Question

A typical vitamin \(\mathrm{C}\) tablet (containing pure ascorbic acid, \(\left.\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}_{6}\right)\) weighs \(500 .\) mg. One vitamin \(\mathrm{C}\) tablet is dissolved in enough water to make \(200.0 \mathrm{mL}\) of solution. Calculate the pH of this solution. Ascorbic acid is a diprotic acid.

Short answer

The pH of the 200.0 mL ascorbic acid solution from a 500 mg vitamin C tablet is approximately 1.85.

Step-by-step solution

Calculate moles of ascorbic acid from mass

First, we need to convert the mass of the vitamin C tablet (500 mg) into moles. To do this, we will use the molar mass of ascorbic acid, which is \(M = 2 \times 1.01 + 6 \times 12.01 + 6 \times 1.01 + 6 \times 16.00 \approx 176.12 \, g/mol\). We also need to convert the mass of the tablet from milligrams to grams: \(m = 500 \, mg \times (1 \, g / 1000 \, mg) = 0.500 \, g\) Now we can find the moles of ascorbic acid: \(n = m / M = 0.500 \, g / 176.12 \, g/mol \approx 0.00284 \, mol\)

Find the molarity of ascorbic acid in the solution

Next, we need to find the molarity of ascorbic acid in the 200.0 mL solution. To do this, we will divide the moles of ascorbic acid by the volume of the solution in liters: \(V = 200.0 \, mL \times (1 \, L / 1000 \, mL) = 0.200 \, L\) The molarity of ascorbic acid in the solution is: \(C = n / V = 0.00284 \, mol / 0.200 \, L = 0.0142 \, M\)

Determine the change in concentration due to dissociation

Ascorbic acid is a diprotic acid, meaning it can release two protons. However, in most cases, only one proton dissociates, so we will consider only the first dissociation step: \(H_2C_6H_6O_6 \rightleftharpoons H^+ + HC_6H_6O_6^-\) Since ascorbic acid is a weak acid, the dissociation is not complete, and the change in concentration due to dissociation is minimal. Thus, for simplicity, we can assume that the concentration of hydrogen ions released is equal to the concentration of ascorbic acid in the solution.

Calculate the pH

Finally, we can use the concentration of hydrogen ions \(H^+\) to calculate the pH of the solution: \[pH = -\log_{10}([H^+]) = -\log_{10}(0.0142)\] Now, using a calculator, we find that \(pH \approx 1.85\). Therefore, the pH of this ascorbic acid solution is approximately 1.85.

Key concepts

Molarity

Molarity is a crucial concept in chemistry that measures the concentration of a substance in a solution. It is expressed as moles of solute per liter of solution, and its unit is moles per liter (M).

To calculate molarity, we simply divide the number of moles of solute by the volume of the solution in liters. The formula is:
\[ C = \frac{n}{V} \] where \( C \) is the molarity, \( n \) is the number of moles, and \( V \) is the volume in liters.

In our exercise, we determined the molarity of ascorbic acid by first finding the moles of ascorbic acid using the tablet's mass and molar mass, then we divided that number by the volume of water in which the tablet dissolved. This simple calculation gave us the molarity of the solution, which is an essential step for calculating the pH.

Ascorbic Acid

Ascorbic acid, also known as Vitamin C, is an essential nutrient for humans, well-known for its antioxidant properties and its role in collagen synthesis. Its chemical formula is \( \mathrm{H}_{2}\mathrm{C}_{6}\mathrm{H}_{6}\mathrm{O}_{6} \), and it is a diprotic acid, meaning it can donate two protons (\(H^+\)) to the solution.

Although ascorbic acid is diprotic, it's important to note that the two protons are not released with equal propensity. The first proton tends to dissociate more readily than the second, so for simplification in pH calculations, especially in a weak acid like ascorbic acid, we often only consider the first dissociation.

Diprotic Acid

Diprotic acids have the ability to donate two protons or hydrogen ions into an aqueous solution. The dissociation of a diprotic acid occurs in two stages, each with its own equilibrium constant denoted as \( K_{a1} \) and \( K_{a2} \) for the first and second dissociation respectively.

For ascorbic acid:

• The first dissociation: \( H_2C_6H_6O_6 \rightleftharpoons H^+ + HC_6H_6O_6^- \)
• The second dissociation: \( HC_6H_6O_6^- \rightleftharpoons H^+ + C_6H_6O_6^{2-} \)

In the context of our exercise, we simplify by considering only the first dissociation since the second dissociation is typically minimal and has less impact on the overall concentration of hydrogen ions, which are critical to pH calculation.

Hydrogen Ion Concentration

The concentration of hydrogen ions, often denoted by \( [H^+] \), is a measure of the acidity of a solution. It is precisely this concentration of hydrogen ions that the pH scale quantifies. pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration:
\[ pH = -\log_{10}([H^+]) \]
Because the pH scale is logarithmic, each whole number on the scale represents a tenfold increase or decrease in hydrogen ion concentration. In the case of our ascorbic acid solution, once we determined the molarity (which we used as an approximation for \( [H^+] \)), we applied this formula to find that the solution has a pH of approximately 1.85, indicating it is quite acidic.