Problem 31 - General Chemistry: Principles and Modern Applications

Question

What are the molarities of the following solutes when dissolved in water? (a) \(2.92 \mathrm{mol} \mathrm{CH}_{3} \mathrm{OH}\) in 7.16 L of solution (b) \(7.69 \mathrm{mmol} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) in \(50.00 \mathrm{mL}\) of solution (c) \(25.2 \mathrm{g} \mathrm{CO}\left(\mathrm{NH}_{2}\right)_{2}\) in \(275 \mathrm{mL}\) of solution

Step-by-Step Solution

Answer

The molarities are (a) \(0.408 \mathrm{M}\), (b) \(0.154 \mathrm{M}\), and (c) \(1.524 \mathrm{M}\).

Step 1: Analyze the Given Values and Necessary Conversions

Check the given amounts of solute and the volume of the solutions. For each case, one needs to make sure that the amount of solute is in moles and the volume of solution is in liters. For (b), the amount of solute is given in millimoles, which need to be converted to moles. For (b) and (c), the volume of the solutions are given in milliliters, which need to be converted to liters. Therefore, the conversions that are necessary are \(1 \mathrm{mmol} = 10^{-3} \mathrm{mol}\) and \(1 \mathrm{mL} = 10^{-3} \mathrm{L}\).

Step 2: Calculate the Molarity (a)

Molarity (M) is calculated as moles of solute divided by liters of solution. For (a), \(2.92 \mathrm{mol} \mathrm{CH}_{3} \mathrm{OH}\) in 7.16 L of solution gives a molarity of \(\frac{2.92 \mathrm{mol}}{7.16 \mathrm{L}} = 0.408 \mathrm{M}\).

Step 3: Calculate the Molarity (b)

First convert the given millimoles to moles and milliliters to liters. Then calculate the molarity. This gives \(7.69 \mathrm{mmol} = 7.69 \times 10^{-3} \mathrm{mol}\) and \(50.00 \mathrm{mL} = 50.00 \times 10^{-3} \mathrm{L}\). Thus, the molarity is \(\frac{7.69 \times 10^{-3} \mathrm{mol}}{50.00 \times 10^{-3} \mathrm{L}} = 0.154 \mathrm{M}\).

Step 4: Calculate the Molarity (c)

First, one needs to convert grams of solute to moles. The molar mass of \(\mathrm{CO}\left(\mathrm{NH}_{2}\right)_{2}\) is 60.06 g/mol. Hence, 25.2 g of \(\mathrm{CO}\left(\mathrm{NH}_{2}\right)_{2}\) is \(\frac{25.2 \mathrm{g}}{60.06 \mathrm{g/mol}} = 0.419 \mathrm{mol}\). Convert milliliters to liters and calculate the molarity. This gives \(275 \mathrm{mL} = 275 \times 10^{-3} \mathrm{L}\) and \(\frac{0.419 \mathrm{mol}}{275 \times 10^{-3} \mathrm{L}} = 1.524 \mathrm{M}\).

Key Concepts

Moles to Liters Conversion

Converting between moles and liters is a crucial step in finding molarity. Molarity, often denoted as "M," measures the concentration of a solute in a solution.
It is determined by how many moles of solute are present in one liter of solution. Thus, any moles to liters conversion directly impacts the calculation of molarity. In this exercise, you will encounter two types of conversions related to volume:

• Moles (amount of substance) to liters (volume of solution). Always ensure the final volume is expressed in liters for accurate molarity calculations.
• Milliliters to liters. Since 1 milliliter (mL) equals 0.001 liters (L), you need to multiply the milliliter amount by 0.001 to convert it into liters.

This conversion is especially important when the given data includes volumes in milliliters, resulting in small number magnitudes suitable for laboratory conditions. By understanding and applying these conversions, you ensure accurate calculations of solute concentration, enabling a true measurement of the solute's presence in the solution.

Solute Concentration

Solute concentration, a fundamental concept in chemistry, tells us how much solute is present in a given volume of solution. Molarity is a specific measure of solute concentration expressed as moles per liter.
Knowing the solute concentration is critical in scientific experiments, as it allows scientists to create solutions with precise chemical properties.The general equation to calculate molarity ( M ) is: \[ \text{Molarity} (M) = \frac{\text{moles of solute}}{\text{liters of solution}} \]When calculating solute concentration, be mindful of the following steps:

• Ensure measurements of solute are in moles. If provided in another unit (like millimoles), convert it to moles by multiplying by \(10^{-3}\).
• Ensure solution volume is given in liters. If the volume is not in liters, convert it from milliliters by multiplying by \(10^{-3}\).
• Use the equation to ascertain the molarity of the solution by dividing the moles of solute by the liters of solution.

Understanding solute concentration allows you to predict the behavior and reactivity of solutions, making it a cornerstone of both qualitative and quantitative chemical analysis.

Molar Mass Calculation

Molar mass is the weight of one mole of a substance and is typically listed on the periodic table for each element. Understanding molar mass enables us to convert a given mass of a compound to moles.
Moles are essential for calculating concentration and predicting the product of a reaction.For instance, to calculate the molar mass of a compound, such as urea ( \mathrm{CO}(\mathrm{NH}_2)_2 ), the following steps are helpful:

• Identify each element's atomic masses in the compound: Carbon (C), Oxygen (O), and Nitrogen (N) from the periodic table.
• Multiply the atomic mass of each element by the number of atoms it contributes to the compound:
• C has an atomic mass of approximately 12.01 g/mol
• O has an atomic mass of approximately 16.00 g/mol
• N has an atomic mass of approximately 14.01 g/mol
• Sum the contributions of each element to find the molar mass: \(60.06 \text{ g/mol} \) for urea.

Calculating the molar mass is essential for converting grams to moles, facilitating the accurate determination of solute concentration in various chemical solutions.