Question

3\. 65 grams of \(\mathrm{HCl}\) is dissolved in \(16.2\) grams of water The mole fraction of \(\mathrm{HCl}\) in the resulting solution is (a) \(0.4\) (b) \(0.3\) (c) \(0.2\) (d) \(0.1\)

Short answer

The mole fraction of HCl is approximately 0.664, which does not match the given options.

Step-by-step solution

Calculate the Moles of HCl

To find the number of moles, use the formula: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). The molar mass of \( \mathrm{HCl} \) is \( 1 + 35.5 = 36.5 \) g/mol. So, \( n_{\text{HCl}} = \frac{65}{36.5} \approx 1.78 \) moles.

Calculate the Moles of Water

The molar mass of water (\( \mathrm{H_2O} \)) is \( 2 + 16 = 18 \) g/mol. Calculate the moles of water: \( n_{\text{H}_2\text{O}} = \frac{16.2}{18} \approx 0.90 \) moles.

Compute Total Moles in Solution

Add the moles of \( \mathrm{HCl} \) and water to get total moles in the solution: \( n_{\text{total}} = n_{\text{HCl}} + n_{\text{H}_2\text{O}} = 1.78 + 0.90 = 2.68 \) moles.

Calculate the Mole Fraction of HCl

Use the formula for mole fraction: \( \chi_{\text{HCl}} = \frac{n_{\text{HCl}}}{n_{\text{total}}} = \frac{1.78}{2.68} \approx 0.664 \).

Key concepts

Understanding Moles of HCl

To find the moles of \( ext{HCl}\), you need to understand the relationship between mass, molar mass, and moles. The formula used for this calculation is: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). The molar mass is essentially the weight of one mole of a substance, given in grams per mole (g/mol). For hydrogen chloride (HCl), the molar mass is determined by adding the atomic masses of hydrogen and chlorine:

• Hydrogen (H) has an atomic mass of 1 g/mol.
• Chlorine (Cl) has an atomic mass of 35.5 g/mol.

So, the molar mass of HCl is \(1 + 35.5 = 36.5\) g/mol. With this information, you can calculate the moles using the given mass of 65 grams of HCl:

• \( n_{\text{HCl}} = \frac{65}{36.5} \approx 1.78 \text{ moles} \)

This calculation allows us to understand how much of the substance is present in the solution on a molecular level.

Finding Moles of Water

Water (\( ext{H}_2 ext{O}\)) is another essential component to consider in mole fraction calculations. Water consists of two hydrogen atoms and one oxygen atom. The molar mass of water can be determined as follows:

• Two hydrogen atoms contribute \(2 \times 1 = 2\) g/mol.
• One oxygen atom adds 16 g/mol.

Thus, the molar mass of water is \(2 + 16 = 18\) g/mol. To find the moles in our specific scenario, you divide the mass of water by its molar mass:

• \( n_{\text{H}_2 ext{O}} = \frac{16.2}{18} \approx 0.90 \text{ moles} \)

Calculating the moles of water helps us understand the relative amount of water compared to other substances in the solution.

Molar Mass and Its Calculation

Molar mass calculation is a foundational concept in chemistry that allows chemists to convert between mass and moles. This calculation is pivotal when determining quantities in reactions and solutions. To calculate the molar mass of any compound, sum up the atomic masses of all atoms in the compound. Here's the process broken down:

• Identify each atom within the compound using the Periodic Table.
• Find the atomic mass of each type of atom.
• Multiply the atomic mass by the number of each type of atom in the molecule.
• Add these values together to get the molar mass of the compound.

For example, calculating the molar mass of HCl involves adding the atomic masses of one hydrogen atom and one chlorine atom, resulting in 36.5 g/mol. This value allows you to convert from grams to moles, making it easier to understand and predict how substances will behave in chemical reactions and solutions.