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On a hot day, a \(200.0-\mathrm{mL}\) sample of a saturated solution of \(\mathrm{Pb} \mathrm{I}_{2}\) was allowed to evaporate until dry. If \(240 \mathrm{mg}\) of solid \(\mathrm{PbI}_{2}\) was collected after evaporation was complete, calculate the \(K_{\mathrm{sp}}\) value for \(\mathrm{PbI}_{2}\) on this hot day.

Short Answer

Expert verified
The Ksp value for PbI鈧 on this hot day is approximately 3.53 脳 10鈦烩伕.

Step by step solution

01

Convert mass of PbI2 to moles

First, we will convert the given mass of PbI2 (240 mg) into moles. This will require the molar mass of PbI2, which can be calculated as follows: Molar mass of PbI2 = (molar mass of Pb) + 2 脳 (molar mass of I) Molar mass of PbI2 = (207.2 g/mol) + 2 脳 (126.9 g/mol) = 460.98 g/mol Now, convert the mass of PbI2 to moles: moles of PbI2 = mass of PbI2 / molar mass of PbI2 moles of PbI2 = 0.240 g / 460.98 g/mol 鈮 5.206 脳 10鈦烩伌 mol
02

Calculate the concentration of ions in the solution

Now, we have the moles of PbI2 and the volume of the solution (200.0 mL). We will need to find the concentration of the ions (Pb虏鈦 and I鈦) in the solution by dividing the moles by the volume in liters: \[Total \ volume \ of \ the \ solution \ (in \ litres)=\frac{200.0 \ mL}{1000} = 0.200 L \] \[ Concentration \ of \ Pb^{2+} \ ions= \frac{ moles \ of \ PbI_{2}}{ Volume \ of \ the \ solution \ (in \ litres)} =\frac{ 5.206 脳 10^{鈦4} \ mol}{ 0.200L } 鈮 2.603 脳 10^{鈦3} \ M\] As one Pb虏鈦 ion releases two I鈦 ions, the concentration of I鈦 ions in the solution is double of Pb虏鈦 ions: \[ Concentration \ of \ I^{-} \ ions = 2 脳 2.603 脳 10^{-3} \ M \ 鈮 5.206 脳 10^{-3} \ M\]
03

Calculate the Ksp value for PbI2

Now that we have the concentration of ions in the solution, we can calculate the Ksp for PbI2 using the solubility product equation: \[ K_{sp} = [Pb^{2+}] [I^{-}]^{2}\] Substitute the concentrations determined in Step 2: Ksp = (2.603 脳 10鈦宦) 脳 (5.206 脳 10鈦宦)虏 Ksp 鈮 3.53 脳 10鈦烩伕. Therefore, the Ksp value for PbI2 on this hot day is approximately 3.53 脳 10鈦烩伕.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Solubility Product Constant (Ksp)
The solubility product constant, commonly abbreviated as Ksp, is a vital concept in chemistry that describes the equilibrium between a solid and its ions in a saturated solution. This constant is unique to each substance and tells us how much of the substance can dissolve in a solvent at a certain temperature. To put it simply, Ksp helps us understand whether a salt will dissolve well in water or not.

When a substance like lead(II) iodide (PbI2) is in a saturated solution, it dissociates into ions (Pb虏鈦 and 2I鈦). The concentrations of these ions at equilibrium multiply together, raised to the power of their respective coefficients in the balanced equation, equal the Ksp. The reaction for PbI2 can be expressed as:
\[ PbI2_{(s)} \rightleftharpoons Pb^{2+}_{(aq)} + 2 I^-_{(aq)} \]
The Ksp expression is then:
\[ K_{sp} = [Pb^{2+}][I^-]^2 \]
Higher Ksp values indicate greater solubility. In our scenario, the calculated Ksp provides insight into the solubility of PbI2 on a hot day, which could differ from its solubility under different temperature conditions.
The Role of Molar Mass Calculation in Chemistry
The molar mass of a substance is the weight of one mole of that substance. It鈥檚 essentially the sum of the atomic masses of all the atoms in a molecule, measured in grams per mole (g/mol). Why is this important? Well, to engage in most quantitative chemistry problems, like calculating the Ksp, we need to convert between the mass of a solid (in grams) and the amount in moles.

