Problem 45 - Precalculus with Calculus Previews

Question

Determine the hydrogen-ion concentration \(\left[\mathrm{H}^{+}\right]\) of a solution with the given \(\mathrm{pH}\). $$ 3.3 $$

Step-by-Step Solution

Answer

The hydrogen-ion concentration is approximately \( 5.01 \times 10^{-4} \text{ M} \).

Step 1: Understanding the pH Formula

The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen-ion concentration. The formula is given by: \( \text{pH} = -\log\left([\mathrm{H}^{+}]\right) \).We need to use this formula to find \([\mathrm{H}^{+}]\) when pH = 3.3.

Step 2: Rearranging the Formula

To find \([\mathrm{H}^{+}]\), we rearrange the pH formula to read the concentration explicitly:Multiply both sides of the equation by -1 and apply the inverse log function:\( [\mathrm{H}^{+}] = 10^{-\text{pH}} \).

Step 3: Substituting the pH Value

Now, substitute the given \( \text{pH} = 3.3 \) into the rearranged formula:\( [\mathrm{H}^{+}] = 10^{-3.3} \).

Step 4: Calculating the Hydrogen-ion Concentration

Calculate using the formula:\[ [\mathrm{H}^{+}] = 10^{-3.3} \approx 5.01 \times 10^{-4} \]Thus, the hydrogen-ion concentration for a solution with a pH of 3.3 is approximately \( 5.01 \times 10^{-4} \text{ M} \).

Key Concepts

Logarithms in pH Calculations

Let's start with understanding what logarithms are and why they appear in pH calculations. Logarithms are mathematical operations that are the inverse of exponentiation. If you think about an exponent, such as \(2^3\), where 2 is raised to the power of 3, it means \(2 \times 2 \times 2 = 8\). A logarithm answers the question: "To what power must the base be raised, to produce a given number?" So, with \(\log_{10}(x) = y\), it means \(10^y = x\).

In the context of pH, the formula \( \text{pH} = -\log([\mathrm{H}^+]) \) means the pH is the exponent you get when you express the hydrogen-ion concentration in terms of a power of 10, with a negative sign indicating the inverse relationship. By understanding this, you can see why higher acidic concentrations result in lower pH values. The logarithmic scale compresses the range of possible values, making it easier to manage and interpret within a smaller set of numbers.

Understanding Hydrogen-Ion Concentration

Hydrogen-ion concentration \([\mathrm{H}^{+}]\) plays a vital role in determining the pH of a solution. It tells us how many hydrogen ions are present in one liter of solution. More ions mean higher acidity. In pure water at 25°C, \([\mathrm{H}^{+}]\) is typically around \(1.0 \times 10^{-7}\) M, which corresponds to a neutral pH of 7.

For the exercise, with a given pH of 3.3, we rearranged the formula to find \([\mathrm{H}^{+}] = 10^{-3.3}\). By calculating, we find this is approximately \(5.01 \times 10^{-4}\) M. This result tells us there are \(5.01 \times 10^{-4}\) moles of hydrogen ions per liter of solution, indicating a more acidic environment than neutral water. Remember, as pH values drop below 7, acidity increases.

Inverse Functions in Action

The concept of inverse functions is crucial in solving for \([\mathrm{H}^{+}]\) from the pH. An inverse function essentially "undoes" what the original function does. If you apply a function to a number and then its inverse function, you return to the initial number.

In our case, the pH function \(\text{pH} = -\log([\mathrm{H}^+])\) calculates a pH value from the hydrogen-ion concentration. To reverse this and find \([\mathrm{H}^{+}]\) from a given pH, we use the inverse of the logarithm: the exponential function. By rewriting the equation with \([\mathrm{H}^{+}] = 10^{-\text{pH}}\), we effectively "un-log" the pH value to discover the concentration. This manipulation showcases how inverse functions provide powerful tools for decoding relationships in equations, allowing us to switch back and forth between different forms of data.