) is the initial amount, and \( n \) is the number of half-lives elapsed:
\[ n = \frac{\text{Total Time } (t)}{\text{Half-life } (t_{1/2})} \]

Step 2: Detailed Explanation:
Let us perform the calculations:
Given:
Initial amount \( N_0 = 1\text{ mole} \)
Half-life \( t_{1/2} = 46\text{ days} \)
Total time \( t = 184\text{ days} \)
1. Calculate the number of half-lives elapsed (\( n \)):
\[ n = \frac{184}{46} = 4 \]
2. Calculate the remaining amount (\( N_t \)):
\[ N_t = 1 \times \left( \frac{1}{2} \right)^4 = \frac{1}{16} = 0.0625\text{ mole} \]
Looking at the options, we note that the value \( 0.2625\text{ mole} \) (Option A) has a small typographical error in the paper (printing '2' instead of '0').
Mathematically, the intended and correct answer is indeed \( 0.0625\text{ mole} \), represented by option (A).

Step 3: Final Answer:
The mathematically correct remaining amount is 0.0625 mole, represented as 0.2625 mole in the options due to a typographical print, corresponding to option (A).