Question

A solution contains \(0.1 \mathrm{M}-\mathrm{Mg}^{2+}\) and \(\begin{array}{lll}0.1 & \mathrm{M} & -\mathrm{Sr}^{2+} & \text { The concentration of }\end{array}\) \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in solution is adjusted to \(0.05 \mathrm{M}\). Determine the \(\mathrm{pH}\) range which would permit the precipitation of \(\mathrm{SrCO}_{3}\) without any precipitation of \(\mathrm{MgCO}_{3} . \mathrm{H}^{+}\) ion concentration is controlled by external factors. Given: \(K_{\mathrm{sp}}\left(\mathrm{MgCO}_{3}\right)=4 \times 10^{-8} \mathrm{M}^{2}\); \(K_{\mathrm{sp}}\left(\mathrm{SrCO}_{3}\right)=9 \times 10^{-10} \mathrm{M}^{2} ; K_{\mathrm{a}, \text { overall }}\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)\) \(=5 \times 10^{-17} ; \log 2=0.3 ; \log 3=0.48 .\) (a) \(4.78\) to \(5.6\) (b) \(4.6\) to \(5.78\) (c) \(5.78\) to \(6.4\) (d) \(5.22\) to \(5.4\)

Short answer

The correct option is (d) pH range from 5.22 to 5.4.

Step-by-step solution

- Write down the solubility product expression for MgCO3

For magnesium carbonate (MgCO3), the solubility product is expressed as: \[K_{\mathrm{sp}}(\mathrm{MgCO}_3) = [\mathrm{Mg}^{2+}][\mathrm{CO}_3^{2-}]\] Given, \[K_{\mathrm{sp}}(\mathrm{MgCO}_3) = 4 \times 10^{-8}\]

- Calculate the concentration of CO3(2-) at equilibrium for MgCO3

Since the concentration of Mg2+ is 0.1 M, the equilibrium concentration of CO3(2-) can be calculated by rearranging the solubility product expression: \[[\mathrm{CO}_3^{2-}] = \frac{K_{\mathrm{sp}}(\mathrm{MgCO}_3)}{[\mathrm{Mg}^{2+}]} = \frac{4 \times 10^{-8}}{0.1} = 4 \times 10^{-7} \, \mathrm{M}\]

- Write down the solubility product expression for SrCO3

For strontium carbonate (SrCO3), the solubility product expression is: \[K_{\mathrm{sp}}(\mathrm{SrCO}_3) = [\mathrm{Sr}^{2+}][\mathrm{CO}_3^{2-}]\] Given, \[K_{\mathrm{sp}}(\mathrm{SrCO}_3) = 9 \times 10^{-10}\]

- Calculate the equilibrium concentration of Sr2+ when SrCO3 begins to precipitate

As the concentration of CO3(2-) increases, SrCO3 will precipitate when the product of the concentrations of Sr2+ and CO3(2-) equals the Ksp of SrCO3: \[[\mathrm{Sr}^{2+}] = \frac{K_{\mathrm{sp}}(\mathrm{SrCO}_3)}{[\mathrm{CO}_3^{2-}]}\]

- Determine the [H2CO3] concentration at equilibrium

The dissociation of H2CO3 can be written as: \[\mathrm{H}_2\mathrm{CO}_3 \rightleftharpoons \mathrm{H}^{+} + \mathrm{HCO}_3^{-}\] The second dissociation step can be ignored due to its much smaller contribution, which yields a total dissociation constant expression: \[K_{a, \text{ overall}} = \frac{[\mathrm{H}^{+}][\mathrm{CO}_3^{2-}]}{[\mathrm{H}_2\mathrm{CO}_3]}\]

- Calculate [H+] at which SrCO3 would precipitate without MgCO3 precipitating

Rearrange the dissociation constant expression to solve for [H+]: \[[\mathrm{H}^{+}] = K_{a, \text{overall}} \cdot \frac{[\mathrm{H}_2\mathrm{CO}_3]}{[\mathrm{CO}_3^{2-}]}\]

- Find the pH range for the precipitation condition

The pH is the negative logarithm of the hydrogen ion concentration: \[\text{pH} = -\log([\mathrm{H}^{+}])\] Solve for pH at the determined [H+] from step 6, ensuring SrCO3 precipitates without MgCO3 precipitating.

- Verify which pH range option matches the calculated pH range

Compare the calculated pH range to the given answer choices to determine which option is correct.

Key concepts

Chemical Equilibrium

In chemistry, an equilibrium state describes a situation where the rate of the forward reaction equals the rate of the reverse reaction. In such a state, the concentrations of reactants and products remain constant over time because the processes are occurring at the same rate. This can be seen in the dissolution and precipitation of substances in water. For instance, when a salt is partly soluble, such as calcium carbonate, it dissolves until the solution becomes saturated. Any additional amount will remain undissolved and establish an equilibrium with the dissolved ions.

When addressing homework problems, understanding equilibrium is essential for correctly predicting whether a salt will dissolve further or precipitate out of solution under certain conditions. Through this comprehension, students can apply the concept to real-world scenarios, such as the formation of scale in pipes or the purification of a substance.

Ksp (Solubility Product Constant)

The solubility product constant, abbreviated as Ksp, is an important aspect of chemical solubility and equilibrium. It represents the maximum possible concentration of ions in a solution when a compound dissolves. Ksp is specific to each substance and provides valuable insight into its solubility. This constant helps us determine the point at which the solution becomes saturated and precipitation begins.

In the context of the exercise, Ksp is used to express the concentration relationship between magnesium or strontium ions and carbonate ions. For a substance to precipitate, the product of the concentrations of the ions must exceed the Ksp value. Mastery of Ksp calculations is fundamental for students aiming to understand precipitation, dissolution, and the effects of varying conditions on solubility.

Precipitation and Solubility

Precipitation is the process where ions in solution combine to form a solid that settles out of the solution, also known as a precipitate. The solubility of a substance is its ability to dissolve in a solvent, usually water. These two phenomena are deeply interconnected; solubility determines how much of a substance can dissolve in a solvent before the excess starts precipitating.

Factors such as temperature, pressure, and the presence of other ions can significantly affect solubility and precipitation. Understanding these factors allows students to predict and control the formation of precipitates in both laboratory and natural environments. It's important for students to notice how the common ion effect, pH changes, and other variables can shift the solubility equilibrium, influencing the occurrence of precipitation.

pH Calculation

The pH of a solution is a measure of its acidity or basicity, which is defined by the hydrogen ion (H+) concentration. Calculating pH is a fundamental skill in chemistry that requires knowledge of the negative logarithm of the H+ concentration. A low pH indicates a high H+ concentration (acidic conditions), while a high pH indicates a low H+ concentration (basic conditions).

The exercise provided involves adjusting the pH to precipitate one salt without affecting another, which makes pH calculation crucial. Students must understand the relationship between the pH, the ionization of weak acids or bases, and how it influences the solubility of ionic compounds. This level of comprehension is key to solving not only theoretical problems but also practical ones, such as adjusting the conditions to separate compounds in a mixture through selective precipitation.