Question

The empirical formula of a commercial ion exchange resin is \(\mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na}\). The resin can be used to soften water according to the reaction \(\mathrm{Ca}^{+2}+2 \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na} \rightarrow\) \(\left(\mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3}\right)_{2} \mathrm{Ca}+2 \mathrm{Na}^{+} .\)What would be the maximum uptake of \(\overline{\mathrm{Ca}}^{+2}\) by the resin expressed in mole/g resin? (a) \(0.0024\) (b) \(0.0246\) (c) \(0.246\) (d) \(24.6\)

Short answer

The maximum uptake of \( \mathrm{Ca}^{+2} \) by the resin is 0.0024 mole/g.

Step-by-step solution

Determine Molar Mass of Resin

Calculate the molar mass of the empirical formula \( \mathrm{C}_{8}\mathrm{H}_{7}\mathrm{SO}_{3}\mathrm{Na} \). - Carbon (C) has an atomic mass of 12.01 g/mol, so \( 8 \times 12.01 = 96.08 \). - Hydrogen (H) has an atomic mass of 1.01 g/mol, so \( 7 \times 1.01 = 7.07 \). - Sulfur (S) has an atomic mass of 32.07 g/mol, so the mass is 32.07. - Oxygen (O) has an atomic mass of 16.00 g/mol, so \( 3 \times 16.00 = 48.00 \). - Sodium (Na) has an atomic mass of 22.99 g/mol.Add these together to get the total molar mass: \[ 96.08 + 7.07 + 32.07 + 48.00 + 22.99 = 206.21 \, \text{g/mol} \]

Reaction Stoichiometry

According to the equation, 1 mole of \( \mathrm{Ca}^{+2} \) reacts with 2 moles of the resin, \( \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na} \). This means 2 moles of resin is equivalent in terms of mole exchange to 1 mole of \( \mathrm{Ca}^{+2} \).

Calculate Maximum Uptake in Moles

From the stoichiometry, 1 mole of \( \mathrm{Ca}^{+2} \) corresponds to 2 moles of resin. Therefore, the mole uptake by the resin is given as \[ \frac{1}{2} \text{ mole of \( \mathrm{Ca}^{+2} \) per mole of resin} \].

Express Uptake in Mole/Gram

To express this uptake in terms of moles of \( \mathrm{Ca}^{+2} \) per gram of resin, use the molar mass of the resin:\[ \frac{1}{2} \text{ mole of \( \mathrm{Ca}^{+2} \) per 206.21 g of resin} \].Calculate the uptake per gram:\[ \frac{1/2}{206.21} = 0.002424 \, \text{mole/g} \].

Choose the Correct Option

The calculated uptake is approximately 0.00242 mole/g. The closest option given is: (a) 0.0024.

Key concepts

Understanding Empirical Formulas

Empirical formulas are a fundamental concept in chemistry. They represent the simplest whole-number ratio of elements in a compound. This means the formula gives the proportion of atoms of each element, but not necessarily the exact number of atoms. It's the most reduced version of the formula.
For example, the empirical formula for glucose (C_6H_{12}O_6) is CH_2O, because it simplifies the number of each element. In the case of the ion exchange resin in the exercise, C_8H_7SO_3Na, each unit contains:

• 8 carbon atoms
• 7 hydrogen atoms
• 1 sulfur atom
• 3 oxygen atoms
• 1 sodium atom

This formula provides a simple snapshot of the resin's composition and is pivotal for calculations in stoichiometry and molar mass.

Reaction Stoichiometry Explained

Reaction stoichiometry involves the precise balancing of the quantities of reactants and products in a chemical reaction. It's about knowing the exact proportions needed to allow a reaction to proceed without any leftover reactants. This is based on the concept of the conservation of mass, where the mass of the reactants equals the mass of the products.
In the exercise's reaction, Ca^{+2}+2 C_8H_7SO_3Na  (C_8H_7SO_3)_2 Ca + 2 Na^{+}, this means:

• 1 mole of Ca^{+2} reacts with 2 moles of the resin C_8H_7SO_3Na.
• This stoichiometry reflects that each calcium ion replaces two sodium ions in the resin.

This balance helps in determining how much of a substance will react and form when starting with a known quantity of reactants.

Calculating Molar Mass

Molar mass is a crucial term that refers to the mass of one mole of a substance, expressed in grams per mole (g/mol). It's the sum of the atomic masses of all atoms in a molecule. This concept allows chemists to convert between the mass of a substance and the number of moles, facilitating a deeper understanding of chemical reactions. To determine the molar mass of a compound:

• Identify the molar mass of each element in the compound from the periodic table (e.g., Carbon is 12.01 g/mol).
• Multiply the atomic mass by the number of atoms of that element in the empirical formula.
• Add the resulting values for all elements together.

In our example, the molar mass of C_8H_7SO_3Na is calculated by adding the masses from its constituent elements, yielding 206.21 g/mol.

Ion Exchange and Water Softening

Ion exchange resins are widely used in water softening processes. The resin beads contain functional groups that replace hard ions, such as calcium (Ca^{+2}) and magnesium (Mg^{+2}), with softer ions like sodium (Na^{+}). This is essential for treating hard water, which can cause scale buildup in pipes and appliances. In the given reaction, calcium ions are exchanged with sodium ions from the resin:

• The resin bonds with the calcium ions, turning them into a removed by-product, while releasing sodium ions into the water.
• This lowers the concentration of calcium and helps to "soften" the water.

This is an example of ion exchange, where the resin takes advantage of its affinity for acting as a medium to swap out specific ions, making water easier to manage for household and industrial applications.