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10.6: Working with pH

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    Learning Objectives
    • Calculate pH from \([H_3O^+]\) and \([H_3O^+]\) from pH.

    Calculating pH from Hydronium Concentration

    The pH of solutions can be determined by using logarithms as illustrated in the next example for stomach acid. Stomach acid is a solution of \(HCl\) with a hydronium ion concentration of \(1.2 \times 10^{−3}\; M\), what is the \(pH\) of the solution?

    \[ \begin{align} \mathrm{pH} &= \mathrm{-\log [H_3O^+]} \nonumber \\ &=-\log(1.2 \times 10^{−3}) \nonumber \\ &=−(−2.92)=2.92 \nonumber \end{align}\]

    Logarithms

    To get the log value on your calculator, enter the number (in this case, the hydronium ion concentration) first, then press the LOG key.

    If the number is 1.0 x 10-5 (for [H3O+] = 1.0 x 10-5 M) you should get an answer of "-5".

    If you get a different answer, or an error, try pressing the LOG key before you enter the number.

    Example \(\PageIndex{2}\): Converting Ph to Hydronium Concentration

    Find the pH, given the \([H_3O^+]\) of the following:

    1. 1 ×10-3 M
    2. 2.5 ×10-11 M
    3. 4.7 ×10-9 M
    Solution

    Steps for Problem Solving
    Identify the "given" information and what the problem is asking you to "find."

    Given:

    1. [H3O+] =1 × 10−3 M
    2. [H3O+] =2.5 ×10-11 M
    3. [H3O+] = 4.7 ×10-9 M

    Find: ? pH
    Plan the problem.

    Need to use the expression for pH (Equation \ref{pH}).

    pH = - log [H3O+]
    Calculate.

    Now substitute the known quantity into the equation and solve.

    1. pH = - log [1 × 10−3 ] = 3.0 (1 decimal places since 1 has 1 significant figure)
    2. pH = - log [2.5 ×10-11] = 10.60 (2 decimal places since 2.5 has 2 significant figures)
    3. pH = - log [4.7 ×10-9] = 8.30 (2 decimal places since 4.7 has 2 significant figures)

    The other issue that concerns us here is significant figures. Because the number(s) before the decimal point in a logarithm relate to the power on 10, the number of digits after the decimal point is what determines the number of significant figures in the final answer:

    alt
    Exercise \(\PageIndex{2}\)

    Find the pH, given [H3O+] of the following:

    1. 5.8 ×10-4 M
    2. 1.0×10-7
    Answer a
    3.22
    Answer b
    7.00

    Calculating Hydronium Concentration from pH

    Sometimes you need to work "backwards"—you know the pH of a solution and need to find \([H_3O^+]\), or even the concentration of the acid solution. How do you do that? To convert pH into \([H_3O^+]\) we solve Equation \ref{pH} for \([H_3O^+]\). This involves taking the antilog (or inverse log) of the negative value of pH .

    \[[\ce{H3O^{+}}] = \text{antilog} (-pH)\]

    or

    \[[\ce{H_3O^+}] = 10^{-pH} \label{ph1}\]

    As mentioned above, different calculators work slightly differently—make sure you can do the following calculations using your calculator.

    Calculator Skills

    We have a solution with a pH = 8.3. What is [H3O+] ?

    With some calculators you will do things in the following order:

    1. Enter 8.3 as a negative number (use the key with both the +/- signs, not the subtraction key).
    2. Use your calculator's 2nd or Shift or INV (inverse) key to type in the symbol found above the LOG key. The shifted function should be 10x.
    3. You should get the answer 5.0 × 10-9.

    Other calculators require you to enter keys in the order they appear in the equation.

    1. Use the Shift or second function to key in the 10x function.
    2. Use the +/- key to type in a negative number, then type in 8.3.
    3. You should get the answer 5.0 × 10-9.

    If neither of these methods work, try rearranging the order in which you type in the keys. Don't give up—you must master your calculator!

    Example \(\PageIndex{3}\): Calculating Hydronium Concentration from pH

    Find the hydronium ion concentration in a solution with a pH of 12.6. Is this solution an acid or a base? How do you know?

    Solution
    Steps for Problem Solving
    Identify the "given" information and what the problem is asking you to "find."

    Given: pH = 12.6

    Find: [H3O+] = ? M
    Plan the problem.

    Need to use the expression for [H3O+] (Equation \ref{ph1}).

    [H3O+] = antilog (-pH) or [H3O+] = 10-pH
    Calculate.

    Now substitute the known quantity into the equation and solve.

    [H3O+] = antilog (12.60) = 2.5 x 10-13 M (2 significant figures since 4.7 has 12.60 2 decimal places)

    or

    [H3O+] = 10-12.60 = 2.5 x 10-13 M (2 significant figures since 4.7 has 12.60 2 decimal places)

    The other issue that concerns us here is significant figures. Because the number(s) before the decimal point in a logarithm relate to the power on 10, the number of digits after the decimal point is what determines the number of significant figures in the final answer:

    alt
    Exercise \(\PageIndex{3}\)

    If moist soil has a pH of 7.84, what is [H3O+] of the soil solution?

    Answer
    1.5 x 10-8 M

    10.6: Working with pH is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by LibreTexts.

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