Question

How many grams of solute are in each of the following solutions? a. \(2.5 \mathrm{~L}\) of a \(3.0 \mathrm{M} \mathrm{Al}\left(\mathrm{NO}_{3}\right)_{3}\) solution b. \(75 \mathrm{~mL}\) of a \(0.50 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) solution c. \(235 \mathrm{~mL}\) of a \(1.80 \mathrm{M} \mathrm{LiCl}\) solution

Short answer

a. 1597.58 g, b. 6.76 g, c. 17.92 g

Step-by-step solution

Understand molarity definition

Molarity (\text{M}) is defined as the number of moles of solute per liter of solution. The formula to find the number of moles is: \[\text{Moles of solute} = \text{Molarity (M)} \times \text{Volume of solution (L)}\]

Calculate moles of solute for solution a

Given: \(2.5 \text{ L}\) of a \(3.0 \text{ M} \text{Al}\text{(NO}_3\text{)}_3\) solution.\[\text{Moles of Al(NO}_3\text{)}_3 = 3.0 \text{ M} \times 2.5 \text{ L} = 7.5 \text{ moles}\]

Convert moles to grams for solution a

The molar mass of \(\text{Al(NO}_3\text{)}_3\): \(Al: 1 \times 26.98 \text{ g/mol}\)\(N: 3 \times 14.01 \text{ g/mol}\)\(O: 9 \times 16.00 \text{ g/mol}\)\[\text{Molar mass} = 26.98 + 42.03 + 144.0 = 213.01 \text{ g/mol} \]\[\text{Mass of Al(NO}_3\text{)}_3 = 7.5 \times 213.01 \text{ g/mol} = 1597.58 \text{ g}\]

Calculate moles of solute for solution b

Given: \(75 \text{ mL} = 0.075 \text{ L}\) of a \(0.50 \text{ M} \text{C}_6\text{H}_{12}\text{O}_6\) solution.\[\text{Moles of C}_6\text{H}_{12}\text{O}_6 = 0.50 \text{ M} \times 0.075 \text{ L} = 0.0375 \text{ moles}\]

Convert moles to grams for solution b

The molar mass of \(\text{C}_6\text{H}_{12}\text{O}_6\): \(C: 6 \times 12.01 \text{ g/mol}\)\(H: 12 \times 1.01 \text{ g/mol}\)\(O: 6 \times 16.00 \text{ g/mol}\)\[\text{Molar mass} = 72.06 + 12.12 + 96.00 = 180.18 \text{ g/mol} \]\[\text{Mass of C}_6\text{H}_{12}\text{O}_6 = 0.0375 \times 180.18 \text{ g/mol} = 6.76 \text{ g}\]

Calculate moles of solute for solution c

Given: \(235 \text{ mL} = 0.235 \text{ L}\) of a \(1.80 \text{ M} \text{LiCl}\) solution.\[\text{Moles of LiCl} = 1.80 \text{ M} \times 0.235 \text{ L} = 0.423 \text{ moles}\]

Convert moles to grams for solution c

The molar mass of \(\text{LiCl}\): \(Li: 1 \times 6.94 \text{ g/mol}\)\(Cl: 1 \times 35.45 \text{ g/mol}\)\[\text{Molar mass} = 6.94 + 35.45 = 42.39 \text{ g/mol} \]\[\text{Mass of LiCl} = 0.423 \times 42.39 \text{ g/mol} = 17.92 \text{ g}\]

Key concepts

Molarity

Molarity is a way to express the concentration of a solution. It's defined as the number of moles of solute found in one liter of solution. The formula we use is: \[ M = \frac{n}{V} \] where \( M \) is the molarity, \( n \) is the number of moles of solute, and \( V \) is the volume of the solution in liters. Understanding molarity is crucial because it tells us how concentrated a solution is. Whether we’re preparing chemical reactions, diluting samples, or studying solution properties, molarity provides a consistent way to quantify concentrations. To calculate molarity:

• Measure the amount of solute in moles.
• Measure the volume of the solution in liters.
• Divide the moles of solute by the volume of solution in liters.

Molarity is widely used in chemistry labs and industries, making it a fundamental concept for students to grasp.

Moles of Solute

The number of moles of a solute (\( n \)) is essential for calculating molarity and performing various chemical calculations. A mole is a unit that measures the amount of a substance and is defined as exactly \( 6.022 \times 10^{23} \) particles (Avogadro's number). To find the number of moles, use this formula: \[ n = M \times V \] where \( M \) is the molarity of the solution and \( V \) is the volume of the solution in liters. For instance, for a solution where \( M = 3.0 \) and \( V = 2.5 \), the number of moles of solute can be calculated as \( 3.0 \) moles/L \( \times 2.5 \) L = \( 7.5 \) moles. Understanding how to calculate moles solidifies the foundation for further calculations, including converting moles to grams.

Molar Mass

Molar mass is the mass of one mole of a substance, usually measured in g/mol. It is calculated by summing the atomic masses of all atoms present in a molecule. For example, to find the molar mass of \( \text{Al(NO}_3\text{)}_3 \):

• Al: 1 atom \( \times 26.98 \) g/mol
• N: 3 atoms \( \times 14.01 \) g/mol
• O: 9 atoms \( \times 16.00 \) g/mol

The total molar mass = 26.98 g/mol + 42.03 g/mol + 144.00 g/mol = 213.01 g/mol. Molar mass links the macroscopic quantities we measure in the lab to the microscopic quantities (moles) that dictate chemical behavior. It's a vital step in converting moles to grams: \[ \text{Mass (g)} = \text{Moles} \times \text{Molar Mass} \] This conversion is particularly useful in real-world applications like chemical manufacturing and analysis.

Solution Concentration

Solution concentration describes how much solute is present in a given quantity of solvent or solution. Molarity is one way to express concentration, but we can also use grams per liter or percentage composition depending on the context. For an accurate calculation, follow these steps:

• Calculate the moles of solute using molarity and volume.
• Find the molar mass of the solute to convert moles to grams.
• Divide the mass of the solute by the volume of the solution to get concentration in grams per liter, if needed.

Let's consider a \( 0.50 \text{ M} \text{C}_6\text{H}_{12}\text{O}_6 \) solution with a volume of \( 0.075 \) L. The moles of solute are calculated as \( 0.50 \) M \( \times 0.075 \) L = 0.0375 moles. The molar mass of \( \text{C}_6\text{H}_{12}\text{O}_6 \) is \( 180.18 \) g/mol. Therefore, the mass of solute is \( 0.0375 \) moles \( \times 180.18 \) g/mol = 6.76 g. Understanding solution concentration helps in predicting reaction outcomes and scaling up experimental results to industrial processes.