pH and pOH of Strong Acids and Bases强酸强碱的 pH 与 pOH

Big Idea · Structure and Properties (SAP)AP Science Practice 5 · Mathematical RoutinesAP link · Unit 8~40 min

Learning goals

  • Recall the common strong acids and strong bases, and write equations for their complete ionization in water
  • Calculate [H₃O⁺], [OH⁻], pH and pOH from the concentration of a strong acid or base, noting that M(OH)₂ gives 2 OH⁻
  • Calculate the pH after mixing a strong acid with a strong base by finding the excess H⁺ or OH⁻ and dividing by the total volume

Before you start

  • Strong and weak acids and bases (2.12.2)
  • pH and pOH (2.12.3)
  • Introduction to acids and bases (U8.1)

Strong acids and bases ionize completely

A strong acid强酸 ionizes essentially completely in water, so almost no un-ionized acid molecules remain. For AP, know the six strong acids: HCl, HBr, HI, HNO₃, HClO₄ and H₂SO₄ (only its first ionization is complete).

HNOX3+HX2O→HX3OX++NOX3X−\ce{HNO3 + H2O -> H3O+ + NO3-}

So in a solution of a monoprotic strong acid, [H₃O⁺] = [acid].

The strong bases强碱 are the group 1 hydroxides (LiOH, NaOH, KOH and so on) and Ca(OH)₂, Sr(OH)₂ and Ba(OH)₂. Each formula unit of the last three gives 2 OH⁻:

Ba(OH)X2→BaX2++2 OHX−[OHX−]=2[Ba(OH)X2]\begin{gathered} \ce{Ba(OH)2 -> Ba^2+ + 2OH-} \\ [\ce{OH-}] = 2[\ce{Ba(OH)2}] \end{gathered}

Given any one of [H⁺], [OH⁻], pH and pOH, you can find the other three. Click “×10” or “÷10” and watch the pH change by exactly 1 each time:

Try it: convert between [H⁺], [OH⁻], pH, and pOH (25 °C)
Given
× 10^M

    Mixing a strong acid with a strong base

    When a strong acid and a strong base are mixed, H₃O⁺ and OH⁻ react completely (H₃O⁺ + OH⁻ → 2H₂O). The calculation takes three steps:

    1. Find n(H₃O⁺) and n(OH⁻) separately (multiply by 2 for Ba(OH)₂).
    2. Subtract to see which one is in excess, and by how much.
    3. Divide the excess amount by the total volume after mixing, then find the pH. If the two amounts are equal, pH = 7.00 at 25 °C.

    Drag the volumes and concentrations, and compare NaOH with Ba(OH)₂:

    Try it: pH after mixing a strong acid and a strong base

    When the leftover H₃O⁺ or OH⁻ is close to 10⁻⁷ M, the ions from water’s own ionization can’t be ignored. So diluting a strong acid further and further only brings the pH closer to 7; it never goes past 7.

    Example 1: a Ca(OH)₂ solution

    Find the pH of 0.0050 M Ca(OH)₂ at 25 °C.

    Step-by-step solution

    1. A strong base dissociates completely, so count OH⁻ from the formula (in M). Each Ca(OH)₂ gives 2 OH⁻:

      Ca(OH)X2→CaX2++2 OHX−[OHX−]=2×0.0050=0.010\begin{gathered} \ce{Ca(OH)2 -> Ca^2+ + 2OH-} \\ [\ce{OH-}] = 2 \times 0.0050 = 0.010 \end{gathered}

    2. Calculate the pOH.

      pOH=−log⁡0.010=−log⁡(10−2)=2.00\begin{aligned} \text{pOH} &= -\log 0.010 \\ &= -\log(10^{-2}) = 2.00 \end{aligned}

    3. Calculate the pH.

      pH=14.00−2.00=12.00\text{pH} = 14.00 - 2.00 = \boxed{12.00}

    4. Check: can all the Ca(OH)₂ dissolve? From Ksp = 1.3 × 10⁻⁶ (OpenStax), the solubility of the saturated solution (in M) follows from Ksp = s(2s)² = 4s³:

      s=1.3×10−643=6.9×10−3s = \sqrt[3]{\frac{1.3 \times 10^{-6}}{4}} = 6.9 \times 10^{-3}

      0.0050 < 6.9 × 10⁻³, so the solution isn’t saturated and all the Ca(OH)₂ dissolves, as assumed in step 1.

    Example 2: neutralizing HNO₃ with Ba(OH)₂

    What volume of 0.0250 M Ba(OH)₂ exactly neutralizes 40.0 mL of 0.100 M HNO₃? If only 60.0 mL of the Ba(OH)₂ solution is added, what is the pH of the mixture?

    Step-by-step solution

    1. Write the reaction and find the ratio. The net reaction is H₃O⁺ + OH⁻ → 2H₂O, and each Ba(OH)₂ supplies 2 OH⁻, so the overall equation is 2HNO₃ + Ba(OH)₂ → Ba(NO₃)₂ + 2H₂O.

