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Chapter 4: Problem 106

The recommended procedure for preparing a very dilute solution is not to weigh out a very small mass or measure a very small volume of a stock solution. Instead, it is done by a series of dilutions. A sample of \(0.8214 \mathrm{~g}\) of \(\mathrm{KMnO}_{4}\) was dissolved in water and made up to the volume in a 500 -mL volumetric flask. A 2.000 -mL sample of this solution was transferred to a 1000 -mL volumetric flask and diluted to the mark with water. Next, \(10.00 \mathrm{~mL}\) of the diluted solution were transferred to a 250 -mL flask and diluted to the mark with water. (a) Calculate the concentration (in molarity) of the final solution. (b) Calculate the mass of \(\mathrm{KMnO}_{4}\) needed to directly prepare the final solution.

Short Answer

Expert verified
The molarity of the final solution is \(8.315 x 10^-5 mol/L\) and the amount of KMnO4 needed to directly prepare the final solution would be \(3.287 x 10^-3 g\).

Step by step solution

01

Calculate the number of moles of KMnO4

The molar mass of KMnO4 (Potassium permanganate) is 158.034 g/mol. Divide the mass used by the molar mass to find the number of moles: Number of moles of KMnO4 = \(0.8214 g / 158.034 g/mol = 0.005197 mol\)
02

Calculate the original concentration

The molarity (concentration in moles/L) of the original solution can be calculated using number of moles and volume. The volume should be in liters, so convert 500 mL to liters: Volume = 500 mL = 0.5 L. Concentration of original solution = \((0.005197 mol) / (0.5 L) = 0.010394 M\) or moles/L
03

Apply the dilution formula

Apply the formula for the dilution process: \(M1 x V1 = M2 x V2\). Here \(M1\) and \(V1\) represent the initial molarity and volume, \(M2\) and \(V2\) are the final molarity and volume. The original solution was diluted twice, so this step needs to be repeated twice. First dilution: Concentration after the first dilution: \((0.010394 M) x (0.002 L) = M2 x (1 L)\) hence \(M2 = 0.020788 x 10^-2 M\). Second dilution: Again apply the formula: \((0.020788 x 10^-2 M) x (0.01 L) = M2 x (0.25 L)\) hence \(M2 = 0.008315 x 10^-2 M\) or \(8.315 x 10^-5 mol/L\) which is the final concentration of the solution.
04

Calculate the required mass of KMnO4

To prepare a solution with final molarity directly, first calculate the required number of moles using the formula of molarity \(M = n/V\) where \(n\) is the number of moles and \(V\) is the volume in Liters. Hence \(n = M x V = 0.008315 x 10^-2 mol/L x 0.25 L = 2.079 x 10^-5 mol\). Then, Multiply this by the molar mass of KMnO4 to obtain the mass. Required mass = \(2.079 x 10^-5 mol x 158.034 g/mol = 3.287 x 10^-3 g.\)

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Most popular questions from this chapter

Magnesium is a valuable, lightweight metal. It is used as a structural metal and in alloys, in batteries, and in chemical synthesis. Although magnesium is plentiful in Earth's crust, it is cheaper to "mine" the metal from seawater. Magnesium forms the second most abundant cation in the sea (after sodium); there are about \(1.3 \mathrm{~g}\) of magnesium in \(1 \mathrm{~kg}\) of seawater. The method of obtaining magnesium from seawater employs all three types of reactions discussed in this chapter: precipitation, acid-base, and redox reactions. In the first stage in the recovery of magnesium, limestone \(\left(\mathrm{CaCO}_{3}\right)\) is heated at high temperatures to produce quicklime, or calcium oxide \((\mathrm{CaO})\) : $$ \mathrm{CaCO}_{3}(s) \longrightarrow \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) $$ When calcium oxide is treated with seawater, it forms calcium hydroxide \(\left[\mathrm{Ca}(\mathrm{OH})_{2}\right]\), which is slightly soluble and ionizes to give \(\mathrm{Ca}^{2+}\) and \(\mathrm{OH}^{-}\) ions: $$ \mathrm{CaO}(s)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}^{2+}(a q)+2 \mathrm{OH}^{-}(a q) $$ The surplus hydroxide ions cause the much less soluble magnesium hydroxide to precipitate: $$ \mathrm{Mg}^{2+}(a q)+2 \mathrm{OH}^{-}(a q) \longrightarrow \mathrm{Mg}(\mathrm{OH})_{2}(s) $$ The solid magnesium hydroxide is filtered and reacted with hydrochloric acid to form magnesium chloride \(\left(\mathrm{MgCl}_{2}\right)\) \(\mathrm{Mg}(\mathrm{OH})_{2}(s)+2 \mathrm{HCl}(a q) \longrightarrow\) $$ \mathrm{MgCl}_{2}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l) $$ After the water is evaporated, the solid magnesium chloride is melted in a steel cell. The molten magnesium chloride contains both \(\mathrm{Mg}^{2+}\) and \(\mathrm{Cl}^{-}\) ions. In a process called electrolysis, an electric current is passed through the cell to reduce the \(\mathrm{Mg}^{2+}\) ions and oxidize the \(\mathrm{Cl}^{-}\) ions. The halfreactions are $$ \begin{aligned} \mathrm{Mg}^{2+}+2 e^{-} \longrightarrow \mathrm{Mg} \\ 2 \mathrm{Cl}^{-} \longrightarrow \mathrm{Cl}_{2}+2 e^{-} \end{aligned} $$ The overall reaction is $$ \mathrm{MgCl}_{2}(l) \longrightarrow \mathrm{Mg}(s)+\mathrm{Cl}_{2}(g) $$ This is how magnesium metal is produced. The chlorine gas generated can be converted to hydrochloric acid and recycled through the process. (a) Identify the precipitation, acid-base, and redox processes. (b) Instead of calcium oxide, why don't we simply add sodium hydroxide to precipitate magnesium hydroxide? (c) Sometimes a mineral called dolomite (a combination of \(\mathrm{CaCO}_{3}\) and \(\mathrm{MgCO}_{3}\) ) is substituted for limestone \(\left(\mathrm{CaCO}_{3}\right)\) to bring about the precipitation of magnesium hydroxide. What is the advantage of using dolomite? (d) What are the advantages of mining magnesium from the ocean rather than from Earth's crust?

How would you prepare \(60.0 \mathrm{~mL}\) of \(0.200 \mathrm{M} \mathrm{HNO}_{3}\) from a stock solution of \(4.00 M \mathrm{HNO}_{3} ?\)

Based on oxidation number, explain why carbon monoxide (CO) is flammable but carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is not.

Identify the following as a weak or strong acid or base: (a) \(\mathrm{NH}_{3},\) (b) \(\mathrm{H}_{3} \mathrm{PO}_{4},\) (c) \(\mathrm{LiOH},\) d) \(\mathrm{HCOOH}\) (formic acid), (e) \(\mathrm{H}_{2} \mathrm{SO}_{4},\) (f) \(\mathrm{HF},(\mathrm{g}) \mathrm{Ba}(\mathrm{OH})_{2}\)

Write the equation that enables us to calculate the concentration of a diluted solution. Give units for all the terms.
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