Question

Calculate the molality, molarity, and mole fraction of \(\mathrm{NH}_{3}\) in ordinary household ammonia, which is an 8.00 mass \% aqueous solution \((d=0.9651 \mathrm{~g} / \mathrm{mL})\).

Short answer

Molality = 5.11 m, Molarity = 4.54 M, Mole Fraction of \(\text{NH}_{3}\) = 0.084

Step-by-step solution

- Calculate the mass of \(\text{NH}_{3}\)

Given an 8.00 mass \% solution, this means there are 8.00 grams of \(\text{NH}_{3}\) in 100 grams of solution. So, mass of \(\text{NH}_{3}\) is 8.00 grams.

- Calculate the mass of water

The mass of water in the solution is the remaining mass after considering the \(\text{NH}_{3}\). \[ \text{Mass of water} = 100.0 - 8.0 = 92.0 \text{ grams} \]

- Calculate the moles of \(\text{NH}_{3}\)

Using the molar mass of \(\text{NH}_{3}\) (17.03 g/mol), calculate the moles: \[ n_{\text{NH}_{3}} = \frac{8.00 \text{ g}}{17.03 \text{ g/mol}} = 0.470 \text{ moles} \]

- Calculate the moles of water

Using the molar mass of water (18.02 g/mol), calculate the moles: \[ n_{\text{H}_{2}\text{O}} = \frac{92.0 \text{ g}}{18.02 \text{ g/mol}} = 5.106 \text{ moles} \]

- Calculate the molality (m)

Molality is defined as moles of solute per kilogram of solvent: \[ m = \frac{n_{\text{NH}_{3}}}{\text{kg of water}} = \frac{0.470 \text{ moles}}{0.092 \text{ kg}} = 5.11 \text{ m} \]

- Calculate the volume of the solution

Using the density and mass of the solution, \[ \text{Volume} = \frac{\text{mass}}{\text{density}} = \frac{100 \text{ g}}{0.9651 \text{ g/mL}} = 103.6 \text{ mL} = 0.1036 \text{ L} \]

- Calculate the molarity (M)

Molarity is defined as moles of solute per liter of solution: \[ M = \frac{n_{\text{NH}_{3}}}{\text{L of solution}} = \frac{0.470 \text{ moles}}{0.1036 \text{ L}} = 4.54 \text{ M} \]

- Calculate the mole fraction of \(\text{NH}_{3}\)

Mole fraction is defined as the ratio of moles of solute to the total moles of the solution: \[ \text{Mole fraction of } \text{NH}_{3} = \frac{n_{\text{NH}_{3}}}{n_{\text{NH}_{3}} + n_{\text{H}_{2}\text{O}}} = \frac{0.470}{0.470 + 5.106} = 0.084 \]

Key concepts

Understanding Molality

Molality is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per kilogram of solvent. To calculate molality, first determine the moles of the solute and then divide it by the mass (in kg) of the solvent.
In our case, the solute is \text{NH}_{3}\, and the solvent is water.
Here's how you do it:

• Identify the mass of \text{NH}_{3}\, which is 8 grams.
• Identify the mass of water, which is 92 grams.
• Convert the mass of water from grams to kilograms: \(0.092\) kg.
• Calculate the moles of \text{NH}_{3}\ using its molar mass (17.03 g/mol), resulting in 0.470 moles.

Finally, calculate molality using the formula:
\[ m = \frac{n_{\text{NH}_{3}}}{\text{kg of water}} = \frac{0.470 \text{ moles}}{0.092 \text{ kg}} = 5.11 \text{ m} \] This gives us a molality of 5.11 m.

Exploring Molarity

Molarity is another commonly used measure of concentration, defined as the number of moles of solute per liter of solution. Here's how you approach the calculation of molarity:

• First, calculate the mass of the solution, which is 100 grams.
• Next, use the density of the solution (\text{0.9651 g/mL}) to find the volume: \( \text{Volume} = \frac{\text{mass}}{\text{density}} = \frac{100 \text{ g}}{0.9651 \text{ g/mL}} = 103.6 \text{ mL} \).
• Convert this volume into liters: \(0.1036 \text{ L}\).
• Once again, the moles of \text{NH}_{3}\ is 0.470.

With these values, you can find the molarity:
\[ M = \frac{n_{\text{NH}_{3}}}{\text{L of solution}} = \frac{0.470 \text{ moles}}{0.1036 \text{ L}} = 4.54 \text{ M} \] Thus, the molarity comes out to be 4.54 M.

Understanding Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of the moles of a solute to the total number of moles of all components in the solution. Here's the breakdown for calculating the mole fraction:

• We know the moles of \text{NH}_{3} are 0.470.
• The moles of water, calculated with a molar mass of 18.02 g/mol, is 5.106 moles.

To find the total moles in the solution, add the moles of \text{NH}_{3}\ and the moles of water: \(0.470 + 5.106 = 5.576\).
Finally, calculate the mole fraction of \text{NH}_{3}\:
\[ \text{Mole fraction of } \text{NH}_{3} = \frac{0.470}{0.470 + 5.106} = 0.084 \] This gives a mole fraction of 0.084 for \text{NH}_{3}.