Problem 61 - A Graphical Approach to College Algebra

Question

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\) Soda pop, 2.7

Step-by-Step Solution

Answer

\([\mathrm{H}_3\mathrm{O}^+] \approx 2.0 \times 10^{-3} \text{ M}\)

Step 1: Understand the relationship between pH and hydronium ion concentration

The pH of a solution is defined as the negative logarithm (base 10) of the hydronium ion concentration. Mathematically, it is expressed as: \[ \text{pH} = -\log_{10}[\mathrm{H}_3\mathrm{O}^+] \] Our task is to rearrange this formula to find \([\mathrm{H}_3\mathrm{O}^+]\).

Step 2: Rearrange the pH equation for hydronium ion concentration

To isolate \([\mathrm{H}_3\mathrm{O}^+]\), we need to solve the equation for this variable. By exponentiating both sides with base 10, we obtain:\[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}} \]

Step 3: Substitute the given pH value

Now we'll substitute the given pH value of 2.7 into the equation:\[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-2.7} \]

Step 4: Calculate the hydronium ion concentration

Calculate \(10^{-2.7}\) using a calculator:\[ [\mathrm{H}_3\mathrm{O}^+] \approx 2.0 \times 10^{-3} \text{ M} \] This is the hydronium ion concentration for the given pH.

Key Concepts

Logarithms in Chemistry

Logarithms are essential in chemistry for simplifying calculations with extremely small or large numbers, particularly when dealing with concentrations like hydronium ions . The use of logarithms allows scientists to transform multiplicative processes into additive ones, making them easier to work with. In the context of pH, the logarithm is a mathematical function that helps express the concentration of hydrogen ions in a solution on a more manageable scale . Since the pH is defined as the negative logarithm of the hydronium ion concentration, it converts the often very small molarity values into a simple number.

• The formula for pH is: \( \text{pH} = -\log_{10} [\mathrm{H}_3\mathrm{O}^+] \)
• This means that for each whole number change in pH, there is a ten-fold change in the concentration.

Understanding logarithms helps in calculating both pH and the original ion concentrations, making it an indispensable tool in chemistry.

Acid-Base Equilibrium

Acid-base equilibrium is a crucial concept in understanding how acids and bases behave in solutions. It is about reaching a state where concentrations of reactants and products remain constant over time . When we deal with solutions like soda pop which have a pH, we are often interested in how these solutions form or neutralize hydronium ions . Acids are substances that increase the concentration of hydronium ions ( \([\mathrm{H}_3\mathrm{O}^+]\) ). Bases , on the other hand, , decrease it by providing hydroxide ions ( \([\mathrm{OH}^-] \)). In a solution, equilibrium is maintained by the dynamic and reversible nature of the processes involved:

• The dissociation of an acid increases \([\mathrm{H}_3\mathrm{O}^+]\).
• The reaction of a base neutralizes some hydronium ions, reducing their concentration.

Maintaining equilibrium is central to understanding how acids and bases react, which in turn helps in calculating pH and concentrations.

Calculating Concentrations

Calculating the concentration of certain ions in a solution is a fundamental skill in chemistry. It involves using mathematical tools, such as logarithms, to determine the amount of a particular ion present in a given volume . In the case of hydronium ions, once we know the pH, we can rearrange the pH equation to find their concentration . Rearranging involves using the fundamental inverse operation of the logarithm, exponentiation, to solve for the concentration.

• The pH equation \( \text{pH} = -\log_{10}[\mathrm{H}_3\mathrm{O}^+] \) can be rearranged to find the hydronium ion concentration as \( [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}} \).
• To find the concentration, simply substitute the pH value into the equation and solve.

For example, for a pH of 2.7, calculating \( 10^{-2.7} \) provides the hydronium ion concentration . Using a calculator, it comes out to approximately \( 2.0 \times 10^{-3} \text{ M} \). This concentration indicates the strength of the acid in the solution, and is key to understanding its chemical properties.