Question

A radioactive nuclei has a half life of 693s. The activity of one mole of that nuclei sample is:(Avogadro's number=6.023 x10²³ and In(2)=0.693) 

• A. 2×10¹⁰ Bq
• B. 3.7×10¹⁰ Bq
• C. 6.023×10²⁰ Bq
• D. 0.5×10^-10 Bq
• E. 1×10²⁰ Bq

Answer 1

The Correct Option is C

Given:

• Half-life, \( T_{1/2} = 693 \, \text{s} \)
• Avogadro's number, \( N_A = 6.023 \times 10^{23} \, \text{mol}^{-1} \)
• \(\ln(2) = 0.693\)

Step 1: Calculate the Decay Constant (\( \lambda \))

The decay constant is given by:

\[ \lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{693} = 0.001 \, \text{s}^{-1} \]

Step 2: Determine the Number of Nuclei (\( N \))

For 1 mole of nuclei:

\[ N = N_A = 6.023 \times 10^{23} \]

Step 3: Compute Activity (\( A \))

Activity is defined as:

\[ A = \lambda N = 0.001 \times 6.023 \times 10^{23} = 6.023 \times 10^{20} \, \text{Bq} \]

Conclusion:

The activity of one mole of the radioactive nuclei is \( 6.023 \times 10^{20} \, \text{Bq} \).

Answer: \(\boxed{C}\)

Answer 2

1. Define variables and given information:

• t_1/2 = 693 s (half-life)
• N_A = 6.023 × 10²³ (Avogadro's number)
• ln(2) = 0.693
• Activity (A) = ?

2. Calculate the decay constant (λ):

The decay constant is related to the half-life by the following equation:

\[t_{1/2} = \frac{\ln(2)}{\lambda}\]

\[\lambda = \frac{\ln(2)}{t_{1/2}} = \frac{0.693}{693 \, s} = 10^{-3} \, s^{-1}\]

3. Calculate the number of nuclei (N) in one mole:

One mole of any substance contains Avogadro's number of particles (in this case, nuclei):

\[N = N_A = 6.023 \times 10^{23}\]

4. Calculate the activity (A):

Activity is defined as the rate of decay and is given by:

\[A = \lambda N\]

\[A = (10^{-3} \, s^{-1})(6.023 \times 10^{23}) = 6.023 \times 10^{20} \, Bq\]