Question

\( ^{24}\text{Na} \) decays to one-fourth of its initial amount in 29.8 hours. Its decay constant is ........... hour\(^{-1}\) (rounded up to four decimal places).

Hint:

For first-order decay, use the equation \( \lambda = \frac{0.693}{t_{1/2}} \) to calculate the decay constant.

Answer 1

Correct Answer: 0.046 - 0.047

Step 1: Understanding the decay process. 
The decay of a substance follows first-order kinetics, where the decay constant \( \lambda \) is related to the half-life (\( t_{1/2} \)) by the equation: \[ t_{1/2} = \frac{0.693}{\lambda} \] The decay of \( ^{24}\text{Na} \) reaches one-fourth of its initial amount in 29.8 hours, which corresponds to 2 half-lives. Thus, we have: \[ t_{1/2} = \frac{29.8}{2} = 14.9 \, \text{hours} \]

Step 2: Calculating the decay constant. 
Using the equation for half-life, we can solve for \( \lambda \): \[ \lambda = \frac{0.693}{t_{1/2}} = \frac{0.693}{14.9} = 0.0465 \, \text{hour}^{-1} \]

Step 3: Conclusion. 
The decay constant \( \lambda \) is \( 0.0465 \, \text{hour}^{-1} \).