Multiple Choice
How long will it take for 80% of a 1.20-mg sample of Rn-220 to decay? The half-life of Rn-220 is 55.6 s.
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21. Nuclear Chemistry
Rate of Radioactive Decay
Multiple Choice
During World War I radium-226 was used in the manufacturing of luminous paint. If it takes 2.12 × 104 days for its degradation to be 2.49% complete, what is its decay constant?
A
1.74 × 10–4 days–1
B
1.08 × 10–5 days–1
C
1.19 × 10–6 days–1
D
2.14 × 10–5 days–1
Verified step by step guidance1
Identify the type of decay process involved. Since radium-226 undergoes radioactive decay, it follows first-order kinetics, where the decay constant (\( k \)) relates to the fraction of substance remaining over time.
Express the given information in terms of the fraction remaining. If 2.49% has decayed, then the fraction remaining is \( 1 - 0.0249 = 0.9751 \).
Use the first-order decay formula:
\[ N = N_0 e^{-k t} \]
where \( N/N_0 \) is the fraction remaining, \( k \) is the decay constant, and \( t \) is the time elapsed.
Rearrange the formula to solve for the decay constant \( k \):
\[ k = -\frac{1}{t} \ln\left( \frac{N}{N_0} \right) \]
Substitute \( t = 2.12 \times 10^{4} \) days and \( N/N_0 = 0.9751 \) into the equation.
Calculate \( k \) using the values substituted. This will give the decay constant in units of days\(^{-1}\).
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