Question

A solution of phosphoric acid was made by dissolving \(10.0 \mathrm{~g}\) \(\mathrm{H}_{3} \mathrm{PO}_{4}\) in \(100.0 \mathrm{~mL}\) water. The resulting volume was \(104 \mathrm{~mL}\) Calculate the density, mole fraction, molarity, and molality of the solution. Assume water has a density of \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\).

Short answer

The density of the phosphoric acid solution is 1.058 g/mL, the mole fraction of H₃PO₄ is 0.0180, the molarity of the solution is 0.981 M, and the molality is 1.02 mol/kg.

Step-by-step solution

1. Calculate the moles of H₃PO₄

First, we need to calculate the number of moles of H₃PO₄ in the solution. We will use the formula: moles = mass of solute / molar mass. The molar mass of H₃PO₄ = 3(1.01) + 1(15.99) + 4(16.00) = 97.99 g/mol. So, moles of H₃PO₄ = \( \frac{10.0 \ g}{97.99 \ g/mol} = 0.102 \ \text{moles} \)

2. Calculate the mass of water and the mass of the solution

We are given that the mass of H₃PO₄ was 10.0 g and the volume of water is 100.0 mL. Assuming the density of water is 1.00 g/cm³, we can convert the volume of water to mass: Mass of water = density × volume = 1.00 g/cm³ × 100.0 cm³ = 100.0 g The total mass of the solution will be the sum of the mass of H₃PO₄ and the mass of water: Total mass of solution = 10.0 g + 100.0 g = 110.0 g

3. Calculate the density of the solution

We are given that the volume of the solution is 104 mL. To find the density, we will use the formula: density = mass / volume. So, density of the solution = \( \frac{110.0 \ g}{104.0 \ mL} = 1.058 \ \frac{g}{mL} \)

4. Calculate the mole fraction of H₃PO₄

To find the mole fraction of H₃PO₄, we need to find the moles of water and then use the formula: mole_fraction = moles of solute / (moles of solute + moles of solvent). First, we find the moles of water using the formula: moles = mass of solvent / molar mass. The molar mass of water = 18.015 g/mol. Moles of water = \( \frac{100.0 \ g}{18.015 \ g/mol} = 5.553 \ \text{moles} \) Now, we find the mole fraction of H₃PO₄: Mole fraction of H₃PO₄ = \( \frac{0.102 \ \text{moles}}{0.102 \ \text{moles} + 5.553 \ \text{moles}} = 0.0180 \)

5. Calculate the molarity of the solution

Molarity is defined as the number of moles of solute per liter of solution. We know that the volume of the solution is 104 mL. We can convert this into liters: Volume of solution = 104 mL × \( \frac{1 \ \text{L}}{1000 \ \text{mL}} \) = 0.104 L Now, we find the molarity using the formula: molarity = moles of solute / volume of solution (in liters): Molarity of H₃PO₄ = \( \frac{0.102 \ \text{moles}}{0.104 \ L} \) = 0.981 M

6. Calculate the molality of the solution

Molality is defined as the number of moles of solute per kilogram of solvent. We have already found the moles of H₃PO₄ (0.102 moles) and we know the mass of water (100.0 g). We can convert the mass of water to kilograms: Mass of water = 100.0 g × \( \frac{1 \ \text{kg}}{1000 \ \text{g}} \) = 0.100 kg Now, we find the molality using the formula: molality = moles of solute / mass of solvent (in kg): Molality of H₃PO₄ = \( \frac{0.102 \ \text{moles}}{0.100 \ \text{kg}} \) = 1.02 mol/kg In conclusion, the density of the solution is 1.058 g/mL, the mole fraction of H₃PO₄ is 0.0180, the molarity is 0.981 M, and the molality is 1.02 mol/kg.

Key concepts

Molarity

Molarity is a way to express the concentration of a solution. It tells us how many moles of a solute are present in one liter of solution. The formula to calculate molarity (M) is given by:\[\text{Molarity (M)} = \frac{\text{moles of solute}}{\text{volume of solution in liters}}\]In our exercise, we learned that the moles of H₃PO₄ were 0.102, and the volume of the solution was 104 mL, which we converted to 0.104 liters. Using these values in the formula, we computed the molarity to be 0.981 M. This means there are approximately 0.981 moles of H₃PO₄ in every liter of the solution.

Molarity is very practical for reactions that occur in a liquid phase, such as titrations or saturation issues, because it relates to the volume of the solution directly.

Molality

Molality is another measurement of a solution's concentration, which is slightly different from molarity. It is defined as the moles of solute per kilogram of solvent, not the total solution. Here’s how you can calculate molality (m):\[\text{Molality (m)} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}}\]In the given exercise, the mass of water, acting as the solvent, was calculated to be 0.100 kg (since 100.0 g of water is 0.100 kg). We used the moles of H₃PO₄ previously calculated (0.102 moles) to find the molality, resulting in 1.02 mol/kg.

Molality is useful in scenarios where temperature changes can cause expansion or contraction of volumes, as its value stays constant with temperature fluctuations because mass does not change.

Density of Solution

Density is a measure of how much mass is contained in a given volume. For solutions, density helps in understanding how concentrated a solution is when comparing to pure solvents. The formula for density (\(\rho\)) is:\[\text{Density (}\rho\text{)} = \frac{\text{mass of solution}}{\text{volume of solution}}\]In our example, we first calculated the total mass of the solution by adding the mass of the solute (10.0 g of H₃PO₄) and the mass of the solvent (100.0 g of water), resulting in 110.0 g. The total volume, given as 104 mL, allowed us to compute the density as 1.058 g/mL.

High density can indicate a more concentrated solution, which can have implications in various applications, such as determining the mass required for further reactions.

Mole Fraction

Mole fraction is a way of expressing the concentration of an ingredient by comparing the moles of the solute to the total moles of all components. It is useful in vapor-liquid equilibrium calculations:\[\text{Mole Fraction of a component (X)} = \frac{\text{moles of component}}{\text{total moles in the solution}}\]In our case, first, we calculated the moles of water (solvent) as 5.553 moles. We then added these to the moles of H₃PO₄ (0.102 moles), obtaining a total of 5.655 moles in the solution. Using these values, the mole fraction of H₃PO₄ was found to be 0.0180.

Mole fractions are dimensionless and help in analyzing mixtures where the interactions are based on the count of molecules rather than their volumes or masses.