Problem 31 - Algebra 2

Question

The pH of each food is given. Find the concentration of hydrogen ions \(\left[\mathrm{H}^{+}\right] .\) soy sauce, 4.9

Step-by-Step Solution

Answer

The concentration of hydrogen ions in soy sauce is approximately \( \[\mathrm{H}^{+} = 1.2589254 × 10^{-5}\]\).

Step 1: Understanding the Concepts

Recall the definition of pH: it's the negative logarithm (base 10) of the concentration of hydrogen ions. This relationship is expressed as \(pH = -log[\mathrm{H}^{+}]\). So, we'll reverse this equation to find \([\mathrm{H}^{+}]\) using the given pH. That is \([\mathrm{H}^{+}] = 10^{-pH}\).

Step 2: Plug the Known Value into the Equation

Substitute the given pH (4.9 for soy sauce) into the equation \([\mathrm{H}^{+}] = 10^{-pH}\). Therefore, \([\mathrm{H}^{+}] = 10^{-4.9}\).

Step 3: Solving for the Concentration of Hydrogen Ions

Calculate the above expression to find the exact concentration. Hence, \([\mathrm{H}^{+}] = 1.2589254 × 10^{-5}\).

Key Concepts

Logarithms

Logarithms offer a way to simplify complex multiplication or division calculations. They take exponential numbers and express them in a manageable format. For example, the equation \( pH = -\log[\mathrm{H}^+] \) uses a logarithm to describe hydrogen ion concentration in terms of pH.

A key property of logarithms is that they are the inverse of exponentials. This means if you know the pH of a solution, you can quickly reverse the process to find the hydrogen ion concentration by raising 10 to the power of the negative pH, like so: \([\mathrm{H}^+] = 10^{-\mathrm{pH}}\).

• Logarithms give us a tool to transform multiplicative relationships into additive ones.
• By using the property of inversion, they allow us to solve exponential equations easily.

Understanding logarithms is crucial when dealing with equations involving pH, as they help in transitioning from a logarithmic to an exponential form.

Hydrogen Ion Concentration

Hydrogen ion concentration \([\mathrm{H}^+]\) is a measure of the acidity or alkalinity of a solution. The pH scale, which ranges from 0 to 14, is used to express this concentration.Lower pH values indicate a higher concentration of hydrogen ions, which means the solution is more acidic.

For a neutral solution, like pure water, the pH is around 7, indicating an equal balance of H\(^+\) ions and hydroxide ions (OH\(^{-}\)). In highly acidic environments, such as lemon juice or vinegar, pH values drop, signifying a greater presence of H\(^+\) ions.

• To find the concentration from pH, we reverse the logarithmic form: \([\mathrm{H}^+] = 10^{-\mathrm{pH}}\).
• This transformation allows you to quantify hydrogen ion concentration accurately.

Knowing the exact concentration helps us understand the chemical behavior and potential reactivity of a solution.

Exponential Equations

Exponential equations involve variables in the exponent, such as \([\mathrm{H}^+] = 10^{-\mathrm{pH}}\). These are vital in chemistry to depict natural behaviors like reaction rates and concentrations.

In our context, it translates the logarithmic form of pH back into a concentration value.Here's how it works:

• You start with the pH value, which is a logarithm of the hydrogen ion concentration.
• Convert it using an exponential function: the base (10) raised to the power of the negative pH.

This provides you with the actual concentration of hydrogen ions in a straightforward numerical value. Understanding exponential equations enables effective manipulation of these forms to unlock real-world data from theoretical values.