Problem 100 - Algebra A Combined Function

Question

The formula \(\mathrm{pH}=-\log _{10}\left(\mathrm{H}^{+}\right)\) gives the \(\mathrm{pH}\) for a liquid, where \(\mathrm{H}^{+}\) stands for the concentration of hydronium ions. Find the \(\mathrm{pH}\) of lemonade, whose concentration of hydronium ions is 0.0050 moles/liter.

Step-by-Step Solution

Answer

The pH of the lemonade is approximately 2.30.

Step 1: Identify the Given Information

We are given the concentration of hydronium ions, denoted as \(\mathrm{H}^{+}\), which is 0.0050 moles/liter. We need to find the pH using the formula \(\mathrm{pH} = -\log_{10}(\mathrm{H}^{+})\).

Step 2: Substitute the Values

Substitute the concentration \(\mathrm{H}^{+} = 0.0050\) into the pH formula: \(\mathrm{pH} = -\log_{10}(0.0050)\).

Step 3: Calculate the Logarithm

Calculate the base-10 logarithm of 0.0050. Using a calculator, \(\log_{10}(0.0050) \approx -2.3010\).

Step 4: Determine the pH Value

Apply the negative sign from the pH formula to the logarithm result: \(\mathrm{pH} = -(-2.3010) = 2.3010\).

Key Concepts

Understanding Logarithms in pH Calculation

Logarithms are powerful mathematical tools that transform multiplicative relationships into additive ones. In chemistry, logarithms often simplify calculations involving orders of magnitude, like concentrations of ions in solutions. The pH scale, which runs from 0 to 14, depends on a base-10 logarithm.
The pH formula is \[\mathrm{pH} = -\log_{10}(\mathrm{H}^+)\]. This formula uses the logarithm to convert the concentration of hydronium ions into a more manageable number.
Negative signs in logarithms like this one are crucial because they invert the logarithm's output. This is important because it allows us to express very small concentrations (like 0.0050 moles/liter) as a simple positive pH number. With a basic understanding of logarithms, you can easily navigate through pH calculations.

The Role of Hydronium Ion Concentration

In the context of pH, hydronium ions (\( \mathrm{H}^{+} \)) play a central role. These ions are present in all aqueous solutions and are crucial when determining acidity or basicity. The concentration of hydronium ions directly influences the pH value.
Higher hydronium ion concentrations indicate more acidic solutions and result in lower pH values, while lower concentrations signal more basic solutions and lead to higher pH values.

• A concentration of \(0.0050\) moles/liter indicates a relatively acidic solution based on the pH scale.
• The pH scale is logarithmic and inverse; thus, every unit decrease in pH corresponds to a tenfold increase in acidity.

When performing pH calculations, knowing the hydronium ion concentration means you can compute the acidity level of the solution—which is an essential skill in chemistry.

Chemistry Calculations Made Simple

Chemistry calculations, especially those involving acids and bases, often use the pH formula to determine a liquid's acidity. These calculations are straightforward once you understand the relationship between hydronium ion concentration and pH.

• Begin with identifying the ion concentration in the solution.
• Use the formula \(\mathrm{pH} = -\log_{10}(\mathrm{H}^+)\) to find the pH.
• Calculate the base-10 logarithm with a calculator for precision.

For example, to find the pH of lemonade with \(0.0050\) moles/liter hydronium ion concentration, substitute this concentration into the formula: \(\mathrm{pH} = -\log_{10}(0.0050)\). Calculating this gives a pH of around \(2.3010\), which indicates an acidic nature. By understanding the foundational chemistry principles, these calculations become a useful tool for various scientific fields.