Problem 107 - Elementary and Intermediate Algebra

Question

Find the pH of a solution with a hydrogen ion concentration of \(1.7 \times 10^{-5}\) gram-ions per liter.

Step-by-Step Solution

Answer

The pH of the solution is approximately 4.77.

Step 1: Understanding pH

pH is a measure of the acidity or basicity of a solution, calculated using the formula \( \text{pH} = -\log[H^+] \), where \([H^+]\) is the hydrogen ion concentration in moles per liter.

Step 2: Convert Gram-Ions to Moles

Since the hydrogen ion concentration is given in gram-ions per liter, we need to convert it to moles per liter. There are Avogadro's number of ions (approximately \(6.022 \times 10^{23}\) ions) in one mole of any substance, so 1 gram-ion per liter is equivalent to 1 mole per liter. Hence, the concentration in moles per liter is also \(1.7 \times 10^{-5}\) moles per liter.

Step 3: Apply the pH Formula

Using the concentration in moles per liter, we apply the formula for pH. Thus, \( \text{pH} = -\log[1.7 \times 10^{-5}] \).

Step 4: Calculate the pH

Calculate the pH using a calculator. \(-\log[1.7 \times 10^{-5}]\) is approximately equal to \(4.77\).

Step 5: Solution Verification

Verify the calculation to ensure accuracy: \(-\log_{10}[1.7 \times 10^{-5}]\) using a standard calculator should again yield a pH of approximately 4.77. This confirms the pH of the solution.

Key Concepts

Understanding Hydrogen Ion Concentration

Hydrogen ion concentration is central to understanding the concept of pH. It refers to the amount of hydrogen ions, [H extsuperscript{+}], present in a solution. In this case, the concentration is given as gram-ions per liter, denoting the number of hydrogen ions in grams present in one liter of solution.

To work with pH, we typically need this concentration in moles per liter. A mole is a unit that helps in quantifying substances in chemistry, especially at the atomic level. Knowing the hydrogen ion concentration in moles per liter is crucial because the pH calculation relies on this metric.

• Hydrogen ion concentration shows how acidic or basic a solution is.
• Gram-ion signifies the mass of ions in grams.
• Conversion to moles per liter allows for pH calculation using logarithmic functions.

Moles per Liter: The Standard Measurement

Chemists often use moles per liter, also known as molarity, as a standard measurement of concentration. This popularity comes from its convenience in chemical equations and reactions where balanced moles help ensure proper interaction between substances.

In this context, one gram-ion per liter equates precisely to one mole per liter due to the relation with Avogadro's number, the number of atoms or ions in one mole of a substance. This standardization makes calculations straightforward when dealing with atomic-scale quantities.

• A mole represents approximately \(6.022 \times 10^{23}\) entities, known as Avogadro's number.
• Converting measurements to moles per liter aids in uniformity and accuracy for chemical calculations.

Logarithmic Functions in pH Calculation

Logarithmic functions play a pivotal role in scientific calculations, providing a way to deal with the vast range of concentrations found in chemical solutions. The pH scale itself is a logarithmic scale, meaning each whole pH value below 7 is ten times more acidic than the next higher value.

The formula used for calculating pH, \(\text{pH} = -\log[H^+]\), utilizes the logarithm to scale down the typically small values of hydrogen ion concentration into a more manageable form. This transformation allows scientists to easily express and compare acidity and basicity of different solutions.

• A logarithm represents the exponent needed to produce a given number.
• The "\(-\log\)" operation on \([H^+]\) helps express concentration in a smaller, more interpretable scale.