Determining the molar mass of PbI2 is crucial for our calculations. It's a two-step calculation:
  • Add the molar mass of one atom of lead (Pb).
  • Add twice the molar mass of iodine (I) because there are two iodine atoms in the molecule.

With the molar mass, we can figure out how many moles of PbI2 we have based on the mass provided. This step is foundational for calculating the Ksp because we need to know the quantity of PbI2 in moles to proceed with determining the saturation level of the solution.
Concentration of Ions in Solution
In chemistry, the concentration of ions in a solution is a measurement of how much of a particular ion is present in a given volume of solution. It鈥檚 usually expressed in moles per liter (M), which makes it a molarity concentration. After finding the moles of PbI2 from our previous calculations, we then turned our attention to the concentration of ions that resulted from the dissolution of PbI2.

The concentration of each ion is directly linked to the stoichiometry of the dissociation reaction. Since PbI2 breaks down into one Pb虏鈦 ion and two I鈦 ions, the concentration of I鈦 will always be twice that of Pb虏鈦. In the context of the exercise, this knowledge is fundamental to deriving the Ksp value accurately, because if our ion concentrations are incorrect, our final Ksp will be off. Understanding the relationship between dissolved substances and their constituent ions is pivotal for any work involving solutions in chemistry.
Stoichiometry: The Mathematics of Chemical Reactions
Stoichiometry is the aspect of chemistry that involves the quantitative relationships between the substances as they participate in chemical reactions. It enables chemists to predict the amounts of substances consumed and produced in a given reaction. When we use stoichiometry in the context of calculating the Ksp, we are relying on the coefficients from the balanced chemical equation to guide us.

In our PbI2 example, stoichiometry tells us that for every mole of PbI2 that dissolves, one mole of Pb虏鈦 and two moles of I鈦 are produced in the solution. This stoichiometric relationship directly informs the exponents in the Ksp expression. Knowing how to apply stoichiometry is essential in not just finding out how much of a compound will dissolve, but also understanding the proportions of ions in solution at equilibrium. It鈥檚 a fundamental skill that is absolutely necessary for working through problems involving solubility equilibria and beyond.

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Most popular questions from this chapter

For each of the following pairs of solids, determine which solid has the smallest molar solubility. a. \(\mathrm{CaF}_{2}(s), K_{\mathrm{sp}}=4.0 \times 10^{-11}\), or \(\mathrm{BaF}_{2}(s), K_{\mathrm{sp}}=2.4 \times 10^{-5}\) b. \(\mathrm{Ca}_{3}\left(\mathrm{PO}_{4}\right)_{2}(s), K_{\infty}=1.3 \times 10^{-32}\), or \(\mathrm{FePO}_{4}(s), K_{\mathrm{sp}}=1.0 \times 10^{-22}\)

Write equations for the stepwise formation of each of the following complex ions. a. \(\mathrm{Ni}(\mathrm{CN})_{4}^{2-}\) b. \(\mathrm{V}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}{ }^{3-}\)

A mixture contains \(1.0 \times 10^{-3} \mathrm{M} \mathrm{Cu}^{2+}\) and \(1.0 \times 10^{-3} \mathrm{M} \mathrm{Mn}^{2+}\) and is saturated with \(0.10 M \mathrm{H}_{2} \mathrm{~S}\). Determine a \(\mathrm{pH}\) where CuS precipitates but MnS does not precipitate. \(K_{\text {op }}\) for \(\mathrm{CuS}=8.5 \times 10^{-45}\) and \(K_{\mathrm{s}}\) for \(\mathrm{MnS}=2.3 \times 10^{-1.3}\)

When \(\mathrm{Na}_{3} \mathrm{PO}_{4}(a q)\) is added to a solution containing a metal ion and a precipitate forms, the precipitate generally could be one of two possibilities. What are the two possibilities?

Calcium oxalate \(\left(\mathrm{CaC}_{2} \mathrm{O}_{4}\right)\) is relatively insoluble in water \(\left(K_{\mathrm{xp}}=\right.\) \(\left.2 \times 10^{-9}\right)\). However, calcium oxalate is more soluble in acidic solution. How much more soluble is calcium oxalate in \(0.10 \mathrm{M}\) \(\mathrm{H}^{+}\) than in pure water? In pure water, ignore the basic properties of \(\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\)

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