Problem 30 - Chemistry: Atoms First

Question

Calculate the molarity of each of the following solutions: (a) 0.195 g of cholesterol, \(\mathrm{C}_{27} \mathrm{H}_{46} \mathrm{O}\), in 0.100 \(\mathrm{L}\) of serum, the average concentration of cholesterol in human serum. (b) 4.25 g of NH 3 in 0.500 L of solution, the concentration of NH 3 in household ammonia. (c) 1.49 kg of isopropyl alcohol, C \(_{3} \mathrm{H}_{7} \mathrm{OH}\), in 2.50 \(\mathrm{L}\) of solution, the concentration of isopropyl alcohol in rubbing alcohol. (d) 0.029 g of I_ in 0.100 L of solution, the solubility of I \(_{2}\) in water at \(20^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The molarities are (a) 0.00504 M, (b) 0.4988 M, (c) 9.9164 M, (d) 0.00114 M.

Step 1: Calculate Molarity of Cholesterol Solution

To find the molarity of the cholesterol solution, use the formula molarity (M) = moles of solute / liters of solution. First, calculate the moles of cholesterol (C27H46O) by dividing the given mass (0.195 g) by its molar mass. The molar mass of cholesterol can be calculated as follows: (27 * 12.01 g/mol for Carbon) + (46 * 1.008 g/mol for Hydrogen) + (16.00 g/mol for Oxygen) = 386.65 g/mol. Moles of cholesterol = 0.195 g / 386.65 g/mol = 0.000504 moles. Now, divide the moles by the volume of serum in liters: Molarity = 0.000504 moles / 0.100 L = 0.00504 M.

Step 2: Calculate Molarity of Ammonia Solution

For NH3 solution, follow the same approach. Calculate the moles of NH3 (molar mass = 14.01 g/mol for Nitrogen + 3 * 1.008 g/mol for Hydrogen = 17.034 g/mol) using mass and molar mass. Moles of NH3 = 4.25 g / 17.034 g/mol = 0.2494 moles. Then, calculate its molarity: Molarity = 0.2494 moles / 0.500 L = 0.4988 M.

Step 3: Calculate Molarity of Isopropyl Alcohol Solution

For the isopropyl alcohol (C3H7OH), calculate moles using the molar mass (3 * 12.01 g/mol for Carbon + 7 * 1.008 g/mol for Hydrogen + 16.00 g/mol for Oxygen + 1.008 g/mol for attached Hydrogen = 60.096 g/mol). Moles of isopropyl alcohol = 1.49 kg (or 1490 g) / 60.096 g/mol = 24.791 moles. Divide by the volume to find molarity: Molarity = 24.791 moles / 2.50 L = 9.9164 M.

Step 4: Calculate Molarity of Iodine Solution

For I2 solution, calculate the number of moles: Molar mass of I2 = 2 * 126.90 g/mol for Iodine = 253.8 g/mol. Moles of I2 = 0.029 g / 253.8 g/mol = 0.000114 moles. Calculate its molarity: Molarity = 0.000114 moles / 0.100 L = 0.00114 M.

Key Concepts

Molarity

Understanding molarity is fundamental in chemistry, specifically when dealing with solution concentration. Molarity, denoted by the symbol 'M', is the measure of the concentration of a solute in a solution. It is defined as the number of moles of solute present in one liter of solution.

The formula to calculate molarity is: ewline ewline \[ M = \frac{{\text{moles of solute}}}{{\text{liters of solution}}} \] ewline ewline Knowing the molarity of a solution is essential because it allows chemists to quantify the strength of the solution, which is critical in both academic studies and practical applications such as medicine, where precise dosages are required.

Solution Concentration

Solution concentration refers to the amount of solute that is dissolved in a solvent to form a solution. There are several ways to express this concentration, but molarity is one of the most common methods used in chemistry. Concentration can affect the properties of a solution, including boiling and freezing points, vapor pressure, and osmotic pressure.

When calculating molarity, it's crucial to measure accurately the volume of the solution; usually in liters, and the amount of solute in moles. The relation of these two gives us a clear and quantifiable measure of solution concentration, allowing for predictable reactions and formulations in chemistry.

Moles of Solute

The concept of moles of solute is central to the study of solution chemistry. A mole represents Avogadro's number (approximately \(6.022 \times 10^{23}\)) of particles, whether they are atoms, molecules, ions, or electrons.

To find the moles of a solute in a solution, you need the mass of the solute and its molar mass. The calculation involves dividing the mass of the solute (usually in grams) by the substance's molar mass: ewline ewline \[ \text{Moles of solute} = \frac{{\text{mass of solute}}}{{\text{molar mass}}} \] ewline ewline This essential step enables us to use the molarity formula and thus calculate the solution's concentration accurately.

Molar Mass

Molar mass is the mass of one mole of a substance and is usually measured in grams per mole (g/mol). It's the sum of the atomic masses of all the atoms in the chemical formula of a substance. This value is indispensable when converting between the mass of a substance and the amount in moles.

For instance, the molar mass of a compound like cholesterol (\(\mathrm{C}_{27}\mathrm{H}_{46}\mathrm{O}\)) is calculated by adding up the atomic masses of all carbons, hydrogens, and oxygen in its formula. Having the correct molar mass is crucial because even a small discrepancy can lead to significant errors in calculating molarity and, consequently, in experimentation and industry applications.