Problem 85 - Chemistry: The Central Science in SI Units, Global Edition

Question

A \(4.36-g\) sample of an unknown alkali metal hydroxide is dissolved in \(100.0 \mathrm{~mL}\) of water. An acid-base indicator is added, and the resulting solution is titrated with \(2.50 \mathrm{MHCl}(a q)\) solution. The indicator changes color, signaling that the equivalence point has been reached, after \(17.0 \mathrm{~mL}\) 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 alkali metal cation: \(\mathrm{Li}^{+}, \mathrm{Na}^{+}, \mathrm{K}^{+}, \mathrm{Rb}^{+},\) or \(\mathrm{Cs}^{+} ?\)

Step-by-Step Solution

Answer

The molar mass of the unknown alkali metal hydroxide is approximately 102.588 g/mol, and the identity of the alkali metal cation is \(\mathrm{Rb}^{+}\).

Step 1: Calculate moles of HCl added

We are given that 17.0 mL of a 2.50 M HCl solution is added to reach the equivalence point. To find the moles of HCl added, we use this formula: moles of HCl = volume of HCl (in liters) × molarity of HCl (in mol/L). moles of HCl = (0.017 L) × (2.50 mol/L) = 0.0425 mol

Step 2: Calculate moles of alkali metal hydroxide

At the equivalence point, the moles of HCl will be equal to the moles of the alkali metal hydroxide (MOH). Thus, moles of MOH = 0.0425 mol

Step 3: Calculate molar mass of alkali metal hydroxide

Now that we have the moles of the alkali metal hydroxide, we can calculate its molar mass by dividing the given mass (4.36 g) by the moles of MOH (0.0425 mol). Molar mass of MOH = (4.36 g)/(0.0425 mol) = 102.588 g/mol

Step 4: Identify the alkali metal cation

We are given the potential cations as Li+, Na+, K+, Rb+, and Cs+. We can find the molar mass of the hydroxides for each of these cations: - For LiOH: Molar mass = 6.939 (Li) + 15.999 (O) + 1.008 (H) = 23.946 g/mol - For NaOH: Molar mass = 22.990 (Na) + 15.999 (O) + 1.008 (H) = 39.997 g/mol - For KOH: Molar mass = 39.098 (K) + 15.999 (O) + 1.008 (H) = 56.105 g/mol - For RbOH: Molar mass = 85.468 (Rb) + 15.999 (O) + 1.008 (H) = 102.475 g/mol - For CsOH: Molar mass = 132.905 (Cs) + 15.999 (O) + 1.008 (H) = 149.912 g/mol The closest molar mass to our calculated value of 102.588 g/mol is RbOH with a molar mass of 102.475 g/mol. Thus, the identity of the alkali metal cation is Rb+. The final answers are: (a) Molar mass of the metal hydroxide is 102.588 g/mol, and (b) The identity of the alkali metal cation is \(\mathrm{Rb}^{+}\).

Key Concepts

Moles Calculation

In the world of chemistry, understanding the concept of moles is crucial for analyzing reactions and finding equivalency points. When performing a titration, as in our exercise, calculating the moles of the titrant is an essential step. Here, we are given that 17.0 mL of a 2.50 M HCl solution is used to reach the equivalence point in the titration of an unknown alkali metal hydroxide. To find the moles of HCl, we use the simple formula:
- Moles of HCl = Volume (L) × Molarity (mol/L)
Converting 17.0 mL to liters, we get 0.017 L. Multiplying this with the molarity (2.50 mol/L), we find that 0.0425 mol of HCl was used. At the equivalence point in an acid-base titration, the moles of acid will equal the moles of the base it reacts with. Thus, for the unknown metal hydroxide, this means:

• Moles of MOH = Moles of HCl = 0.0425 mol

This concept ensures accuracy in knowing exactly how much of a substance is involved in the reaction.

Molar Mass Calculation

Once we know the number of moles of the alkali metal hydroxide, we can use that information to calculate its molar mass. The molar mass is a physical property that refers to the mass of a given substance (chemical element or chemical compound) divided by the amount of substance (moles). In our exercise, we have a sample weighing 4.36 g and we previously calculated that it contains 0.0425 mol. Thus, the molar mass can be calculated with the formula:
- Molar Mass = Mass (g) / Moles (mol)
For the alkali metal hydroxide, substituting the known values gives us:

• Molar Mass = 4.36 g / 0.0425 mol = 102.588 g/mol

This result is essential for identifying the compound, as each alkali metal hydroxide will have a characteristic molar mass. By comparing the calculated molar mass to known values of potential alkali metal hydroxides, we can determine the specific compound present in the sample.

Alkali Metal Hydroxide Identification

Identifying an unknown alkali metal hydroxide involves comparing the calculated molar mass to known values. In the exercise, we determined the molar mass to be approximately 102.588 g/mol. Different alkali metal cations form hydroxides with distinct molar masses. We have the following possibilities:

• LiOH: 23.946 g/mol
• NaOH: 39.997 g/mol
• KOH: 56.105 g/mol
• RbOH: 102.475 g/mol
• CsOH: 149.912 g/mol

Comparing these to our calculated value, the molar mass of RbOH is the closest match at 102.475 g/mol, confirming that RbOH, or rubidium hydroxide, is the identity of the unknown compound. This kind of analysis is a common task in chemistry, often used to either verify the composition of a sample or identify unknown compounds based on their properties. Understanding this process allows chemists to confidently determine the substances they are working with.