Question

Before the advent of \(\mathrm{pH}\) meters, a urologist collected \(1.3\) liters of urine from a hospitalized patient over the course of a day. In order to calculate what the \(\mathrm{pH}\) was a laboratory technician determined the number of equivalents of acid present in the sample. A given sample of urine contains \(1.3 \times 10^{-6}\) equivalents of dissociated acid. What is the \(\mathrm{pH}\) of this sample?

Short answer

The pH of the urine sample is 6.

Step-by-step solution

Finding the concentration of hydrogen ions

First, we need to find the concentration of hydrogen ions. To do that, we will use the formula: \(Concentration\) (Hydrogen ion) \( = \dfrac{\text{Equivalents of Acid}}{\text{Volume of Sample}}\) We are given: Equivalents of Acid \(= 1.3 \times 10^{-6} \) equivalents Volume of Sample \(= 1.3\) liters Now, let's calculate the concentration of hydrogen ions: \(Concentration\) (Hydrogen ion) \( = \dfrac{1.3 \times 10^{-6}}{1.3}\)

Simplifying the concentration of hydrogen ions

Now, we simplify the concentration of hydrogen ions. \(Concentration\) (Hydrogen ion) \( = 1 \times 10^{-6}\) The concentration of hydrogen ions is \(1 \times 10^{-6}\) molar.

Calculating the pH

Now, we will calculate the pH using the pH formula: \(pH = -\log_{10}[\mathrm{H}^+]\) Where \([\mathrm{H}^+]\) represents the concentration of hydrogen ions. \(pH = -\log_{10}(1 \times 10^{-6})\)

Simplifying the pH value

Finally, we simplify the pH value. \(pH = -\log_{10}(1 \times 10^{-6}) = -(-6) = 6\) So, the pH of the urine sample is 6.

Key concepts

Understanding Hydrogen Ion Concentration in Solutions

Hydrogen ion concentration is a measure of the amount of hydrogen ions present in a solution. It is crucial in determining the acidity or basicity of the solution. For example, when a laboratory technician measures the equivalents of dissociated acid in a urine sample, the result shows the number of hydrogen ions available.

It's essential to understand that hydrogen ions (H+) are released when an acid dissolves in water. The higher the concentration of hydrogen ions, the more acidic the solution is. Conversely, a lower concentration indicates a more basic or alkaline solution. In the given exercise, the technique to find this concentration involves dividing the number of equivalents of acid by the volume of the sample. The result, expressed in moles per liter (molarity), is pivotal for the next steps in pH calculation.

Importance of Accurate Measurement

Precision in measuring hydrogen ion concentration is vital since even small errors can lead to substantial changes in pH. This accuracy impacts not only academic exercises but also real-world applications, such as medical diagnoses and treatment plans.

The Role of Acid-Base Equilibrium in pH

Acid-base equilibrium refers to the state of balance between the concentration of acids and bases in a solution. This balance is the foundation for understanding pH levels. In any given solution, there is a dynamic equilibrium between the ionization of acids and bases, resulting in the formation of their respective ions.

The concept of equilibrium is highlighted in the exercise where we see the dissociated acid in the form of hydrogen ions. This scenario is a snapshot of the acid-base equilibrium in the urine sample. The equilibrium constant for this ionization process helps predict the concentrations of the various ions in the solution.

Importance of Equilibrium in Biological Systems

Maintaining a stable acid-base equilibrium is crucial for biological systems, such as the human body. For instance, the pH of blood needs to remain within a narrow range for optimal physiological function, and the kidneys play a significant role in managing this balance by excreting or reabsorbing acids and bases.

Calculating pH Using the Logarithmic pH Scale

The pH scale is a logarithmic measure used to specify the acidity or basicity of an aqueous solution. It's logarithmic because each whole pH value below 7 is ten times more acidic than the next higher value. For example, a pH of 3 is ten times more acidic than a pH of 4.

Following the steps given for the pH calculation, we apply the formula \(pH = -\log_{10}[\mathrm{H}^+]\) to the concentration of hydrogen ions. By using the properties of logarithms, we find that a hydrogen ion concentration of \(1 \times 10^{-6}\) molar corresponds to a pH of 6, which is acidic but close to neutral pH 7.

Understanding the pH Formula

The negative in front of the logarithm indicates that a higher concentration of hydrogen ions (more acidity) will result in a lower pH value, and vice versa. Because the pH scale is logarithmic, it provides a convenient way of expressing hydrogen ion concentrations without resorting to very small numbers.