Question

Calculate the molarity and molality of an \(\mathrm{NH}_{3}\) solution made up of \(35.0 \mathrm{~g}\) of \(\mathrm{NH}_{3}\) in \(75.0 \mathrm{~g}\) of water. The density of the solution is \(0.982 \mathrm{~g} / \mathrm{mL}\).

Short answer

Molarity is 18.32 M, and molality is 27.39 m.

Step-by-step solution

Calculate moles of NH3

First, find the molar mass of \(\mathrm{NH}_3\). The molar mass of nitrogen (N) is approximately \(14.01 \, \mathrm{g/mol}\) and hydrogen (H) is approximately \(1.01 \, \mathrm{g/mol}\). So, \(\text{Molar Mass of } \mathrm{NH}_3 = 14.01 + 3 \times 1.01 = 17.04 \, \mathrm{g/mol}\). Divide the mass of \(\mathrm{NH}_3\) by its molar mass:\[\text{Moles of } \mathrm{NH}_3 = \frac{35.0 \, \mathrm{g}}{17.04 \, \mathrm{g/mol}} = 2.054 \mathrm{~mol}\].

Calculate total mass of solution

Add the mass of \(\mathrm{NH}_3\) and the mass of water to find the total mass of the solution:\[\text{Total Mass} = 35.0 \, \mathrm{g} \text{ (NH}_3\text{) } + 75.0 \, \mathrm{g} \text{ (water)} = 110.0 \, \mathrm{g}\].

Calculate volume of solution

Using the density of the solution \(0.982 \, \mathrm{g/mL}\), calculate the volume:\[\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{110.0 \, \mathrm{g}}{0.982 \, \mathrm{g/mL}} = 112.11 \, \mathrm{mL} = 0.11211 \, \mathrm{L}\].

Calculate molarity (M)

Molarity is defined as moles of solute per liter of solution. Use the moles of \(\mathrm{NH}_3\) and the volume of solution in liters to find molarity:\[\text{Molarity (M)} = \frac{2.054 \, \mathrm{mol}}{0.11211 \, \mathrm{L}} = 18.32 \, \mathrm{M}\].

Calculate molality (m)

Molality is defined as moles of solute per kilogram of solvent. Divide the moles of \(\mathrm{NH}_3\) by the mass of water (solvent) in kilograms:\[\text{Mass of water} = 75.0 \, \mathrm{g} = 0.0750 \, \mathrm{kg}\]\[\text{Molality (m)} = \frac{2.054 \, \mathrm{mol}}{0.0750 \, \mathrm{kg}} = 27.39 \, \mathrm{m}\].

Key concepts

Moles of Solute

Moles are a way to express the amount of a chemical substance. In calculations like molarity and molality, knowing the number of moles of solute is crucial.
To find the moles of a solute, like \(\mathrm{NH}_3\), you must first know its molar mass.
This is found by adding the atomic masses of each element in the molecule. In \(\mathrm{NH}_3\), there is one nitrogen atom (\(14.01 \, \mathrm{g/mol}\)) and three hydrogen atoms (\(3 \times 1.01 \, \mathrm{g/mol}\)).
Adding these gives \(17.04 \, \mathrm{g/mol}\) as the molar mass of \(\mathrm{NH}_3\).

• Use the formula: \(\text{Moles of solute} = \frac{\text{Mass of solute (g)}}{\text{Molar Mass (g/mol)}}\)

For example, dividing \(35.0 \, \mathrm{g}\) of \(\mathrm{NH}_3\) by \(17.04 \, \mathrm{g/mol}\) results in approximately \(2.054 \, \text{mol}\) of solute.

Density of Solution

Density is a property that describes how much mass is contained in a given volume. It's a crucial factor in converting mass into volume, which helps in determining concentration parameters.
For our solution, the density is given as \(0.982 \, \mathrm{g/mL}\).
This means that each milliliter of the solution has a mass of \(0.982 \, \mathrm{g}\).

• To find the volume from mass and density, use the formula: \(\text{Volume (mL)} = \frac{\text{Mass (g)}}{\text{Density (g/mL)}}\)

For instance, when the total mass of \(110.0 \, \mathrm{g}\) is divided by the density, you get a volume of \(112.11 \, \mathrm{mL}\). This conversion is vital for other calculations like molarity.

Molar Mass Calculation

Molar mass is essential for converting grams to moles, which is necessary in concentration calculations.
To calculate the molar mass of a compound like \(\mathrm{NH}_3\), consider the periodic table and the atomic masses.
Add the atomic masses of all the atoms in the molecule. For \(\mathrm{NH}_3\):

• Nitrogen = 14.01 g/mol
• Hydrogen = 1.01 g/mol per atom, and there are three hydrogen atoms

By computing: \(\text{Molar Mass of } \mathrm{NH}_3 = 14.01 + (3 \times 1.01) = 17.04 \, \text{g/mol}\), one ensures the correct conversion from mass in grams to moles.

Mass of Solvent

The mass of the solvent (in this case, water) is essential for calculating molality, a concentration measure different from molarity.
Molality considers the mass of the solvent in which the solute is dissolved, expressed in kilograms.
This is why water's mass (\(75.0 \, \mathrm{g}\)) is converted to kilograms by dividing by \(1000\), resulting in \(0.0750 \, \mathrm{kg}\).

• Molality formula: \(\text{molality (m) = } \frac{\text{moles of solute (mol)}}{\text{mass of solvent (kg)}}\)

Using this method, we consider only the solvent's mass, not the entire solution's.

Volume Conversion

In chemistry, dealing with solutions often requires converting between units of volume.
The volume can be in milliliters (mL) or liters (L).
To calculate molarity, which is moles of solute per liter of solution, converting mL to L is necessary.
Remember that \(1 \, \mathrm{L} = 1000 \, \mathrm{mL}\).

• Formula: \(\text{Volume in liters (L)} = \frac{\text{Volume in milliliters (mL)}}{1000}\)

For the example where the solution's volume is \(112.11 \, \mathrm{mL}\), divide by \(1000\) to convert to liters: \(0.11211 \, \mathrm{L}\) ensures accurate molarity calculations.