In order to truly understand what a ” pH ” means, one must first understand how exponents are used in mathematics. Afterward, the usefulness of logarithms, which are the opposite operation of exponents, will be greatly appreciated.
Ten times one can be written as 10 X 1. Likewise, it may be written as 10*1. Most commonly, 10 times 1 would be expressed using curved brackets. Therefore, 10 times 1 is most commonly expressed as ( 10 )( 1 ) = 10. Using exponents, 10 times 1 = 10^1 ( 10 to the first power ). I’ll use both of the above notations for multiplication from here on out.
So, 10 times 1 = ( 10 )( 1 ) = 10^1. ( 10 )( 10 ) = 100. Since 10 was multiplied by itself, the notation 10^2 is used. 10^3 = 1000, because ( 10 )( 10 )( 10 ) = 1000.
NOTICE THAT THESE CIRCUMSTANCES REGARD THE SAME NUMBER BEING MULTIPLIED BY ITSELF. ( 10 )( 3 ) = 30, AND IT IS NOT REPRESENTED BY 10^3. 10^3 = ( 10 )(10 )( 10 ) = 1000.
Very small numbers are represented using negative exponents. The number ” one tenth ” is expressed as 1 divided by 10 = ( 1 / 10 ). One hundredth = ( 1 / 100 ). One thousandth = ( 1 / 1000 ). Using exponents, ( 1 / 10 ) = one tenth = 10^-1. Negative exponents are used to form fractions. To form the number 100, we previously used ( 10 )( 10 ) = 10^2 = 100. To form the fractional opposite ( one hundredth ), ( 1 / 100 ) = 10^-2. 10^-3 = ( 1 / 1000 ) = one-thousandth. Obviously, if the number ten is used to express concentrations of things, the range can be as large as a metric ton of pollution or as small as a fraction of acid ( hydrogen ions ) in water ( 10^-7 ).
Thankfully, things are simplified using logarithms. On a calculator, logarithms are simply expressed by the LOG key. A logarithm answers the following important question: What exponent was used to get this number? Example : ( 10 )( 10 ) = 100 = 10^2. If we wanted to know what exponent is used to convert ( or raise ) a base of 10 to get 100, using a calculator, we would enter LOG 100, and the answer = 2. The LOG 1000 = 3, because ( 10 )( 10 )( 10 ) = 1000, and 1000 = 10^3.
Logarithms can be used to accomplish the same task that compression of a computer file accomplishes. LOGs save space. Is it more cumbersome to plot a graph with inputs from 10^6, 10^5, 10^4, 10^3, 10^2, 10^1, ( 10^0 = 1 ), 10^-1, 10^-2, 10^-3, 10^-4, 10^-5, to 10^-6 in order to represent a range of [+ H ] ion concentrations, or would it be easier to plot the numbers 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6 ? If the latter method is used, it is called a logarithmic chart.
Reading and interpreting a LOG chart is relatively easy. In the above LOG sequence from 6 to -6, choose any number. For example, what would a concentration of 2 represent? 10^2 = ( 10 )( 10 ) = 100 units of something.
Finally, what is pH? pH is a measure of acidity or alkalinity of a solution. An acidic solution contains a greater concentration of [+ H ] ions than pure water, and an alkaline ( or basic ) solution contains less [+ H ] ions than pure water. The midpoint between acidity and alkalinity is defined as the concentration of [+ H ] ions in pure water ( 10^-7 ). How might we use the logarithmic function to express the fact that ( 10^-7 ) [+ H ] ions are in a solution of water? Well, the difference between the number 10 and 10^-7 is the -7 exponent. Therefore, LOG 10^-7 = -7.
Why not make the outcome of most acid and basic pH calculations have a positive value? Using the rules of algebra, -1 times the LOG 10^-7 = 7. This is the definition of the pH of a solution. pH = – LOG [ + H concentration ]. Since the concentration of hydrogen ions in water is 10^-7, pH = – LOG [ 10^-7 ] = 7.
Furthermore, it is understood that something with a pH of 6 is 10 times more acidic than water ( 10^-6 vs. 10^-7 ). A pH of 5 is interpreted as a solution that is 100 times more acidic than water, etc.
RENAISSANCE PHYSICS 