To solve the problem, we need to determine which expression correctly represents the mass of sodium-22 remaining after [tex]\( h \)[/tex] half-lives, given the initial mass is 800 mg.
1. Understanding Half-life:
- The half-life of a substance is the time it takes for half of the substance to decay.
- For sodium-22, the half-life is 2.6 years.
2. Exponential Decay:
- When dealing with half-lives, the decay process follows exponential decay, where the mass halves every half-life.
- If you start with a mass [tex]\( M \)[/tex] and experience [tex]\( h \)[/tex] half-lives, the formula for the remaining mass [tex]\( S \)[/tex] is:
[tex]\[
S = M \times \left(\frac{1}{2}\right)^h
\][/tex]
3. Applying the Initial Mass:
- The initial mass [tex]\( M \)[/tex] is given as 800 mg.
- Plugging this into the formula, we get:
[tex]\[
S = 800 \times \left(0.5\right)^h
\][/tex]
4. Checking the Options:
- [tex]\( S = 0.5(800)^h \)[/tex]: This suggests the base of the exponentiation is 800, which is incorrect. The factor [tex]\( 0.5 \)[/tex] should affect the [tex]\( h \)[/tex] half-lives, not the 800.
- [tex]\( S = 800(0.5)^h \)[/tex]: This aligns with our derived formula and correctly models exponential decay.
- [tex]\( S = 800^{0.5h} \)[/tex]: This incorrectly applies the exponent [tex]\( 0.5h \)[/tex] to the initial mass directly, not considering the halving process correctly.
- [tex]\( S = 800^{h+0.5} \)[/tex]: This incorrectly adds [tex]\( 0.5 \)[/tex] to the exponent of 800, which does not align with the concept of half-life decay.
Correct Expression:
The correct expression that represents the mass of sodium-22 remaining after [tex]\( h \)[/tex] half-lives is:
[tex]\[
S = 800 \times (0.5)^h
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{S = 800(0.5)^h}
\][/tex]
