Question

The fraction of $^{24}𝑁𝑎_{11}$ atoms remaining after a decay interval of 5.0 hours will be ______ . (Round off to three decimal places) (Use $t_{1/2}$ = 15.0 hours)

Answer 1

Correct Answer: 0.792

Calculating the Fraction of Atoms Remaining After a Given Time 

Step 1: Understanding the Formula

The fraction of atoms remaining after a given time can be calculated using the radioactive decay formula:

\[ \frac{N}{N_0} = \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]

Where:

• \(\frac{N}{N_0}\) is the fraction of remaining atoms.
• t is the time elapsed (5.0 hours).
• t₁/₂ is the half-life of the isotope (15.0 hours).

Step 2: Substituting the Given Values

Substitute the given values into the formula:

\[ \frac{N}{N_0} = \left( \frac{1}{2} \right)^{\frac{5.0}{15.0}} = \left( \frac{1}{2} \right)^{\frac{1}{3}} \approx 0.796 \]

Final Answer:

The fraction of ²⁴11Na atoms remaining is approximately: 0.796 (rounded to three decimal places).