Question

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

Short answer

The pH of the solution containing the dissolved vitamin C tablet is approximately 1.85. This calculation takes into account the diprotic nature of ascorbic acid and assumes the first dissociation step is more prominent than the second one.

Step-by-step solution

Determine the number of moles of ascorbic acid in one tablet

First, we need to convert the mass of the tablet into moles using the molar mass of ascorbic acid (H2C6H6O6). The tablet weighs 500 mg, and we need to convert that into grams by dividing by 1000: \[ 0.5 \ \mathrm{g} \] The molar mass of ascorbic acid is: \[ 2(1 \ \frac{\mathrm{g}}{\mathrm{mol}}) + 6(12.01 \ \frac{\mathrm{g}}{\mathrm{mol}}) + 6(1 \ \frac{\mathrm{g}}{\mathrm{mol}}) + 6(16 \ \frac{\mathrm{g}}{\mathrm{mol}}) \approx 176.12 \ \frac{\mathrm{g}}{\mathrm{mol}} \] Then, the number of moles of ascorbic acid in one tablet would be: \[ n = \frac{0.5 \ \mathrm{g}}{176.12 \ \frac{\mathrm{g}}{\mathrm{mol}}} \approx \ 0.00284 \ \mathrm{mol} \]

Calculate the concentration of ascorbic acid in the solution

The tablet is dissolved in 200 mL of water. To find the concentration of ascorbic acid in the solution, we need to divide the number of moles by the volume of the solution in liters: \[ c = \frac{0.00284 \ \mathrm{mol}}{0.2 \ \mathrm{L}} = 0.0142 \ \mathrm{M} \]

Apply the concept of pH, considering the diprotic nature of ascorbic acid

Since ascorbic acid is a diprotic acid, meaning it has two acidic protons, it would go through two steps of dissociation: \[ \mathrm{H2C6H6O6} \rightleftharpoons \mathrm{HC6H6O6^-} + \mathrm{H^+} \] \[ \mathrm{HC6H6O^6-} \rightleftharpoons \mathrm{C6H6O6^{2-}} + \mathrm{H^+} \] Since we don't have information about the dissociation constants for ascorbic acid, we assume that the first dissociation step is much more prominent than the second step. It means that the solution will have a concentration of H+ ions essentially equal to the concentration of ascorbic acid, i.e., 0.0142 M. Now we can calculate the pH of the solution using the formula: \[ \mathrm{pH} = -\log_{10} [\mathrm{H^+}] = -\log_{10} (0.0142) \approx 1.85 \] Thus, the pH of the solution is approximately 1.85.

Key concepts

Ascorbic Acid

Ascorbic acid, more commonly known as vitamin C, is an essential nutrient for humans and certain animals. It helps in various bodily functions and is a crucial antioxidant. The chemical formula for ascorbic acid is \(\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}_{6}\), indicating it is composed of hydrogen, carbon, and oxygen. As a diprotic acid, ascorbic acid can donate two protons or hydrogen ions, \(\mathrm{H^+}\), in reactions. This characteristic affects how it dissociates in solution and contributes to its acidic nature.
Understanding its structure and properties helps in calculating related chemistry problems, such as concentration and pH.

Molar Mass Calculation

To find the molar mass of a compound, like ascorbic acid, you must add up the atomic masses of all the atoms in its molecular formula. The molecular formula for ascorbic acid is \(\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}_{6}\). Here's how you calculate its molar mass step by step:

• Hydrogen: 2 atoms \(\times\) 1 g/mol = 2 g/mol
• Carbon: 6 atoms \(\times\) 12.01 g/mol = 72.06 g/mol
• Oxygen: 6 atoms \(\times\) 16 g/mol = 96 g/mol

Add these amounts together to get the total molar mass:
\[2 + 72.06 + 96 = 176.12 \text{ g/mol}\] This value is used to convert between grams and moles, aiding in solving various chemistry problems.

Concentration Calculation

Concentration measures how much solute is dissolved in a solvent, typically expressed in molarity (\(\mathrm{M}\)). Molarity is the number of moles of solute per liter of solution. For the ascorbic acid problem, follow these steps:

• First, calculate the number of moles using molar mass. With 0.5 g of ascorbic acid and a molar mass of 176.12 g/mol, number of moles \(n\) is: \(n = \frac{0.5}{176.12} \approx 0.00284 \, \mathrm{mol}\)
• Dissolve this in 200 mL of water. Convert mL to L by dividing by 1000, giving 0.2 L.
• Finally, calculate concentration \(c\) using: \(c = \frac{0.00284 \, \mathrm{mol}}{0.2 \, \mathrm{L}} = 0.0142 \, \mathrm{M}\)

Understanding concentration helps predict how a solution will behave in reactions, like acidity.

pH Calculation

pH measures the acidity or basicity of a solution, with lower values indicating higher acidity. For ascorbic acid, its diprotic nature means it can release two \(\mathrm{H^+}\) ions, but typically the first dissociation is most significant. In this case, the concentration of hydrogen ions is approximately equal to the concentration of ascorbic acid, \(0.0142 \, \mathrm{M}\).
Using the pH formula:
\[\mathrm{pH} = -\log_{10} [\mathrm{H^+}]\]
Substitute the concentration:
\[\mathrm{pH} = -\log_{10} (0.0142) \approx 1.85\]
This calculation shows that ascorbic acid solutions are fairly acidic, corresponding to their nutritional and chemical properties.