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I have $0.0585$ moles of $\ce{Al^{3+}}$, and need to find the number of actual ions. I thought I just divided by $6.022\times 10^{23}$ but I get $9.7\times 10^{-26}$, and the book says it should be $3.5\times 10^{22}$.

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  • $\begingroup$ Welcome to Chemistry.SE! To acquaint yourself with this page, take the tour and visit the help center. Furthermore this tutorial shows you how math and chemical formulae can be nicely formatted on this site. Hint: Think about the unit of avogadro's number $6.022\times 10^{23}$; if you divide your number of moles by avogadro's number, what unit do you get and what unit should you actually get? $\endgroup$ – Philipp Nov 29 '14 at 19:17
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You should multiply by Avogadro's number $6.022\times 10^{23}$ instead of dividing by it. To make sure you always use a conversion factor (Avogadro's number is a conversion factor between moles and numbers) correctly, you need to think about units.

Step 1 - What are the units of the answer?

You want to find the number of ions. The units of the answer is number.

Step 2 - What are the units of the quantities you were given?

You were given 0.0585 moles of $\ce{Al^{3+}}$. The units you are given is "moles".

Step 3 - What are the units of the conversion factor?

Avogadro's number (with units) is $6.022\times 10^{23} \text{ mol}^{-1}$. The units of Avogadro's number are "number per mole".

Step 4 - Set up the problem so that units cancel and you are left with only the units you need.

Multiplying by Avogadro's number

$$\require{cancel} \cancel{\text{mol}}\times \dfrac{\text{number}}{\cancel{\text{mol}}}=\text{number}$$ The "moles" units cancel, leaving you with "number".

Dividing by Avogadro's number

$$\dfrac{\text{mol}}{\frac{\text{number}}{\text{mol}}}=\text{mol}\times \dfrac{\text{mol}}{\text{number}}=\dfrac{\text{mol}^2}{\text{number}}$$

The units do not cancel appropriately, leaving us with units that do not match the desired outcome. This approach is an application of dimensional analysis commonly called the factor-label method. With practice and conscientious use of this method, you should not make unit conversion errors.

Step 5 - Check to see if your answer makes sense.

$0.0585$ moles is about $\frac{1}{20}$ of a mole (or one half of one tenth). Your answer should be about $\frac{1}{20}$ of Avogadro's number.

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