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

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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, in order to eliminate the unnecessary placeholder zeroes that must be present when these numbers are written in decimal format. 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. Additionally, because these concentrations collectively span across an exponential range of values, correctly interpreting the difference in magnitude between these molarities is prohibitively-challenging.

## The pH Equation

In 1909, Danish biochemist Søren Sørenson recognized that these hydronium ion, H_{3}O^{+1}, and hydroxide ion, OH^{–1}, concentrations could be represented using a logarithmic scale, which changes the numerical interval between sequential values and, consequently, compresses the range that is spanned by the corresponding data. Because scientific notation is written using a *base 10 exponential* scale, Sørenson used a *base 10 logarithmic* scale when defining his system. As a result of this conversion, the coefficient and power that are written when expressing a number using scientific notation can be directly related to one another and, therefore, reported using a single value. Sørenson chose to incorporate the molarity of hydronium ions, H_{3}O^{+1}, into his logarithmic equation and named the resultant mathematical relationship, which is shown below, the "**pH Equation**." This title was originally chosen as an abbreviation for the Latin phrase *"pondus hydrogenii*,

*"*which translates to "quantity of hydrogen," because a hydronium ion, H

_{3}O

^{+1}, the particle that is represented in this equation, is generated when a water molecule absorbs a proton, H

^{+1}, which, in turn, is formed from the ionization of a

*hydrogen*atom. Scientists also use the letters "pH" to represent the phrase "power of hydrogen," in reference to the

*exponent*that is written when expressing a number in scientific notation. 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.

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

When converting a number that is written in scientific notation to a logarithmic value, the sign of the exponent is reflected in the sign of the corresponding calculated value. Therefore, because hydronium ion, H_{3}O^{+1}, concentrations are very small values and, therefore, are written in scientific notation using a *negative* exponent, the logarithmic equivalent to this value is also *negative*. Consequently, in order to calculate a *positive* pH value, Sørenson defined pH to be the *negative* of this logarithmic concentration, since multiplying two negative quantities generates a positive numerical solution. Finally, since, by mathematical definition, a value that is calculated using a logarithmic scale cannot have an associated unit, the pH of a solution must be expressed as a unitless quantity.

## pH Calculations and the pH Scale

Unlike all of the mathematical equations that have been previously presented in this textbook, no algebra is required to calculate a pH value. When using a calculator to determine a pH, the negative key, (–), not the subtraction button, must be used, and, since most calculators are automatically-programmed to calculate logarithms on a "base 10" scale, the subscript value of "10" should not be entered. Additionally, any quantity that is expressed in scientific notation should be offset by parentheses. While most calculators allow for logarithmic expressions to be typed exactly as shown above, a few less-sophisticated models require that the hydronium ion, H_{3}O^{+1}, concentration be entered first, and the "log" and negative keys, (–), respectively, are pressed afterward. Finally, in order to align with the standards that were established in the previous sections of this chapter, chemists required that pH values, which are usually rounded quantities, be recorded to the hundredths place.

For example, calculate the pH of a solution that has a 1.00 x 10^{–7} * M* concentration of hydronium ions.

The molarity of hydronium ions, H_{3}O^{+1}, that is present in a solution is related to the pH of that solution by the pH Equation. Therefore, in order to calculate the pH of a solution, the molar concentration of hydronium ions, H_{3}O^{+1}, 1.00 x 10^{–7} *M*, is incorporated into this equation, as shown below. When using a calculator, the negative key, (–), must be used, and, since most calculators are automatically-programmed to calculate logarithms on a "base 10" scale, the subscript value of "10" should not be entered. Additionally, any quantity that is expressed in scientific notation should be offset by parentheses. Furthermore, in order to align with the standards that were established in the previous sections of this chapter, a pH value, which is usually a rounded quantity, must be recorded to the hundredths place. Finally, because, by mathematical definition, a value that is calculated using a logarithmic scale cannot have an associated unit, the resultant pH is expressed as a unitless number.

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

\(\rm{pH}\) = \({–\rm{log_{10}}}\)(\({1.00 \times 10^{–7} M}\))

\(\rm{pH}\) = \({7} ≈ {7.00}\)

As exemplified by the problem that is shown above, a pH value is straight-forward to interpret, relative to a molar concentration, which must, as stated previously, be written in scientific notation. The given molarity, 1.00 x 10^{–7} * M*, corresponds to the concentration of hydronium ions, H

_{3}O

^{+1}, that are present in a neutral solution. Because a pH value is, mathematically, an

*equivalent expression*of a molar concentration, only the

*scale*, not the

*relative meaning*, of a value is changed through the application of the pH Equation. Therefore, the pH value of a neutral solution is, as calculated above,

*exactly*7.00.

In the previous section of this chapter, this given molar concentration of hydronium ions, H_{3}O^{+1}, 1.00 x 10^{–8} *M*, was compared to the quantity of hydroxide ions, OH^{–1}, 1.00 x 10^{–6} *M*, that were present that solution. Because the hydroxide ion, OH^{–1}, concentration had the larger numerical value, the solution was classified as basic. As stated above, because a pH value is a *mathematically-equivalent expression* of a molar concentration, only the *scale*, not the *relative meaning*, of a value is changed through the application of the pH Equation. Therefore, a solution that has a pH value of 8.00 must be classified as basic. By analyzing all possible hydronium, H_{3}O^{+1}, and hydroxide, OH^{–1}, ion concentration combinations, chemists established that *all* solutions that have pH values that are *greater than* 7.00 can be categorized as basic, and all solutions that have pH values that are *less** than than* 7.00 can be classified as acidic.

Recall that the purpose of representing the molar concentrations of hydronium ions, H_{3}O^{+1}, using a logarithmic scale is to change the numerical interval between sequential values and, consequently, compress the range that is spanned by the corresponding data. The hydronium ion, H_{3}O^{+1}, concentrations that are given above, 1.00 x 10^{–7} *M* and 1.00 x 10^{–8} *M*, respectively, are different from one another by a *power of 10*. Furthermore, since the magnitude of a number that is written in scientific notation *increases *as the value of a * negative power* decreases, the solution that has a 1.00 x 10

^{–7}

*M*hydronium ion, H

_{3}O

^{+1}, concentration

*contains ten times more*hydronium ions, H

_{3}O

^{+1}, than the other solution. Because the corresponding pH values, 7.00 and 8.00, respectively, only differ by a 1.00 "unit" increment, representing the molar concentrations of hydronium ions, H

_{3}O

^{+1}, using a logarithmic scale

*does*compress the range that is spanned by the corresponding data, as intended. Finally, since 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, the solution that has a 1.00 x 10

^{–7}

*M*hydronium ion, H

_{3}O

^{+1}, concentration is

*ten times more acidic*than the other solution. Therefore, because the corresponding pH value of this solution, 7.00, is 1.00 "units"

*smaller*than the pH of the other solution, 8.00, as the molar concentrations of hydronium ions, H

_{3}O

^{+1},

*increases*and a solution becomes

*more acidic*, the corresponding pH value of that solution

*decreases*.

Since a pH is a mathematically-equivalent expression of a molar concentration of hydronium ions, H_{3}O^{+1}, the range of values that can be calculated using the pH Equation are not bound by defined upper- or lower-limits. The lowest and, therefore, most acidic, pH measurement that has been recorded is –3.6, and the highest, most basic, reported pH value is 17.6. However, most solutions, like those that are referenced below in Figure \(\PageIndex{1}\), have pH values between 0.00 to 14.00, which are collectively-known as the **pH Scale**.