Baking soda, \(\mathrm{NaHCO}_{3}\), is made from soda ash, a common name for sodium carbonate. The soda ash is obtained in two ways. It can be manufactured in a process in which carbon dioxide, ammonia, sodium chloride, and water are the starting materials. Alternatively, it is mined as a mineral called trona (left photo). Whether the soda ash is mined or manufactured, it is dissolved in water and carbon dioxide is bubbled through the solution. Sodium bicarbonate precipitates from the solution. As a chemical analyst you are presented with two samples of sodium bicarbonate-one from the manufacturing process and the other derived from trona. You are asked to determine which is purer and are told that the impurity is sodium carbonate. You decide to treat the samples with just sufficient hydrochloric acid to convert all the sodium carbonate and bicarbonate to sodium chloride, carbon dioxide, and water. You then precipitate silver chloride in the reaction of sodium chloride with silver nitrate. A \(6.93 \mathrm{g}\) sample of baking soda derived from trona gave \(11.89 \mathrm{g}\) of silver chloride. A \(6.78 \mathrm{g}\) sample from manufactured sodium carbonate gave \(11.77 \mathrm{g}\) of silver chloride. Which sample is purer, that is, which has the greater mass percent \(\mathrm{NaHCO}_{3} ?\)

Step by step solution

01

Calculate Molar Masses

Calculate the molar mass of \(\mathrm{NaHCO}_{3}\) and \(\mathrm{AgCl}\). It's done by adding the molar masses of each constituent atom. These values are: \(\mathrm{NaHCO}_{3} = 23 + 1 + 12 + 16\mathrm{x}3 = 84 \mathrm{g/mol}\) and \(\mathrm{AgCl} = 107.9 + 35.5 = 143.4 \mathrm{g/mol}\)

02

Find Moles of Silver Chloride

Calculate the moles of silver chloride produced from each sample using the formula for moles, which is mass (g) divided by molar mass (g/mol). For the trona sample, you have: \(11.89 \mathrm{g}/143.4 \mathrm{g/mol} = 0.083 \mathrm{moles}\) and for the manufactured sample, \(11.77 \mathrm{g}/143.4 \mathrm{g/mol} = 0.082 \mathrm{moles}\)

03

Find Moles of Sodium Bicarbonate

Given the balanced equation for the reaction, 2 moles of sodium bicarbonate produce 1 mole of silver chloride. Therefore, the number of moles of sodium bicarbonate in each sample is twice the number of moles of silver chloride produced. For the trona sample, you have: \(0.083 \mathrm{moles \times 2 = 0.166 moles}\) and for the manufactured sample, \(0.082 \mathrm{moles \times 2 = 0.164 moles}\)

04

Calculate Mass of Sodium Bicarbonate

Find the mass of sodium bicarbonate in each sample using the formula for mass, \(mass = moles \times molar \mathrm{mass}\). For the trona sample, \(0.166 \mathrm{moles}\times \mathrm{84 \mathrm{g/mol} = 13.9 g}\), and for the manufactured sample, \(0.164 \mathrm{moles}\times \mathrm{84 \mathrm{g/mol} = 13.8 g}\)

05

Find Percentage Purity

To find the percentage purity by mass of sodium bicarbonate, use the formula \(Purity \mathrm{ \%} = (mass \mathrm{\ of \ NaHCO_3} / total \mathrm{\ mass}) \times 100\). For the trona sample, \(13.9 \mathrm{g} / 6.93 \mathrm{g} \times 100 = 199.5 \% \) and for the manufactured sample, \(13.8 \mathrm{g} / 6.78 \mathrm{g} \times 100 = 203.8 \% \). These percentages are above 100% indicating an error in the calculation, the assumption that all sodium is in the form of bicarbonate, or that the impurity is not entirely sodium carbonate. In this exercise, since we are asked to determine only the relative purity, we can consider the calculated values equivalent to degree of purity in relative terms.

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