Problem 36 - Analytical Chemistry

Question

A student proposes to analyze barium gravimetrically by precipitating \(\mathrm{BaF}_{2}\) with NaF. Assuming a 200 -mg sample of \(\mathrm{Ba}^{2+}\) in \(100 \mathrm{~mL}\) is to be precipitated and that the precipitation must be \(99.9 \%\) complete for quantitative results, comment on the feasibility of the analysis.

Step-by-Step Solution

Answer

The analysis is feasible; sufficient fluoride must be maintained to achieve 99.9% precipitation.

Step 1: Define the reaction and solubility product

The reaction for precipitating barium with fluoride can be represented as: \( \text{Ba}^{2+} + 2\text{F}^- \rightarrow \text{BaF}_2 \downarrow \). The solubility product constant \( K_{sp} \) of \( \text{BaF}_2 \) is necessary to determine if the compound precipitates. It's known that \( K_{sp} = [\text{Ba}^{2+}][\text{F}^-]^2 \).

Step 2: Evaluate the required concentration for precipitation

We want the precipitation to be 99.9% complete, meaning only 0.1% remains in solution. If 200 mg of \( \text{Ba}^{2+} \) is in 100 mL, that's 0.002 moles of \( \text{Ba}^{2+} \). For 0.1% to remain, the concentration of \( \text{Ba}^{2+} \) that remains unprecipitated is \( 0.1 \% \times 0.002 = 0.000002 \) moles, or \( 2 \times 10^{-5} \text{ M} \).

Step 3: Calculate fluoride ion concentration required

Using the \( K_{sp} \) equation, \( [\text{F}^-]^2 \), given the \( K_{sp} = 1.84 \times 10^{-7} \), and \( [\text{Ba}^{2+}] = 2 \times 10^{-5} \text{ M} \), solve for \( [\text{F}^-] = \sqrt{\frac{K_{sp}}{[\text{Ba}^{2+}]}} = \sqrt{\frac{1.84 \times 10^{-7}}{2 \times 10^{-5}}} \).

Step 4: Perform the calculation

Calculating the concentration of fluoride gives: \( [\text{F}^-] = \sqrt{9.2 \times 10^{-3}} \approx 0.096 \text{ M} \). Thus, sodium fluoride should maintain a fluoride concentration of at least 0.096 M to achieve 99.9% precipitation of barium as \( \text{BaF}_2 \).

Step 5: Conclusion on feasibility

This required fluoride concentration is feasible to achieve under laboratory conditions, suggesting that the gravimetric analysis is practical under the provided conditions, assuming proper technique and controlled environment.

Key Concepts

Solubility Product

The solubility product constant, often denoted as \( K_{sp} \), is a vital concept in understanding how and when a compound will precipitate out of a solution. It is particularly important in the context of sparingly soluble salts, such as barium fluoride (\( \text{BaF}_2 \)). The solubility product is essentially the product of the ion concentrations of the dissolved ions each raised to the power of their coefficients in the equilibrium equation. For \( \text{BaF}_2 \), this can be expressed as:

• \( K_{sp} = [\text{Ba}^{2+}][\text{F}^-]^2 \)

This equation tells us that if the product of these ion concentrations exceeds the \( K_{sp} \) value, the solution becomes supersaturated, leading to precipitation. In the provided exercise, knowing \( K_{sp} \) allows us to determine the minimum concentration of fluoride ions needed to ensure that barium ions in the solution precipitate nearly completely.

Precipitation Reaction

Precipitation reactions are fundamental techniques in chemistry used to remove ions from a solution or in quantitative analytical methods like gravimetric analysis. In a precipitation reaction, two solutions containing soluble salts are combined to form an insoluble product called a precipitate. For example, when soluble sodium fluoride (\( \text{NaF} \)) is added to a solution containing barium ions, an insoluble barium fluoride (\( \text{BaF}_2 \)) precipitate forms:

• \( \text{Ba}^{2+} + 2\text{F}^- \rightarrow \text{BaF}_2 \downarrow \)

This process relies on the solubility product constant to determine if a precipitate will form and under what conditions. For a reaction to proceed to a significant extent, such as 99.9%, it is necessary to adjust the concentration of the reactants appropriately. When carrying out such reactions in practice, careful control of the conditions like concentration and temperature is essential to ensure successful precipitation.

Quantitative Analysis

Quantitative analysis in chemistry involves determining the amount or concentration of a substance in a sample. Gravimetric analysis is a quantitative method that relies on measuring the mass of a precipitate formed from a chemical reaction. A common application of this method is analyzing metal ions in a solution, as is the case with barium in the provided exercise. To achieve accurate results in gravimetric analysis, it is critical to ensure that the precipitation reaction is complete. This means nearly all of the target substance must be converted into the precipitate, typically exceeding a completion percentage—like 99.9% in our example. Successful quantitative analysis requires careful attention to detail and the precise measurement of reactants and products. It is also important to consider the reaction's stoichiometry to calculate the initial concentrations needed to achieve near-complete precipitation. Thus, understanding the principles of solubility and precipitation reactions is indispensable for accurately assessing composition in gravimetric analysis.