To determine how many moles of potassium (K) are needed to react completely with 12.8 moles of magnesium bromide (MgBr[tex]\(_2\)[/tex]) according to the balanced chemical equation, let’s follow these steps:
1. Write down the balanced chemical equation:
[tex]\[
2 K + MgBr_2 \rightarrow Mg + 2 KBr
\][/tex]
2. Identify the mole ratio of potassium (K) to magnesium bromide (MgBr[tex]\(_2\)[/tex]) from the balanced equation. According to the equation:
[tex]\[
2 \text{ moles of } K \text{ react with } 1 \text{ mole of } MgBr_2
\][/tex]
3. We are given that we have 12.8 moles of MgBr[tex]\(_2\)[/tex]. Using the mole ratio from the balanced equation, we need to calculate how many moles of K are required to react with 12.8 moles of MgBr[tex]\(_2\)[/tex].
4. Multiply the number of moles of MgBr[tex]\(_2\)[/tex] by the mole ratio of K to MgBr[tex]\(_2\)[/tex]:
[tex]\[
\text{moles of } K \text{ needed} = 12.8 \text{ moles of } MgBr_2 \times \left(\frac{2 \text{ moles of } K}{1 \text{ mole of } MgBr_2}\right)
\][/tex]
5. Perform the multiplication:
[tex]\[
\text{moles of } K \text{ needed} = 12.8 \times 2 = 25.6 \text{ moles of } K
\][/tex]
So, the number of moles of potassium required to react completely with 12.8 moles of magnesium bromide is [tex]\(25.6\)[/tex] moles of [tex]\(K\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{25.6 \text{ moles of } K} \][/tex]
