An unknown metal M forms an ionic hydroxide with the formula M(OH)₂ that exhibits the equilibrium aqin a saturated aqueous solution. If the solution pH is 10, the solubility product constant K_sp of the compound is:

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11ecba2d_1240_489c_a3bf_e5155873d2ad_MD0008_00 For an ionic compound Z that dissociates into ions X and Y when placed in water, an equilibrium will be established in which the rate of dissolution (forward reaction) equals the rate of precipitation (reverse reaction). 11ecba2d_1240_6fad_a3bf_37a2ac59ab41_MD0008_00 This equilibrium position is indicated by the solubility product constant K_sp, which provides a measure of the solubility of a compound at a given temperature. The general K_sp expression is written as: 11ecba2d_1240_6fae_a3bf_2b58b6ecd623_MD0008_00 where [X] and [Y] are the molar concentrations of the X and Y ions, and b and c are the respective coefficients of the balanced dissolution reaction. The K_sp is the product of the maximum possible concentrations of each ion in the solution under the given conditions. Applying these concepts to the solubility equilibrium of M(OH)₂, the K_sp expression for the compound is: 11ecba2d_1240_6faf_a3bf_c78182fd8245_MD0008_00 Using the pH of the saturated solution, the [OH⁻] can be determined by first evaluating the pOH. 11ecba2d_1240_6fb0_a3bf_299ae18cc10c_MD0008_00 11ecba2d_1240_6fb1_a3bf_8388b9e954cc_MD0008_00 The [OH⁻] is then determined from its relationship to pOH. 11ecba2d_1240_6fb2_a3bf_5b91284336b1_MD0008_00 11ecba2d_1240_96c3_a3bf_aba5e902bdac_MD0008_00 The equilibrium shows that 2 moles of OH⁻ are produced for every 1 mole of M²⁺. As a result, [M²⁺] is equal to half of [OH⁻]. 11ecba2d_1240_96c4_a3bf_73603e2dd477_MD0008_00 Substituting the values of [M²⁺] and [OH⁻] into the K_sp expression gives: 11ecba2d_1240_96c5_a3bf_31e9d9d4df3a_MD0008_00 (Choice A) A value of 5.0 × 10⁻³¹ is obtained if 10^−pH is used for [OH⁻] instead of 10^−pOH. (Choice C) A value of 1.0 × 10⁻¹² results from assuming that [M²⁺] = [OH⁻]. (Choice D) A value of 1.0 × 10⁻⁸ is calculated by assuming that [M²⁺] = [OH⁻] and also failing to square the [OH⁻] term. Educational objective: The solubility product constant K_sp is defined as the product of the maximum molar concentrations of the dissolved species, each raised to the power of their respective coefficients from the balanced dissolution reaction. An equilibrium involving [OH⁻] can be evaluated from the pH given that pH + pOH = 14 and pOH = −log[OH⁻].

An Unknown Metal M Forms an Ionic Hydroxide with the Formula