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 [H 3 O + ] = 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 ×10 -3 M 2.5 ×10 -11 M 4.7 ×10 -9 M Solution Steps for Problem Solving Unnamed: 1 Identify the "given" information and what the problem is asking you to "find." Given: [H3O+] =1 × 10−3 M [H3O+] =2.5 ×10-11 M [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. pH = - log [1 × 10−3 ] = 3.0 (1 decimal places since 1 has 1 significant figure) pH = - log [2.5 ×10-11] = 10.60 (2 decimal places since 2.5 has 2 significant figures) 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: Exercise \(\PageIndex{2}\) Find the pH, given [H 3 O + ] of the following: 5.8 ×10 -4 M 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 [H 3 O + ] ? With some calculators you will do things in the following order: Enter 8.3 as a negative number (use the key with both the +/- signs, not the subtraction key). 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 10 x . You should get the answer 5.0 × 10 -9 . Other calculators require you to enter keys in the order they appear in the equation. Use the Shift or second function to key in the 10 x function. Use the +/- key to type in a negative number, then type in 8.3. 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 Unnamed: 1 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: Exercise \(\PageIndex{3}\) If moist soil has a pH of 7.84, what is [H 3 O + ] of the soil solution? Answer 1.5 x 10 -8 M