The equilibrium constant for the reaction: H3BO3 + glycerin ⇌ (H3BO3 + glycerin complex) is 0.90. Ho — Chemical Equilibrium Chemistry Question

💡 Solution & Explanation

Step 1 - Define the Chemical Equation and Variables The complexation reaction between boric acid (\ce{H3BO3}) and glycerin in an aqueous solution is represented as: \[\ce{H3BO3 + glycerin <=> [H3BO3 \cdot glycerin\ complex]}\] We are given: * Volume of the solution, $V = 1\text{ L}$ * Initial concentration of boric acid, $[\ce{H3BO3}]_0 = 0.10\text{ M}$ * Equilibrium constant, $K_c = 0.90$ Let the total initial concentration of glycerin added to the solution be $y\text{ M}$. Step 2 - Determine the Equilibrium Concentrations According to the problem, $60\%$ of the initial boric acid must be converted into the boric acid-glycerin complex at equilibrium. The concentration of the complex formed at equilibrium is: \[[\text{Complex}]_{\text{eq}} = [\ce{H3BO3}]_0 \times 60\%\] \[[\text{Complex}]_{\text{eq}} = 0.10\text{ M} \times 0.60 = 0.06\text{ M}\] The concentration of unreacted boric acid remaining at equilibrium is: \[[\ce{H3BO3}]_{\text{eq}} = [\ce{H3BO3}]_0 - [\text{Complex}]_{\text{eq}}\] \[[\ce{H3BO3}]_{\text{eq}} = 0.10\text{ M} - 0.06\text{ M} = 0.04\text{ M}\] Since the stoichiometry of the reactants and products is $1:1$, the concentration of glycerin consumed to form the complex is also $0.06\text{ M}$. Thus, the concentration of free, uncomplexed glycerin remaining at equilibrium is: \[[\text{glycerin}]_{\text{eq}} = y - 0.06\text{ M}\] Step 3 - Write the Equilibrium Constant Expression The equilibrium constant ($K_c$) for the complexation reaction is defined as: \[K_c = \frac{[\text{Complex}]_{\text{eq}}}{[\ce{H3BO3}]_{\text{eq}} [\text{glycerin}]_{\text{eq}}}\] Substitute the values and expressions into the formula: \[0.90 = \frac{0.06\text{ M}}{0.04\text{ M} \times [\text{glycerin}]_{\text{eq}}}\] Step 4 - Calculate the Glycerin Added Now, solve for the concentration of free glycerin at equilibrium ($[\text{glycerin}]_{\text{eq}}$): \[[\text{glycerin}]_{\text{eq}} = \frac{0.06\text{ M}}{0.04\text{ M} \times 0.90}\] \[[\text{glycerin}]_{\text{eq}} = \frac{0.06}{0.036}\text{ M} = 1.667\text{ M}\] To find the total concentration of glycerin added ($y$), we sum the concentration of free glycerin at equilibrium and the concentration of glycerin that reacted to form the complex: \[\text{Total glycerin added } (y) = [\text{glycerin}]_{\text{eq}} + [\text{Complex}]_{\text{eq}}\] \[y = 1.667\text{ M} + 0.06\text{ M} = 1.727\text{ M} \approx \boxed{1.73\text{ M}}\] Step 5 - Evaluate the Options * **Option (A) Infinite**: Incorrect. This is a nonsensical limit, as a finite and achievable concentration of glycerin ($1.73\text{ M}$) is sufficient to convert $60\%$ of the boric acid. * **Option (B) 1.73 M**: Correct. As demonstrated, adding $1.73\text{ M}$ of glycerin results in a free equilibrium concentration of $1.67\text{ M}$ glycerin and $0.06\text{ M}$ complex, satisfying the equilibrium constant of $0.90$. * **Option (C) 0.10 M**: Incorrect. This value matches the initial concentration of boric acid, not the required concentration of glycerin. * **Option (D) 2.27 M**: Incorrect. This value is a calculation error, which can arise if one fails to account for the glycerin consumed during complexation or uses an incorrect stoichiometric ratio.