Problem 9 - Chemistry for Allied Health

Question

Calculate the molarity for each solution. a. \(87.2 \mathrm{~g}\) of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) in enough water to make \(500 . \mathrm{mL}\) of solution b. \(61.8 \mathrm{~g}\) of \(\mathrm{NH}_{3}\) in enough water to make \(7.00 \mathrm{~L}\) of solution c. 100. \(\mathrm{mL}\) of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) in 500. mL of solution (The density of ethanol is \(0.789 \mathrm{~g} / \mathrm{mL}\).)

Step-by-Step Solution

Answer

(a) 1.228 M, (b) 0.519 M, (c) 3.43 M

Step 1: Understanding Molarity

Molarity (M) is defined as the number of moles of solute per liter of solution. It is expressed by the formula: \( M = \frac{n}{V} \), where \( n \) is the number of moles of solute and \( V \) is the volume of the solution in liters.

Step 2: Calculate Moles of Solute for Part (a)

For part (a), first find the molar mass of \( \mathrm{Na}_{2} \mathrm{SO}_{4} \). Na: 23 g/mol, S: 32 g/mol, O: 16 g/mol. Thus, \( \mathrm{Na}_{2} \mathrm{SO}_{4} \) has a molar mass of \( 2(23) + 32 + 4(16) = 142 \text{ g/mol} \). Now, calculate the moles from the given mass: \[ n = \frac{87.2 \text{ g}}{142 \text{ g/mol}} = 0.614 \text{ moles} \]

Step 3: Calculate Molarity for Part (a)

Convert the volume from mL to L: \( 500 \text{ mL} = 0.5 \text{ L} \). Then, apply the molarity formula:\[ M = \frac{0.614 \text{ moles}}{0.5 \text{ L}} = 1.228 \text{ M} \].

Step 4: Calculate Moles of Solute for Part (b)

Find the molar mass of \( \mathrm{NH}_{3} \): N: 14 g/mol, H: 1 g/mol. Thus, \( \mathrm{NH}_{3} \) has a molar mass of \( 14 + 3(1) = 17 \text{ g/mol} \). Calculate the moles: \[ n = \frac{61.8 \text{ g}}{17 \text{ g/mol}} = 3.635 \text{ moles} \]

Step 5: Calculate Molarity for Part (b)

The volume is already in liters: 7.00 L. Use the molarity formula:\[ M = \frac{3.635 \text{ moles}}{7.00 \text{ L}} = 0.519 \text{ M} \].

Step 6: Calculate Mass of Ethanol for Part (c)

Determine the mass of 100 mL of ethanol using its density: \[ \text{Mass} = \text{Volume} \times \text{Density} = 100 \text{ mL} \times 0.789 \text{ g/mL} = 78.9 \text{ g} \]

Step 7: Calculate Moles of Ethanol for Part (c)

Find the molar mass of \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \): C: 12 g/mol, H: 1 g/mol, O: 16 g/mol. Thus, it has a molar mass of \( 2(12) + 6(1) + 16 = 46 \text{ g/mol} \). Calculate the moles: \[ n = \frac{78.9 \text{ g}}{46 \text{ g/mol}} = 1.715 \text{ moles} \]

Step 8: Calculate Molarity for Part (c)

Convert the volume from mL to L: \( 500 \text{ mL} = 0.5 \text{ L} \). Use the molarity formula:\[ M = \frac{1.715 \text{ moles}}{0.5 \text{ L}} = 3.43 \text{ M} \].

Key Concepts

Moles of Solute

Understanding the concept of moles is crucial in chemistry, especially when calculating solution concentrations. A mole is a standard unit in chemistry that quantifies the amount of substance. It is based on Avogadro's number, which is approximately \( 6.022 \times 10^{23} \) particles or entities per mole. To calculate moles from a given mass, you divide the mass of the substance by its molar mass.

• Moles formula: \( n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \)
• Example: If you have 87.2 g of \( \text{Na}_2\text{SO}_4 \) with a molar mass of 142 g/mol, the moles would be \( \frac{87.2}{142} \approx 0.614 \) moles.

This calculation helps define how much of the solute is present at the atomic level and is foundational for further calculations in solution concentration.

Solution Concentration

The concentration of a solution refers to the amount of solute present in a given quantity of solvent or solution. Molarity (M) is the most common way to express concentration in chemistry and is defined as the number of moles of solute per liter of solution.

• Molarity formula: \( M = \frac{n}{V} \), where \( n \) is moles of solute and \( V \) is volume in liters.
• Example: For a solution with 0.614 moles of solute and a volume of 0.5 liters, the molarity is \( \frac{0.614}{0.5} = 1.228 \text{ M} \).

Knowing the molarity of a solution is crucial for predicting how the solution will behave in chemical reactions. Moreover, it helps in calculating the ratios when solutions are mixed or diluted.

Step-by-Step Solution

Breaking down a problem step-by-step is essential in chemistry to avoid confusion and ensure accurate results. Each calculation in molarity problems should follow a logical sequence:
1. **Identify the Given Values:** Start by noting the given mass of the solute and the volume of the solution. Ensure that all units are consistent.
2. **Calculate Moles:** Use the mass and molar mass of the solute to find the moles. This step sets the stage for calculating concentration.
3. **Convert Volume:** Ensure the volume is expressed in liters, as molarity requires liters for its calculation.
4. **Apply the Molarity Formula:** Use the moles and volume to find the molarity.
This structured approach not only simplifies the calculation process but also minimizes the risk of errors. Implementing each step thoughtfully ensures clarity and accuracy.

Molar Mass

Molar mass is a fundamental concept in chemistry that relates the mass of one mole of a substance to its atomic or molecular composition.

• Calculation: Sum the individual atomic masses of all atoms in a molecule. Refer to the periodic table for atomic masses.
• Example: For \( \text{Na}_2\text{SO}_4 \), the molar mass is \( 2(23) + 32 + 4(16) = 142 \text{ g/mol} \).

The molar mass acts as a conversion factor between mass and moles, allowing chemists to quantify substances for reactions. Accurate calculation of molar mass is critical in achieving the correct stoichiometry for reactions and is the starting point for any mole-related calculation.