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(a) 0.008 M $\ce{Ba(OH)2}$
(b) 0.010 M $\ce{KI}$
The answer is "a" but I do not understand why. The book says that "a" is a strong electrolyte with a total ion concentration of 0.024. How did they calculate the concentration from the information that they gave? And why is it a strong electrolyte based on that calculation. I thought strong electrolytes were (solutes) ions that dissolved completely in a solvent/solution?
Barium hydroxide is a slightly soluble salt. It dissociates in water and gives 3 ions (one ion of barium and two ions of hydroxide) with the concentration of $0.008 \ce{ M}$ for each one. This makes the total concentration of ions to be $0.024 \ce{ M}$. While the total concentration of $\ce{KI}$ is $0.02 \ce{ M}$. So the solution of barium hydroxide is a stronger electrolyte as it has a higher concentration of dissociated ions.
In your book, they treat barium hydroxide as a strong electrolyte, i.e. it dissolved completely in water and dissociates to its constituent ions. This is of course an approximation. While $\ce{KI}$ is a true strong electrolyte; it dissolved completely in water and dissociates to its constituent ions. So, I agree with you with the definition of strong electrolyte, and I hope I clarify the ambiguity.
You are right in thinking that a strong electrolyte is something that dissociates completely in solution.$\ce{Ba(OH)2}$ and $\ce{KI}$ then, being a strong electrolytes, will effectively dissociate into $\ce{Ba^2+}$+ $\ce{2OH-}$ and $\ce{K+}$+ $\ce{I-}$, respectively.
Going into your example, it's a just a matter of multiplication. We can assume that the volume of the solution is 1L for simplicity. Since you begin with 0.008 M $\ce{Ba(OH)2}$, we can say that you begin with 0.008 mol of $\ce{Ba(OH)2}$. In solution this becomes 0.008 mol of $\ce{Ba+}$ and 2(0.008) mol of $\ce{OH-}$, or 0.024 mol of ions in total. Likewise for KI, you are left with 0.010 mol of $\ce{K+}$ and 0.010 mol of $\ce{I-}$, or 0.020 mol of ions in total.
Converting these values back to molarity (by dividing by 1), we get that the concetration of ions in the $\ce{Ba(OH)2}$ solution is 0.024 M, whereas it is only 0.020 M in the $\ce{KI}$ solution.
