# Question 87cb7

May 31, 2016

${\text{Na"_2"B"_4"O}}_{7}$

#### Explanation:

Your strategy here will be pick a sample of this compound and use the given percent composition to determine how many grams of each element it contains.

To make calculations easier, pick a $\text{100-g}$ sample. According to the values given to you for the compound's percent composition, this sample will contain

• 22.8% -> "22.6 g Na"
• 21.5% -> "21.5 g B"
• 55.7% -> "55.7 g O"

Next, use the molar mass of each element to determine how many moles of each you have in the sample

$\text{For Na: " 22.6 color(red)(cancel(color(black)("g"))) * "1 mole Na"/(23.0color(red)(cancel(color(black)("g")))) = "0.9826 moles Na}$

$\text{For B: " 21.5 color(red)(cancel(color(black)("g"))) * "1 mole B"/(10.811color(red)(cancel(color(black)("g")))) = "1.989 moles B}$

$\text{For O: " 55.7color(red)(cancel(color(black)("g"))) * "1 mole O"/(15.9994color(red)(cancel(color(black)("g")))) = "3.481 moles O}$

Now, in order to find the compound's empirical formula, you must find the smallest whole number ratio that exists between its constituent elements.

To do that, divide all values by the smallest one to get

"For Na: " (0.9826color(red)(cancel(color(black)("moles"))))/(0.9826color(red)(cancel(color(black)("moles")))) = 1

"For B: " (1.989color(red)(cancel(color(black)("moles"))))/(0.9826color(red)(cancel(color(black)("moles")))) = 2.024 ~~2

"For O: " (3.481color(red)(cancel(color(black)("moles"))))/(0.9826color(red)(cancel(color(black)("moles")))) = 3.543

Since you're looking for the smallest whole number ratio, multiply all the values by $2$ to get

$\text{For Na: } 1 \times 2 = 2$

$\text{For B: } 2 \times 2 = 4$

$\text{For O: } 3.543 \times 2 = 7.09 \approx 7$

The empirical formula for this compound will thus be

("Na"_1"B"_2"O"_3.5)_2 = color(green)(|bar(ul(color(white)(a/a)color(black)("Na"_2"B"_4"O"_7)color(white)(a/a)|)))#