Problem 22 - Chemical Principles: The Quest for Insight

Question

(a) What mass (in grams) of \(\mathrm{NaOH}\) must be mixed with \(250 \mathrm{~g}\) of water to prepare a \(0.22 \mathrm{~m} \mathrm{NaOH}\) solution? (b) Calculate the amount (in moles) of cthylene glycol, \(\mathrm{HOC}_{2} \mathrm{H}_{4} \mathrm{OH}\), that should be added to \(2.0 \mathrm{~kg}\) of water to prepare a \(0.44 \mathrm{~m} \mathrm{HOC}_{2} \mathrm{H}_{4} \mathrm{OH}\) (aq) solution. (c) Determine the amount (in moles) of HCl that must be dissolved in \(500 \mathrm{~g}\) water to prepare a \(0,0010 m\) HClaq) solution.

Step-by-Step Solution

Answer

a) 2.2 grams of NaOH, b) 0.88 moles of ethylene glycol, c) 0.0005 moles of HCl

Step 1: Understand molality and calculate mass of NaOH

molality (m) is defined as the number of moles of solute per kilogram of solvent. The formula to calculate mass (m) of NaOH is: \[ m_{NaOH} = molality \times mass_{solvent \text{ (kg)}} \times molar \text{ mass of NaOH (g/mol)} \] First, convert the mass of solvent to kilograms: \[ 250 \text{ g} = 0.25 \text{ kg} \] Then use the given molality, and molar mass of NaOH (39.997 g/mol) to find the mass of NaOH: \[ m_{NaOH} = 0.22 \text{ m} \times 0.25 \text{ kg} \times 39.997 \text{ g/mol} \]

Step 2: Calculate mass of NaOH required

Now complete the calculation: \[ m_{NaOH} = 0.22 \times 0.25 \times 39.997 \] \[ m_{NaOH} = 2.19985 \text{ g} \] The mass of NaOH needed is approximately 2.2 grams

Step 3: Calculate moles of ethylene glycol

For ethylene glycol (\text{HOC}_{2}\text{H}_{4}\text{OH}), use the same molality definition. Convert the mass of solvent to kilograms: \[ 2.0 \text{ kg} = 2 \text{ kg} \] Molar mass of ethylene glycol is 62.07 g/mol. Calculate the moles using the molality: \[ n_{\text{HOC}_{2}\text{H}_{4}\text{OH}} = 0.44 \text{ m} \times 2 \text{ kg} \] \[ n_{\text{HOC}_{2}\text{H}_{4}\text{OH}} = 0.88 \text{ moles} \]

Step 4: Determine moles of HCl required

Using the definition of molality for HCl solution and converting the mass of water to kg: \[ 500 \text{ g} = 0.5 \text{ kg} \] The molality is given and calculate the moles of HCl: \[ n_{\text{HCl}} = 0.0010 \text{ m} \times 0.5 \text{ kg} \] \[ n_{\text{HCl}} = 0.0005 \text{ moles} \]

Key Concepts

Molality

Molality is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, molality is not affected by changes in temperature or pressure because it relates to the mass of the solvent rather than the volume of the solution. To calculate molality, the formula is:
\[ \text{molality} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]
It's essential to note that the mass of the solvent is always in kilograms when calculating molality, which can be challenging if you're used to other units. Students often forget to convert the solvent's mass from grams to kilograms, which is a critical step in the calculation.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a fundamental concept in chemistry because it links the mass of a substance to the number of moles and ultimately to the number of particles. To calculate molar mass, sum the atomic masses of each element in the compound, as obtained from the periodic table.

• For example, for NaOH (sodium hydroxide), the molar mass is 39.997 g/mol, calculated as the sum of the atomic masses of sodium (Na), oxygen (O), and hydrogen (H).

Understanding molar mass is crucial when working out molality because you need it to convert between grams and moles of the solute.

Moles of Solute

Moles of solute refers to the amount of the substance (solute) present in the solution. A mole is 6.022 x 10²³ particles of the substance—be it atoms, molecules, ions, or other particles. Calculating moles of solute is a common task in chemistry and is achieved by dividing the mass of the substance by its molar mass.

• To calculate, use the formula: \[ \text{moles of solute} = \frac{\text{mass of solute (in g)}}{\text{molar mass (in g/mol)}} \]

For example, in the case of NaOH in the provided exercise, if you know the mass of NaOH needed, you could rearrange the formula to calculate the moles of NaOH, which is then used to determine molality.

Mass of Solvent

The mass of the solvent is the weight of the liquid (usually water) that the solute is dissolved into to form a solution. When measuring for molality, it's vital to measure the mass in kilograms (kg) to ensure that the calculation is correct.

• Always check the unit of mass in a problem: if it’s given in grams (g), convert to kilograms (kg) by dividing by 1000, as shown in the exercise.

A common oversight is using the total mass of the solution or the mass of the solute instead of the mass of the solvent. Only the mass of the solvent is relevant for calculating molality since it’s the solvent that the concentration is being measured against.