Question

Thiamine hydrochloride \(\left(\mathrm{C}_{12} \mathrm{H}_{18} \mathrm{ON}_{4} \mathrm{SCl}_{2}\right)\) is a water- soluble form of thiamine (vitamin \(\mathrm{B}_{1} ; K_{\mathrm{a}}=3.37 \times 10^{-7}\) ). How many grams of the hydrochloride must be dissolved in \(10.00 \mathrm{~mL}\) of water to give a pH of \(3.50 ?\)

Short answer

Approximately 1.00 grams of thiamine hydrochloride must be dissolved.

Step-by-step solution

Calculate the concentration of \text{H}^+ ions

The pH is given as 3.50. Use the formula to calculate the concentration of \text{H}^+ ions: \[ [\text{H}^+] = 10^{-\text{pH}} \] Substitute the given pH value: \[ [\text{H}^+] = 10^{-3.50} \] This gives: \[ [\text{H}^+] \approx 3.16 \times 10^{-4} \text{ M} \]

Use the Ka expression

The dissociation of the hydrochloride in water can be described using the acid dissociation constant, \(\text{K}_\text{a}\): \[ \text{K}_\text{a} = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \] Where \([\text{HA}]\) is the concentration of the acid and \([\text{A}^-]\) is the concentration of the conjugate base. Assume \([\text{H}^+]=[\text{A}^-] \) because for every \text{H}^+ ion produced, one \text{A}^- ion is produced, and substitute the given value of \(\text{K}_\text{a}\): \[ 3.37 \times 10^{-7} = \frac{(3.16 \times 10^{-4})(3.16 \times 10^{-4})}{[\text{HA}]} \]

Solve for the initial concentration of \text{HA}

Rearrange the equation from Step 2 to solve for \[\text{HA} \]: \[ [\text{HA}] = \frac{(3.16 \times 10^{-4})^2}{3.37 \times 10^{-7}} \] This gives: \[ [\text{HA}] \approx 2.96 \times 10^{-1} \text{ M} \]

Calculate moles of \text{HA}

Find the number of moles of the hydrochloride needed for the solution: \[ \text{Moles} = [\text{HA}] \times \text{Volume} \] Where \([\text{HA}]\) is 0.296 M and the volume is 10.00 \text{ mL} (convert to liters): \[ \text{Volume} = 0.010 \text{ L} \] So, \[ \text{Moles} = 2.96 \times 10^{-1} \text{ M} \times 0.010 \text{ L} = 2.96 \times 10^{-3} \text{ moles} \]

Calculate the mass of thiamine hydrochloride needed

Find the molar mass of the hydrochloride \(\text{C}_{12}\text{H}_{18}\text{ON}_{4}\text{SCl}_{2}\): \[ \text{Molar Mass} = (12 \times 12.01 \text{ g/mol}) + (18 \times 1.01 \text{ g/mol}) + (1 \times 16.00 \text{g/mol}) + (4 \times 14.01 \text{ g/mol}) + (1 \times 32.06 \text{ g/mol}) + (2 \times 35.45 \text{ g/mol}) \approx 337.29 \text{ g/mol} \] Multiply the number of moles by the molar mass: \[ \text{Mass} = 2.96 \times 10^{-3} \text{ moles} \times 337.29 \text{ g/mol} \approx 0.999 \text{ grams} \]

Key concepts

acid dissociation constant

The acid dissociation constant, symbolized as \(\text{K}_\text{a}\), provides an essential measure of the strength of an acid in solution. Specifically, it quantifies the extent to which an acid can donate a proton (\(\text{H}^+\)) to the solvent, typically water. For thiamine hydrochloride with \(\text{K}_\text{a} = 3.37 \times 10^{-7}\), this means it is a weak acid because its \(\text{K}_\text{a}\) value is much less than 1.
To use \(\text{K}_\text{a}\) in calculations, establish the equilibrium expression for the dissociation:
\[ \text{K}_\text{a} = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]
Here, \([\text{HA}]\) is the initial concentration of the undissociated acid, \([\text{H}^+]\) is the concentration of hydrogen ions, and \([\text{A}^-]\) is the concentration of the conjugate base.
In our context:
\[3.37 \times 10^{-7} = \frac{(3.16 \times 10^{-4})(3.16 \times 10^{-4})}{[\text{HA}]}\]
Solving for \([\text{HA}]\) helps in determining how much of the initial acid remains undissociated in solution.
This fundamental property guides us through understanding the reactivity and strength of acids we encounter in solutions.

pH concentration

pH is a measure of the hydrogen ion concentration in a solution, indicating its acidity or alkalinity. The pH scale ranges from 0 to 14, with 7 being neutral. A pH less than 7 indicates acidity, while a pH greater than 7 indicates alkalinity.
To find the hydrogen ion concentration from a given pH, use the formula:
\[ [\text{H}^+] = 10^{-\text{pH}} \]
For a pH of 3.50, calculate it as:
\[ [\text{H}^+] = 10^{-3.50} \approx 3.16 \times 10^{-4} \text{ M} \]
This result tells us the number of hydrogen ions per liter. Understanding the pH and \([\text{H}^+]\) relationship is crucial in many chemical and biological processes, like enzymatic activities, drug solubilities, and even everyday phenomena like food acidity.

molar mass calculation

Molar mass is the mass of one mole of a substance, crucial for converting between moles and grams. To find the molar mass of thiamine hydrochloride \(\text{C}_{12}\text{H}_{18}\text{ON}_{4}\text{SCl}_{2}\), sum the atomic masses of each element, multiplied by their respective counts:

• Carbon: \(12 \times 12.01 \text{ g/mol}\)
• Hydrogen: \(18 \times 1.01 \text{ g/mol}\)
• Oxygen: \(1 \times 16.00 \text{ g/mol}\)
• Nitrogen: \(4 \times 14.01 \text{ g/mol}\)
• Sulfur: \(1 \times 32.06 \text{ g/mol}\)
• Chlorine: \(2 \times 35.45 \text{ g/mol}\)

Total it up:
\[ \text{Molar Mass} \approx 337.29 \text{ g/mol} \]
This allows you to convert between the mass of thiamine hydrochloride and the number of moles, aiding in precise solution preparations.

molecular formula

A molecular formula represents the exact number and type of atoms in a molecule, serving as a fundamental identifier for a substance. For thiamine hydrochloride, the molecular formula is \( \text{C}_{12}\text{H}_{18}\text{ON}_{4}\text{SCl}_{2} \).
This formula informs about:

• 12 Carbon atoms
• 18 Hydrogen atoms
• 1 Oxygen atom
• 4 Nitrogen atoms
• 1 Sulfur atom
• 2 Chlorine atoms

Knowing the molecular formula helps in:

• Understanding chemical properties
• Predicting reactivity
• Calculating molar mass

This knowledge is the foundation for advanced topics like molecular structure, bonding, and interactions.