number of ions present from moles [closed]








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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 $3.5\times 10^{22}$. 




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



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> Step 3 - What are the units of the conversion factor?
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Avogadro's number (with units) is $6.022\times 10^{23} \text{ mol}^{-1}$. The units of Avogadro's number are "number per mole". 



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> Step 4 - Set up the problem so that units cancel and you are left with only the units you need.
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**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](http://en.wikipedia.org/wiki/Dimensional_analysis#The_factor-label_method_for_converting_units). With practice and conscientious use of this method, you should not make unit conversion errors.



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> Step 5 - Check to see if your answer makes sense.
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$0.0585$ 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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