The equation for the pH of a substance is [tex]$pH =-\log \left[ H ^{+}\right]$[/tex], where [tex]$H +$[/tex] is the concentration of hydrogen ions. What is the approximate pH of a solution whose hydrogen ion concentration is [tex]$2 \times 10^{-9}$[/tex]?
Answer (2)
Substitute the hydrogen ion concentration into the pH equation: p H = − lo g [ 2 × 1 0 − 9 ] .
Apply logarithm properties to simplify the equation: p H = − ( lo g ( 2 ) + lo g ( 1 0 − 9 )) = − ( lo g ( 2 ) − 9 ) = 9 − lo g ( 2 ) .
Approximate lo g ( 2 ) as 0.3: p H = 9 − 0.3 .
Calculate the approximate pH: p H = 8.7 . Therefore, the approximate pH of the solution is 8.7 .
Explanation
Understanding the Problem We are given the formula for calculating the pH of a substance: p H = − lo g [ H + ] , where H + is the concentration of hydrogen ions. We are also given that the hydrogen ion concentration is 2 × 1 0 − 9 . Our goal is to find the approximate pH of the solution.
Substituting the Value First, we substitute the given hydrogen ion concentration into the pH equation: p H = − lo g [ 2 × 1 0 − 9 ]
Applying Logarithm Properties Next, we use the logarithm property lo g ( ab ) = lo g ( a ) + lo g ( b ) to rewrite the equation: p H = − ( lo g ( 2 ) + lo g ( 1 0 − 9 ))
Simplifying the Equation Then, we use the logarithm property lo g ( 1 0 x ) = x to simplify the equation: p H = − ( lo g ( 2 ) − 9 )
Distributing the Negative Sign Now, we distribute the negative sign: p H = 9 − lo g ( 2 )
Approximating log(2) We know that lo g ( 2 ) is approximately 0.3. Therefore, we substitute this value into the equation: p H = 9 − 0.3
Calculating the pH Finally, we calculate the approximate pH: p H = 8.7 Therefore, the approximate pH of the solution is 8.7.
Examples
The pH scale is used in many real-world applications, such as monitoring water quality in lakes and rivers, testing soil samples for agriculture, and in various chemical processes. For example, if you are testing the acidity of a swimming pool and find the hydrogen ion concentration to be 2 × 1 0 − 9 , you can use the pH formula to determine if the pool water is safe for swimming. A pH of 8.7 indicates that the water is slightly alkaline, which is within the acceptable range for swimming pools.
The pH of a solution with a hydrogen ion concentration of 2 × 1 0 − 9 is approximately 8.7. This is calculated using the formula p H = − lo g [ H + ] . A pH of 8.7 indicates the solution is slightly basic.
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