Question

Calculate each of the following quantities: (a) Volume (mL) of \(2.26 M\) potassium hydroxide that contains \(8.42 \mathrm{~g}\) of solute (b) Number of \(\mathrm{Cu}^{2+}\) ions in \(52 \mathrm{~L}\) of \(2.3 \mathrm{M}\) copper(II) chloride (c) Molarity of \(275 \mathrm{~mL}\) of solution containing \(135 \mathrm{mmol}\) of glucose

Short answer

(a) 66.6 mL KOH (b) \(7.20 \times 10^{25} \) \(\mathrm{Cu}^{2+}\) ions (c) 0.491 M glucose

Step-by-step solution

Understand what is given and what needs to be found

(a) Determine the volume of a \(2.26 M\) potassium hydroxide (KOH) solution that contains \(8.42 \,g\) of solute. (b) Calculate the number of \(\mathrm{Cu}^{2+}\) ions in \(52 \,L\) of \(2.3 \,M\) copper(II) chloride (CuCl\(_2\)).(c) Find the molarity of \(275 \,mL\) of solution containing \(135 \,mmol\) of glucose.

Use molarity formula for part (a)

Molarity (M) = \(\frac{\text{moles of solute}}{\text{volume of solution in liters}}\). Calculate moles: Moles of KOH = \(\frac{\text{Given Mass}}{\text{Molar Mass}}\), where the molar mass of KOH is approximately \(56.1 \,g/mol\).Moles of KOH = \(\frac{8.42 \,g}{56.1 \,g/mol\}\).Then solve for volume (in liters) using \(M = \frac{\text{moles}}{\text{volume}}\).

Calculate volume of KOH solution in part (a)

Volume = \(\frac{moles}{M} = \frac{\frac{8.42}{56.1}}{2.26} \,L = 0.0666 \,L = 66.6 \,mL\).

Use molarity and volume to find number of ions for part (b)

First, calculate moles of CuCl\(_2\): Moles of CuCl\(_2\) = \(\text{Molarity} \times \text{Volume (in L)} = 2.3 \,M \times 52 \,L = 119.6 \,moles\).Since each CuCl\(_2\) molecule provides one Cu\(^{2+}\) ion, there are \(119.6 \,moles \,Cu^{2+}\) ions.Finally, calculate number of ions using Avogadro's number \((6.022 \times 10^{23} \,ions/mol)\):Number of \(Cu^{2+}\) ions = \(119.6 \,moles \times 6.022 \times 10^{23} = 7.20 \times 10^{25} \,ions\).

Calculate Molarity for part (c)

Convert mmol to mol: 135 \,mmol = 0.135 \,molMolarity = \(\frac{\text{moles}}{\text{volume (in liters)}} = \frac{0.135 \,mol}{0.275 \,L} = 0.491 \,M\).

Key concepts

Molarity

Molarity, often represented by \(M\), is a measure of the concentration of a solute in a solution. It's defined as the number of moles of solute present per liter of solution. The formula for molarity is:
\[ \text{M} = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]
Understanding molarity is crucial for calculating how much solute is dissolved in a specific volume of solvent. For instance, in part (a) of our exercise, we needed to determine the volume of a 2.26 M potassium hydroxide (KOH) solution containing 8.42 grams of KOH. Using molarity helps in various applications, such as dilutions and chemical reactions. To solve such problems, always ensure your units are consistent, typically converting volumes to liters and ensuring the mass is converted to moles if necessary.

Moles and Molar Mass

To solve molarity problems, you often need to calculate moles from a given mass. The mole (mol) is a fundamental unit in chemistry for quantifying amounts of a substance. Molar mass is the mass of one mole of a given substance, usually expressed in grams per mole (g/mol).
Here's the basic formula:
\[ \text{Moles} = \frac{\text{Given Mass}}{\text{Molar Mass}} \]
For example, in part (a), we converted the given 8.42 grams of KOH to moles, knowing the molar mass of KOH is approximately 56.1 g/mol: \[ \text{Moles of KOH} = \frac{8.42 \text{ g}}{56.1 \text{ g/mol}} = 0.150 \text{ moles} \]
This step is essential in converting the mass of a substance to the number of moles, which can then be used in molarity calculations and other chemical analyses.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It is based on the conservation of mass and the concept of moles. Stoichiometric calculations ensure that the right amounts of reactants react to form products, maintaining the balance of the chemical equation.
In part (b), we calculated the number of Cu\textsuperscript{2+} ions in a solution. First, we used the molarity and volume to find moles of CuCl\textsubscript{2}: \[ \text{Moles of CuCl}_2 = 2.3 \text{ M} \times 52 \text{ L} = 119.6 \text{ moles} \]
Since each CuCl\textsubscript{2} formula unit provides one Cu\textsuperscript{2+} ion, we then found the number of ions using Avogadro's number (6.022 \times 10^{23} ions/mol): \[ \text{Number of Cu}^{2+} \text{ ions} = 119.6 \text{ moles} \times 6.022 \times 10^{23} = 7.20 \times 10^{25} \text{ ions} \]
This process exemplifies how stoichiometry links the theoretical and practical aspects of chemical reactions using molarity as a starting point.

Chemistry Problem-Solving

Solving chemistry problems often involves multiple steps, requiring a clear understanding of the concepts and how they interrelate. Let's consider part (c) of our exercise, where we need to find the molarity of a solution:
The given information includes 135 mmol of glucose in 275 mL of solution. First, we need to convert mmol to mol:
135 mmol = 0.135 mol
Then, we convert the volume from mL to liters:
275 mL = 0.275 L
Finally, we use the molarity formula: \[ \text{Molarity} = \frac{0.135 \text{ mol}}{0.275 \text{ L}} = 0.491 \text{ M} \]
Effective problem-solving in chemistry requires identifying given data, converting units as necessary, and applying appropriate formulas. Breaking down each step, as shown, makes even complex problems more manageable and understandable.