Questions: The half-life of sodium-22 is 2.6 years. If there are initially 40 grams of sodium-22, find a formula that gives the amount S, in grams, remaining after t years.
Transcript text: The half-life of sodium-22 is 2.6 years. If there are initially 40 grams of sodium-22, find a formula that gives the amount $S$, in grams, remaining after $t$ years.
Solution
Solution Steps
Step 1: Understanding the Half-Life Formula
The half-life formula for radioactive decay is given by:
\[
S(t) = S_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}
\]
where:
\( S(t) \) is the amount of substance remaining after time \( t \),
\( S_0 \) is the initial amount of the substance,
\( T \) is the half-life of the substance,
\( t \) is the time elapsed.
Step 2: Identifying Given Values
From the problem, we have:
The half-life \( T \) of sodium-22 is 2.6 years.
The initial amount \( S_0 \) is 40 grams.
Step 3: Substituting Values into the Formula
Substitute \( S_0 = 40 \) grams and \( T = 2.6 \) years into the half-life formula:
\[
S(t) = 40 \left(\frac{1}{2}\right)^{\frac{t}{2.6}}
\]
Final Answer
The formula that gives the amount \( S \), in grams, remaining after \( t \) years is:
\[
\boxed{S(t) = 40 \left(\frac{1}{2}\right)^{\frac{t}{2.6}}}
\]