Question

How many grams of \(\mathrm{KCl}\) must be added to \(75 \mathrm{~g}\) of water to produce a solution with a molality of \(2.25\) (a) \(1.257 \mathrm{~g}\) (b) \(125.7 \mathrm{~g}\) (c) \(12.57 \mathrm{~g}\) (d) \(25.14 \mathrm{~g}\)

Short answer

The correct answer is (c) \(12.57 \mathrm{~g}\).

Step-by-step solution

Understand Molality

Molality is defined as the number of moles of solute per kilogram of solvent. It is given by the formula: \( m = \frac{\text{moles of solute}}{\text{kg of solvent}} \), where \( m \) is the molality.

Convert Solvent Mass

First, convert the mass of water from grams to kilograms because molality uses kg of solvent. Thus, \( 75 \text{ g} = 0.075 \text{ kg}. \) This is the mass of the water (solvent) in kilograms.

Calculate Moles of KCl Needed

To achieve the desired molality of \(2.25 \), use the formula: \[ m = \frac{\text{moles of solute}}{\text{kg of solvent}} = \frac{x}{0.075} \] Solving for \(x\) gives \[ x = 2.25 \times 0.075 = 0.16875 \text{ moles}. \]This is the number of moles of \( \mathrm{KCl} \) needed.

Convert Moles to Grams

Now, calculate the mass of \( \mathrm{KCl} \). The molar mass of \( \mathrm{KCl} \) is approximately \(39.1 + 35.45 = 74.55 \text{ g/mol}.\)Calculate the mass of \( \mathrm{KCl} \): \[ \text{grams} = 0.16875 \text{ moles} \times 74.55 \text{ g/mol} = 12.57 \text{ g}. \]

Verify the Answer Choice

Compare the calculated mass with the given options:- (a) \(1.257 \mathrm{~g}\) - (b) \(125.7 \mathrm{~g}\) - (c) \(12.57 \mathrm{~g}\) - (d) \(25.14 \mathrm{~g}\)The correct answer is (c) \(12.57 \mathrm{~g}\).

Key concepts

Understanding Moles of Solute

Moles of solute is a key concept in chemistry used to represent the amount of a substance. It's calculated by dividing the mass of the solute by its molar mass. This concept helps us understand how much of a solute is present in a solution. For any solution, knowing the number of moles is crucial because it allows us to use this information in equations and formulas. Moles serve as a bridge between the mass of substances and their amount based on Avogadro's number. Let's say you have been given a particular mass of a compound, like potassium chloride (\(\mathrm{KCl}\)). To find the moles, you need to know the molar mass of the compound. Divide the given mass of \(\mathrm{KCl}\) by its molar mass to find the number of moles of solute present. This value is essential when working with solution concentrations such as molality.

Defining Molar Mass

Molar mass is the weight of one mole of a substance, which is a fundamental chemical concept. It is usually expressed in grams per mole (g/mol) and is essential for performing calculations involving chemical equations. To find the molar mass of a compound, sum the atomic masses of all atoms in the molecule. For example, \(\mathrm{KCl}\) consists of one potassium atom and one chlorine atom. You find their atomic masses from the periodic table: potassium approximately 39.1 g/mol, and chlorine 35.45 g/mol. Add these together to get the molar mass of \(\mathrm{KCl}\): 74.55 g/mol. Once you have the molar mass, you can calculate the moles of the substance present, which is critical in further calculations like those needed for molality. Knowing molar mass allows you to convert between grams and moles correctly, a skill necessary for solving many chemistry problems.

Understanding Solution Concentration and Molality

The concentration of a solution tells us how much solute is present compared to the solvent. There are many ways to express concentration, but molality is particularly useful in situations where temperature changes might occur because molality remains constant with temperature fluctuations. Molality (\(m\)) is defined as the number of moles of solute per kilogram of solvent. This means the relation between the moles of solute and the mass of solvent in kilograms is highlighted, allowing chemists to focus on the substance interactions without worrying about volume changes due to temperature.The formula for molality is \(m = \frac{\text{moles of solute}}{\text{kg of solvent}}\). This unique property of molality makes it a reliable measure for scientific experiments and industrial applications, ensuring precise results unaffected by temperature variations.

Converting Grams to Kilograms

When dealing with chemical solutions, conversions between units are common and necessary. Specifically, when calculating molality, converting grams to kilograms is crucial because molality uses the kilogram as the unit for the solvent. To convert grams to kilograms, divide the number of grams by 1000 because 1 kilogram equals 1000 grams. For example, if you have 75 grams of water, you convert it by performing the calculation 75 g / 1000 = 0.075 kg. This conversion is a straightforward step but vital in achieving the correct solution concentration. Without making this conversion, any calculation for molality would be incorrect, leading to misunderstandings in the properties or behavior of the solution. Mastering this skill ensures accuracy in laboratory and practical applications.