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