A solution is made up by adding \(0.925 \mathrm{~g}\) of silver(I) nitrate and \(6.25 \mathrm{~g}\) of magnesium nitrate to enough water to make \(375 \mathrm{~mL}\) of solution. Solid sodium carbonate is added without changing the volume of the solution. (a) Which salt, \(\mathrm{Ag}_{2} \mathrm{CO}_{3}\) or \(\mathrm{MgCO}_{3}\), will precipitate first? (b) What is the concentration of the carbonate ion when the first salt starts to precipitate?

In this step, we will list down the solubility products (Ksp) of the salts (Ag2CO3 or MgCO3). We can find these Ksp values from a chemistry reference table or textbook. For Ag2CO3: Ksp = 8.1 × 10^-12 For MgCO3: Ksp = 6.8 × 10^-6 Remember these values as we will use them in later steps.

Now, we need to determine the initial concentrations of Ag+, Mg2+, and NO3- ions in the solution. We do this by dividing the mass of each compound by its molar mass, which will give us the moles. Then, we'll divide the moles by the volume of the solution (in liters) to get the molarity. For Ag+: Moles = 0.925 g / (169.87 g/mol) = 0.00544 mol Molarity = 0.00544 mol / 0.375 L = 0.0145 M For Mg2+: Moles = 6.25 g / (148.31 g/mol) = 0.0421 mol Molarity = 0.0421 mol / 0.375 L = 0.112 M Since we do not need to consider the concentration of nitrate ions in this problem, we can now move on to the next step.

Initially, there are no carbonate ions in the solution. As we add sodium carbonate, the concentration of carbonate ions will increase. The ion product (Q) for each possible precipitate can be compared to the respective Ksp values. The first salt that will precipitate is the one for which Q becomes equal to Ksp first. For Ag2CO3: Q = [Ag+]²[CO3²-] Considering only one of the Ag+ ions produced by Ag2CO3. For MgCO3: Q = [Mg²+][CO3²-] As we add sodium carbonate to the solution, carbonate ions concentration ([CO3²-]) will increase, raising the Q value for both salts. The first salt to precipitate will be the one that reaches its Ksp first. Let's calculate the concentration of CO3²- for each salt when Q equals Ksp.

To find the concentration of CO3²- at the point of precipitation, we need to rearrange the ion product equations to solve for [CO3²-]: For Ag2CO3: [CO3²-] = Ksp / [Ag+]² = (8.1 × 10^-12) / (0.0145)^2 = 3.0 × 10^-8 M For MgCO3: [CO3²-] = Ksp / [Mg²+] = (6.8 × 10^-6) / 0.112 = 6.1 × 10^-5 M (a) Since the concentration of carbonate ions required to precipitate Ag2CO3 is lower than that required for MgCO3, Ag2CO3 will precipitate first. (b) The concentration of the carbonate ion when Ag2CO3 starts to precipitate is 3.0 × 10^-8 M.