    2. Amount of H₃O⁺. HNO₃ is a strong acid and ionizes completely; volume (L) × molarity (mol/L) = moles:

      n(HX3OX+)=0.0400×0.100=4.00×10−3 mol\begin{aligned} n(\ce{H3O+}) &= 0.0400 \times 0.100 \\ &= 4.00 \times 10^{-3}\ \text{mol} \end{aligned}

    3. Volume for exact neutralization. An equal amount of OH⁻ is needed, which is half as many moles of Ba(OH)₂:

      n(Ba(OH)X2)=12×4.00×10−3=2.00×10−3 mol\begin{aligned} n(\ce{Ba(OH)2}) &= \tfrac{1}{2} \times 4.00 \times 10^{-3} \\ &= 2.00 \times 10^{-3}\ \text{mol} \end{aligned} V=2.00×10−30.0250=0.0800 L=80.0 mL\begin{aligned} V &= \frac{2.00 \times 10^{-3}}{0.0250} = 0.0800\ \text{L} \\ &= \boxed{80.0\ \text{mL}} \end{aligned}

    4. With only 60.0 mL added: amount of OH⁻.

      n(OHX−)=2×0.0600×0.0250=3.00×10−3 mol\begin{aligned} n(\ce{OH-}) &= 2 \times 0.0600 \times 0.0250 \\ &= 3.00 \times 10^{-3}\ \text{mol} \end{aligned}

    5. BCA table (in mmol). There isn’t enough OH⁻, so H₃O⁺ is left over:

      H₃O⁺OH⁻
      Before4.003.00
      Change−3.00−3.00
      After1.000

    6. Divide by the total volume and find the pH. The total volume is 40.0 + 60.0 = 100.0 mL = 0.1000 L, and [H₃O⁺] is in M:

      [HX3OX+]=1.00×10−30.1000=1.00×10−2pH=−log⁡(1.00×10−2)=2.000\begin{aligned} [\ce{H3O+}] &= \frac{1.00 \times 10^{-3}}{0.1000} \\ &= 1.00 \times 10^{-2} \\ \text{pH} &= -\log(1.00 \times 10^{-2}) \\ &= \boxed{2.000} \end{aligned}

      1.00 × 10⁻² has three significant figures, so the pH gets three decimal places.

    Practice: which kind of substance is it?
    1. HI
    2. HNO₂
    3. HClO₄
    4. HClO
    5. LiOH
    6. Sr(OH)₂
    7. NH₃
    8. HCOOH

    Common mistakes

    Mistake 1: “A strong acid solution always has a lower pH than a weak acid solution.” The pH also depends on concentration. 1.0 × 10⁻⁴ M HCl has pH 4.00, while 0.10 M CH₃COOH has a pH of about 2.88.

    Mistake 2: “In 0.010 M Ba(OH)₂, [OH⁻] = 0.010 M.” Each Ba(OH)₂ gives 2 OH⁻, so [OH⁻] = 0.020 M.

    Mistake 3: “After mixing, use the original volume to find the concentration.” Mixing two solutions increases the volume, so divide the amount of the excess ion by the total volume.

    Quick check

    What is the pH of 0.0040 M HNO₃?

    What is the pH of 0.0025 M Sr(OH)₂?

    Equal volumes of 0.10 M HNO₃ and 0.050 M Ba(OH)₂ are mixed. At 25 °C the resulting solution:

    A solution has [H₃O⁺] = 6.3 × 10⁻³ M. How should its pH be reported?

    Which solution has the highest pH?

    Which equation correctly represents HClO₄ dissolving in water?

    AP-style free response

    AP-style free response

    A lab has a 0.0200 M HClO₄ solution at 25 °C.

    (a) Calculate the pH of this solution.

    (b) 10.0 mL of it is diluted with water to 250.0 mL. Calculate the pH after dilution.

    (c) Another 25.0 mL of the original solution is mixed with 15.0 mL of 0.0400 M NaOH. Calculate the pH of the mixture.

    (d) What volume of 0.0400 M NaOH exactly neutralizes 25.0 mL of the original solution?

    (e) A student concludes that 1.0 × 10⁻⁸ M HClO₄ has pH 8.00. Explain why this conclusion is unreasonable, and estimate the actual pH.

    Model answer and scoring

    (a) [H₃O⁺] = 0.0200 M, so pH = −log 0.0200 = 1.699.

    (b) c = 0.0200 × 10.0 ÷ 250.0 = 8.00 × 10⁻⁴ M, so pH = 3.097.

    (c) n(H₃O⁺) = 0.0250 × 0.0200 = 5.00 × 10⁻⁴ mol; n(OH⁻) = 0.0150 × 0.0400 = 6.00 × 10⁻⁴ mol. OH⁻ is in excess by 1.00 × 10⁻⁴ mol in a total of 40.0 mL, so [OH⁻] = 2.50 × 10⁻³ M, pOH = 2.602 and pH = 11.398.

    (d) 5.00 × 10⁻⁴ mol ÷ 0.0400 M = 0.0125 L = 12.5 mL.

    (e) An acid was added, so the solution can’t be basic. The H₃O⁺ from HClO₄ (10⁻⁸ M) is less than the H₃O⁺ from water (about 10⁻⁷ M), so water’s ionization can’t be ignored. The actual [H₃O⁺] is slightly more than 1.0 × 10⁻⁷ M, giving a pH slightly below 7 (about 6.98).

    Scoring: (a) pH; (b) concentration after dilution; (c) amounts of both ions, the excess ion, and division by the total volume; (d) volume; (e) water’s ionization identified, pH slightly below 7.

    Summary

    • The six strong acids, the group 1 hydroxides and the hydroxides of Ca, Sr and Ba ionize completely in water; M(OH)₂ gives 2 OH⁻.
    • For a strong acid or base, [H₃O⁺] or [OH⁻] comes straight from the concentration, and then you find the pH.
    • Mixing: find n(H₃O⁺) and n(OH⁻), subtract to get the excess ion, and divide by the total volume.

    AP tip: In a mixing problem, show the calculation of n(H₃O⁺) and n(OH⁻), say which one is in excess, and state the total volume. Report the pH with the number of decimal places set by the significant figures.

    Practice this chapter: Unit 8 · Acids and Bases · 12 questionsPractice this topic: U8.2 · pH and pOH of Strong Acids and Bases · 5 questions

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