If the hydroxide concentration in common duct bile is 1 × 10⁻⁶ M, what is the pH of the bile?

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11ecba2d_11ea_11af_a3bf_7b567fc6faea_MD0008_00 The pH of a solution relates directly to the hydrogen ion (or hydronium) concentration [H₃O⁺] by the relationship pH = −log[H₃O⁺]. If the hydroxide concentration [OH⁻] is known instead, it is related to the [H₃O⁺] by the self-ionization constant K_w of water as given by: K_w = [H₃O⁺][OH⁻] = 1.0 × 10⁻¹⁴ M² Accordingly, dividing K_w by the known [OH⁻] permits the calculation of [H₃O⁺] as follows: [H₃O⁺] = K_w /[OH⁻] = (1.0 × 10⁻¹⁴ M²)/(1.0 × 10⁻⁶ M) [H₃O⁺] = 1.0 × 10⁻⁸ M The pH of the solution can then be determined: pH = −log[H₃O⁺] = −log(1.0 × 10⁻⁸) = 8 Alternatively, the pOH can be used to find the pH, given that pOH = −log[OH⁻] and that pH + pOH = 14 (as derived from K_w). By this method, calculation of the pOH is given by: pOH = −log[OH⁻] = −log(1.0 × 10⁻⁶) = 6 and the subsequent evaluation of the pH is given by: pH = (14 − pOH) = (14 − 6) = 8 (Choice A) A value of 6 corresponds to the pOH rather than to the pH. The calculation of pOH = −log[OH⁻] whereas pH = −log[H₃O⁺]. (Choice B) Based on the ionization constant K_w, the product of [OH⁻] and [H⁺] is always equal to 10⁻¹⁴. A neutral pH of 7 requires that [OH⁻] = [H₃O⁺] = 1.0 × 10⁻⁷ M. This is not the case for the bile, which has [OH⁻] = 1.0 × 10⁻⁶ M. (Choice D) The value of pK_w = 14. This is not equivalent to the pH. Educational objective: The [H₃O⁺] and [OH⁻] concentrations are related by the self-ionization constant K_w of water according to the expression K_w = [H₃O⁺][OH⁻] = 1.0 × 10⁻¹⁴ M². Using a logarithmic approach, pH is related to [OH⁻] by the relationship pH + pOH = 14, where pOH = −log[OH⁻].

If the Hydroxide Concentration in Common Duct Bile Is 1