Question

MODELING WITH MATHEMATICS The \(\mathrm{pH}\) value for a substance measures how acidic or alkaline the substance is. It is given by the formula \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\mathrm{H}^{+}\)is the hydrogen ion concentration (in moles per liter). Find the \(\mathrm{pH}\) of each substance. a. baking soda: \(\left[\mathrm{H}^{+}\right]=10^{-8}\) moles per liter b. vinegar: \(\left[\mathrm{H}^{+}\right]=10^{-3}\) moles per liter

Short answer

The pH value of baking soda is 8, and the PH value of vinegar is 3.

Step-by-step solution

Calculation of pH for Baking Soda

Substitute the given hydrogen ion concentration value for baking soda into the pH formula: \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] = -\log(10^{-8})\). Use the property of logarithms that says the logarithm base 10 of 10 to any power is equal to that power. Therefore, \(\mathrm{pH}=-(-8) = 8\).

Calculation of pH for Vinegar

Similarly, substitute the given hydrogen ion concentration value for vinegar into the formula: \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] = -\log(10^{-3})\). Apply the property of logarithms as in Step 1 to find the pH for vinegar: \(\mathrm{pH}=-(-3) = 3\).

Key concepts

Hydrogen Ion Concentration

The hydrogen ion concentration, often represented by \(\left[\mathrm{H}^{+}\right]\), is a crucial concept in chemistry. It signifies the number of hydrogen ions present in a solution, measured in moles per liter. This concentration determines if a solution is acidic, neutral, or basic. The concentration symbol, \(\left[\mathrm{H}^{+}\right]\), specifically denotes the concentration of hydrogen ions, the positively charged ions (protons) in the solution.

- High hydrogen ion concentration means more protons, indicating an acidic solution.- Low concentration indicates fewer protons, suggesting a basic or alkaline solution.

Understanding this concept is key to grasping other topics in chemistry, like acid-base reactions, where hydrogen ion exchange is central. The concentration directly affects the pH levels of a substance, which in turn informs us about the substance's acidity or basicity.

Logarithmic Function

When dealing with pH, the logarithmic function is at play, using base 10 logarithms. The formula for pH is given by \(\mathrm{pH} = -\log \(\left[\mathrm{H}^{+}\right]\)\). This function is essential in compressing the vast range of hydrogen ion concentrations into a manageable scale.

Here's what happens:

• The negative log function flips the scale so that higher ion concentrations yield lower pH values (more acidic).
• Conversely, lower concentrations yield higher pH values (more basic).

The base 10 property of logarithms simplifies calculations. For example, \(\log(10^{-8})\) directly equals \(-8\) because the exponent is the answer when the base is 10. This functional property allows one to easily determine the pH by identifying the power of 10 in the hydrogen ion concentration.

Acid-Base Chemistry

Acid-base chemistry involves studying the properties and reactions of acids and bases. This field is essential in understanding how substances interact in aqueous solutions. Acids donate hydrogen ions (protons) while bases accept them.

The pH scale, ranging from 0 to 14, provides a quantitative measure of a solution's acidity or basicity:

• A pH less than 7 indicates an acidic solution (more hydrogen ions).
• A pH of exactly 7 indicates neutrality, like pure water.
• A pH greater than 7 indicates a basic (or alkaline) solution (fewer hydrogen ions).

This scale helps chemists predict the behavior of substances in chemical reactions. Understanding acid-base chemistry allows students to predict the effects of combining different chemicals and the potential changes in pH, which is vital for fields ranging from medicine to environmental science.