Problem 17 - Chemistry in 30 days

Question

If \(5.85 \mathrm{~g}\) of \(\mathrm{NaCl}\) is dissolved in \(90 \mathrm{~g}\) of water, the mole fraction of solute is (a) \(0.0196\) (b) \(0.01\) (c) \(0.1\) (d) \(0.2\)

Step-by-Step Solution

Answer

The mole fraction of the solute is 0.0196, option (a).

Step 1: Calculate Molar Mass of Solute

To find the mole fraction of solute, we begin by determining the molar mass of the solute, sodium chloride (\(\mathrm{NaCl}\)). Sodium (\(\mathrm{Na}\)) has an atomic mass of approximately \(23\, \mathrm{g/mol}\) and Chlorine (\(\mathrm{Cl}\)) has an atomic mass of approximately \(35.5\, \mathrm{g/mol}\). Thus, the molar mass of \(\mathrm{NaCl}\) is \(23 + 35.5 = 58.5\, \mathrm{g/mol}\).

Step 2: Calculate Moles of Solute

Given that \(5.85\, \mathrm{g}\) of \(\mathrm{NaCl}\) is dissolved, we calculate the moles of \(\mathrm{NaCl}\) by using its molar mass. The number of moles \(n\) is given by:\[n = \frac{\text{mass}}{\text{molar mass}} = \frac{5.85}{58.5} = 0.1\, \text{moles}\].

Step 3: Calculate Moles of Solvent

The solvent is water. We know the molar mass of water \(\mathrm{H_2O}\) is \(18\, \mathrm{g/mol}\). So, the moles of water are:\[n = \frac{90}{18} = 5\, \text{moles}\].

Step 4: Calculate Mole Fraction of Solute

The mole fraction of the solute \(\chi_{\text{solute}}\) is calculated using the formula:\[\chi_{\text{solute}} = \frac{\text{moles of solute}}{\text{moles of solute} + \text{moles of solvent}} = \frac{0.1}{0.1 + 5} = \frac{0.1}{5.1} \approx 0.0196\].

Step 5: Final Answer

After calculating, we find that the mole fraction of the solute is \(0.0196\), which corresponds to option (a).

Key Concepts

Molar Mass Calculation

When determining the mole fraction of a solute in a solution, a crucial first step is to calculate the molar mass of the solute. Let's start by understanding what molar mass is. Molar mass is the weight of one mole of a chemical element or compound expressed in grams per mole (g/mol). This value allows us to convert between the mass of a substance and the number of moles. For sodium chloride (NaCl), you'll find the atomic masses on the periodic table: sodium (Na) has an atomic mass of about 23 g/mol and chlorine (Cl) is approximately 35.5 g/mol.

• By adding these atomic masses, we can calculate the molar mass of NaCl: 23 + 35.5 = 58.5 g/mol.
• Knowing the molar mass is essential as it bridges the gap between mass and the quantity in moles, which is vital for further calculations.

To summarize, for any chemical reaction involving a substance, knowing its precise molar mass ensures accurate conversion and calculation of other properties.

Moles of Solute

Our next step involves calculating the moles of solute using the molar mass previously determined. Knowing the exact number of moles is essential to finding mole fractions and ultimately understanding the concentration of our solution. Here’s how to find that:

• Use the formula: \[ n = \frac{\text{mass}}{\text{molar mass}} \]where 'mass' is the weight of the solute in grams.
• For our example with 5.85 g of NaCl, we substitute in the formula: \[ n = \frac{5.85}{58.5} = 0.1 \text{ moles} \]

By determining that there are 0.1 moles of sodium chloride in the solution, you're one step closer to quantifying the mole fraction.

Moles of Solvent

Now let's shift our focus to the solvent, which in this case is water. Calculating the moles of solvent is necessary to complete our mole fraction formula. Water (H₂O) has a molar mass of 18 g/mol. Here's how to calculate:

• Start with the mass of water, which is 90 g, and use the formula: \[ n = \frac{\text{mass of water}}{\text{molar mass of water}} \]
• Insert the values into the formula: \[ n = \frac{90}{18} = 5 \text{ moles} \]

By calculating that there are 5 moles of water, we now have the necessary information to calculate the mole fraction. Calculating this accurately ensures that our solution concentration can be correctly defined and understood.

Chemical Solution Concentration

The concentration of a chemical solution is evaluated through various means, and the mole fraction is one such approach. Concentration indicates how much of a solute is present in the solution relative to the amount of solvent. In our case, knowing the mole fraction helps us comprehend the proportional makeup of our solution.

• To calculate the mole fraction of the solute, use the relationship:\[\chi_{\text{solute}} = \frac{\text{moles of solute}}{\text{moles of solute} + \text{moles of solvent}} \]
• Substitute in our values: \[ \chi_{\text{solute}} = \frac{0.1}{0.1 + 5} = \frac{0.1}{5.1} \approx 0.0196 \]

This fraction tells us what portion of the total moles in the solution comes from the solute. Thus, in our mixture, NaCl makes up a tiny fraction, about 0.0196 of the whole. Knowing this helps professionals in fields like chemistry and pharmacology design and understand compositions effectively.