# 8.29: Brønsted-Lowry Acids and Bases: Calculating Hydronium Ion Concentrations from pH Values

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As stated previously, chemists have derived two equations that can be used to calculate the amounts of acid and base that are present in a solution. The first of these mathematical statements, which is known as the Ion-Product Constant Equation, relates the molarities of the acidic hydronium ions, H_{3}O^{+1}, and basic hydroxide ions, OH^{–1}, that are present in a solution by equating the mathematical product of these concentrations to a constant, K_{w}, which is a unitless quantity that has a numerical value of 1.00 x 10^{–14}. The concentrations that can be incorporated into, and calculated from, this equation are very small quantities and, therefore, are written in scientific notation. Consequently, hydronium, H_{3}O^{+1}, and hydroxide, OH^{–1}, ion molarities are compared by analyzing the negative exponents that are written in these numerical representations. Finally, because these concentrations collectively span across an exponential range of values, correctly interpreting the difference in magnitude between these molarities is prohibitively-challenging.

Therefore, in order to both change the numerical interval between sequential values and compress the range that is spanned by this data, these hydronium ion, H_{3}O^{+1}, and hydroxide ion, OH^{–1}, concentrations can be represented using a logarithmic scale. When converting a number that is written using scientific notation to a logarithmic quantity, the coefficient and power that are written when expressing a hydronium, H_{3}O^{+1}, concentration using scientific notation are directly related to one another and, therefore, reported using a single value, which is known as a "pH." The pH Equation relates the pH of a solution to the *negative* logarithm of a hydronium ion, H_{3}O^{+1}, concentration. Because the hydronium ion, H_{3}O^{+1}, is the particle that chemists chose to measure, in order to determine the amount of acid that is present in a given solution, pH is defined as a quantitative measurement that indicates the relative amount of *acid* that is present in a system. Finally, since, by mathematical definition, a value that is calculated using a logarithmic scale cannot have an associated unit, the pH of a solution is expressed as a unitless quantity.

As stated in the previous section of this chapter, no algebra is required to calculate a pH value. Therefore, in order to determine a hydronium ion, H_{3}O^{+1}, concentration from a given pH value, the pH Equation, which is reproduced below, must be mathematically-reformatted.

\(\rm{pH}\) = \({–\rm{log_{10}}}\)\(\rm{[H_3O^{+1}]}\)

In order to eliminate the negative sign from the right side of this equation, both the left and right sides of this mathematical statement must be multiplied *or* divided by –1.

\({–}\)\(\rm{pH}\) = \({\rm{log_{10}}}\)\(\rm{[H_3O^{+1}]}\)

Subsequently, the logarithmic value that is present in this equation must be mathematically-canceled by applying an **anti-log**. Just as performing a division cancels the effects of a multiplicative relationship, calculating an anti-log reverses the scaling of a logarithmic value.

\(\rm{antilog(–pH)}\) = \({\cancel{\rm{antilog}}}\)(\({\cancel{\rm{log_{10}}}}\)\(\rm{[H_3O^{+1}]}\))

\(\rm{antilog(–pH)}\) = \(\rm{[H_3O^{+1}]}\)

Recall that representing a number that is written in scientific notation using a logarithmic scale eliminates the "base 10" portion of the initial value, and, consequently, the remaining coefficient and power are reported using a single value. Therefore, calculating the anti-log of a number is equivalent to *reincorporating* this "base 10" value into the mathematical statement.

\({10^{\rm{–pH}}}\) = \(\rm{[H_3O^{+1}]}\)

This equation, which is known as the **anti-log pH Equation**, in reference to the anti-log mathematical operation that was applied in its derivation, can also be written so that the variable quantity is represented on the left side of the equal sign, as shown below.

\(\rm{[H_3O^{+1}]}\) = \({10^{\rm{–pH}}}\)

In order to align with the standards that were established in the previous sections of this chapter, the numerical solution, which corresponds to a concentration of hydronium ions, H_{3}O^{+1}, must be expressed in scientific notation and reported to three significant figures. Finally, the resultant value is a molarity, and, therefore, is labeled with a unit of *"**M*,*"**even though this unit is not established through the mathematical processes that are described above*.