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.

Solution Steps

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.

From the problem, we have:

• The half-life \( T \) of sodium-22 is 2.6 years.
• The initial amount \( S_0 \) is 40 grams.

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