Problem 86 - Chemistry The Central Science

Question

An 8.65 -g sample of an unknown group 2 A 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

Based on the provided step-by-step solution, the short answer to the question is as follows. (a) To find the molar mass of the metal hydroxide, first, calculate the moles of HCl used in the titration: \(Moles_{HCl} = 2.50 M * 0.0569 L\) Next, use stoichiometry to find the moles of metal hydroxide (MH): \(Moles_{MH} = \frac{Moles_{HCl}}{2}\) Then, calculate the molar mass of the metal hydroxide: \(Molar\,mass_{MH} = \frac{8.65 g}{Moles_{MH}}\) (b) To identify the metal cation, compare the calculated molar mass with the known molar masses of Group 2A metal hydroxides (Ca(OH)2, Sr(OH)2, and Ba(OH)2) to find the metal hydroxide with the closest molar mass. This will determine the identity of the metal cation (Ca2+, Sr2+, or Ba2+).

Step 1: Calculate the moles of HCl used in the titration

First, determine the moles of HCl that was used in the titration. We have the volume (56.9 mL) and molarity (2.50 M) of the HCl solution. Moles can be calculated using the formula: Moles of HCl = Molarity * Volume (in liters) \(Moles_{HCl} = 2.50 M * 0.0569 L\)

Step 2: Use stoichiometry to obtain the moles of the metal hydroxide

Since we know the moles of HCl, we can now use stoichiometry to find the moles of metal hydroxide (denoted as MH here) because at the equivalence point, the moles of MH in the original solution should be equal to the moles of HCl used in titration. The balanced chemical equation would look something like this: \(MH + 2 HCl -> MCl_2 + 2 H_2O\) Here, the stoichiometric ratio of the metal hydroxide to the HCl is 1:2 (as shown in the balanced equation). Therefore, to find the moles of metal hydroxide: \(Moles_{MH} = \frac{Moles_{HCl}}{2}\)

Step 3: Calculate the mass of the metal hydroxide

Now that we have the moles of metal hydroxide, we can find the mass of the metal hydroxide from the given sample mass (8.65 g). To do this, we use the formula: \(g_{MH} = Moles_{MH} * Molar\,mass_{MH}\) Here, we need to solve for the molar mass of the metal hydroxide (Molar mass of MH).

Step 4: Determine the molar mass of the metal hydroxide

Rearrange the formula from Step 3 to solve for the molar mass of the metal hydroxide: \(Molar\,mass_{MH} = \frac{g_{MH}}{Moles_{MH}}\) This will give us the molar mass of the metal hydroxide.

Step 5: Identify the metal cation

Now that we have the molar mass of the metal hydroxide, we can compare it with known molar masses of group 2A metal hydroxides (Ca(OH)2, Sr(OH)2, and Ba(OH)2) to identify the metal cation (Ca2+, Sr2+, or Ba2+). Compare the molar mass calculated in Step 4 with the molar masses of known metal hydroxides, and find the metal hydroxide with the closest molar mass. This will give us the identity of the metal cation.

Key Concepts

Molar Mass Calculation

Calculating the molar mass is an essential step in understanding the properties of a substance. In this exercise, we began by determining the amount of HCl used in the titration by using its known molarity and volume. By employing the formula \( \,\text{Moles of HCl} = \text{Molarity} \times \text{Volume (liters)} \,\), we calculated the moles of HCl.
From there, we used the stoichiometric ratios (from the balanced equation) to find the moles of the unknown metal hydroxide. By dividing the known mass of the MH sample by the moles calculated, we were able to find the unknown metal hydroxide’s molar mass with the formula:
\[ \text{Molar Mass}_{MH} = \frac{g_{MH}}{\text{Moles}_{MH}} \] This process laid the foundation for comparing and identifying the metal cation in the sample.

Stoichiometry

Stoichiometry plays a crucial role in chemical calculations, especially in titration exercises such as this one. It involves the use of balanced chemical equations to determine the relationships between reactants and products.
In this scenario, we used the balanced equation \( MH + 2 HCl \rightarrow MCl_2 + 2 H_2O \), where MH represents the unknown metal hydroxide. This equation tells us that one mole of metal hydroxide reacts with two moles of hydrochloric acid. By using this 1:2 ratio, we converted the moles of HCl to moles of metal hydroxide using the formula:

• \( \text{Moles}_{MH} = \frac{\text{Moles}_{HCl}}{2} \)

The stoichiometric coefficients help us bridge the gap between the known quantities from the titration and the unknown substance’s properties, leading us to further insights into the sample.

Group 2A Metal Hydroxide

Group 2A metals, also known as alkaline earth metals, form hydroxides which are often used in neutralization reactions in titration processes. These elements include the likes of calcium, strontium, and barium, all of which can form metal hydroxides: Ca(OH)_2, Sr(OH)_2, and Ba(OH)_2 respectively.
In this exercise, the problem was to determine the molar mass of an unknown group 2A metal hydroxide by using titration data. Once we found the molar mass, we had to identify the specific metal cation by comparing it with known molar masses of calcium, strontium, and barium hydroxides. Through this method, we pinpointed whether the cation was \( \text{Ca}^{2+}, \text{Sr}^{2+}, \text{or Ba}^{2+} \).
Recognizing the characteristics and behavior of group 2A metal hydroxides formed an important step in reaching a conclusion in the identification of the unknown substance.

Acid-Base Indicator

An acid-base indicator is vital in titration experiments to signal the end point of the reaction, known as the equivalence point. These indicators are substances that change color when the pH of the solution reaches a certain value, corresponding to the neutralization of the acid by the base or vice-versa.
For the metal hydroxide titration, an acid-base indicator was used to visually determine when the reaction reached its equivalence point. This point is where equivalent moles of metal hydroxide and hydrochloric acid had reacted, thus confirming the complete conversion of the reactants into products.
The indicator ensures accuracy in the titration, allowing us to confidently calculate the exact moles of reactants and, subsequently, determine properties like the molar mass of the unknown metal hydroxide.