Problem 86 - Chemistry: The Central Science in SI Units, Global Edition

Question

An \(8.65-g\) sample of an unknown group 2 metal hydroxide is dissolved in \(85.0 \mathrm{~mL}\) of water. An acid-base indicator is added and the resulting solution is titrated with \(2.50 \mathrm{M}\) \(\mathrm{HCl}(a q)\) solution. The indicator changes color, signaling that the equivalence point has been reached, after \(56.9 \mathrm{~mL}\) of the hydrochloric acid solution has been added. (a) What is the molar mass of the metal hydroxide? (b) What is the identity of the metal cation: \(\mathrm{Ca}^{2+}, \mathrm{Sr}^{2+},\) or \(\mathrm{Ba}^{2+} ?\)

Step-by-Step Solution

Answer

The molar mass of the unknown metal hydroxide is \(121.64\,g/mol\), and its metal cation is \(Sr^{2+}\), with the metal hydroxide being \(Sr(OH)_2\).

Step 1: Calculate the moles of HCl used in titration

: We use the molarity and volume of the HCl solution to find the moles: Molarity (M) = moles of solute/liters of solution Moles of HCl = Molarity × Volume The given volume of HCl is 56.9 mL. We convert it to liters by dividing it by 1000: \(56.9\, mL \times \frac{1\,L}{1000\,mL} = 0.0569\, L\) Moles of HCl = 2.50 M × 0.0569 L = 0.14225 moles

Step 2: Determine the moles of the metal hydroxide

: The balanced equation for the reaction between the unknown metal hydroxide (M(OH)2) and HCl is: \(M(OH)_{2}(aq) + 2\,HCl(aq) \rightarrow MCl_{2}(aq) + 2\,H_{2}O(l)\) From the balanced equation, we see that 1 mole of metal hydroxide reacts with 2 moles of HCl. We can use the stoichiometric ratios to find moles of the metal hydroxide: Moles of M(OH)2 = (Moles of HCl) / 2 Moles of M(OH)2 = 0.14225 moles / 2 = 0.071125 moles

Step 3: Calculate the molar mass of the metal hydroxide

: We have the mass and moles of the metal hydroxide, so we can find its molar mass: Molar mass = mass / moles Molar mass = 8.65 g / 0.071125 moles = 121.64 g/mol

Step 4: Identify the metal cation

: The molar mass of the metal hydroxide is 121.64 g/mol. We can identify the metal cation by comparing with the known molar mass of Group 2 metal hydroxides: Ca(OH)2: 40.08 (Ca) + 2(15.999 (O) + 1.00784 (H)) = 74.10 g/mol Sr(OH)2: 87.62 (Sr) + 2(15.999 (O) + 1.00784 (H)) = 121.64 g/mol Ba(OH)2: 137.33 (Ba) + 2(15.999 (O) + 1.00784 (H)) = 171.35 g/mol Based on the molar mass, the metal hydroxide is Sr(OH)2 and hence, the metal cation is Sr²⁺.

Key Concepts

Molar Mass Calculation

Calculating the molar mass is a fundamental concept in stoichiometry. It involves determining the mass of a single mole of a compound or an element. In this example, we are looking to calculate the molar mass of an unknown group 2 metal hydroxide. The formula for molar mass is quite simple:
Molar Mass = \( \frac{\text{mass}}{\text{moles}} \)
To apply this, we first find the number of moles of the substance using stoichiometry. In our solution, we found that there were 0.071125 moles of our unknown metal hydroxide. With a given mass of 8.65 grams, the formula tells us that the molar mass of the compound is \( \frac{8.65 \, \text{g}}{0.071125 \, \text{moles}} \), which equals 121.64 g/mol.
This calculation is pivotal as it allows us to compare the calculated molar mass with known molar masses of various metal hydroxides, ultimately helping identify the unknown metal.

Titration Process

The titration process plays a crucial role in this exercise, particularly in determining the amount of acid needed to react with a base completely. It is an analytical technique where a solution of known concentration is used to determine the concentration of an unknown solution.

In the scenario described, titration is used to find out how much hydrochloric acid (HCl) is needed to neutralize the unknown metal hydroxide completely. The process involves slowly adding the acid to the solution containing the metal hydroxide until the endpoint is reached, which is indicated by a color change due to an added indicator.

• The endpoint is when the number of moles of HCl completely reacts with the moles of OH⁻ ions, signifying the reaction's completion.
• The volume of HCl solution used was 56.9 mL, which, when converted, is 0.0569 L.
• With a known molarity of 2.50 M, the calculation for moles of HCl is straightforward: 2.50 M × 0.0569 L = 0.14225 moles of HCl.

This amount of HCl shows the precise proportion needed for the neutralization, crucial for further calculations.

Group 2 Metal Identification

Identifying a group 2 metal cation involves comparing the calculated molar mass of its hydroxide to known values. Group 2 elements, or alkaline earth metals, react predictably and form hydroxides with consistent molar masses.

In this exercise, the unknown metal hydroxide showed a molar mass of 121.64 g/mol. To identify the metal cation, we aligned this value with the molar masses of known group 2 metal hydroxides:

• Calcium hydroxide, \( \text{Ca(OH)}_2 \), has a molar mass of approximately 74.10 g/mol.
• Strontium hydroxide, \( \text{Sr(OH)}_2 \), has a molar mass of about 121.64 g/mol.
• Barium hydroxide, \( \text{Ba(OH)}_2 \), presents a molar mass near 171.35 g/mol.

Since the calculated molar mass matches that of \( \text{Sr(OH)}_2 \), the unknown metal hydroxide is therefore strontium hydroxide, indicating the metal cation is \( \text{Sr}^{2+} \). This identification method is simple yet powerful, leveraging stoichiometry to deduce the identity of metals in chemical compounds.