Problem 85 - Chemistry The Central Science

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 \(2.50 \mathrm{M} \mathrm{HCl}(a q)\) 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 metal hydroxide is 102.6 g/mol. The identified alkali metal cation is K+ (potassium).

Step 1: Since the HCl solution is 2.50 M, we can use moles = concentration \(\times\) volume to find the number of moles of HCl used: \( moles_{HCl} = ( 2.50 \ mol/L ) \times ( 17.0 \times 10^{-3} L ) \) \( moles_{HCl} = 0.0425 \ mol \) #Step 2: Determine the number of moles of alkali metal hydroxide present #

At the equivalence point, the amount of acid and base will be equal: \( moles_{HCl} = moles_{MOH} \) \( moles_{MOH} = 0.0425 \ mol \) #Step 3: Calculate the molar mass of the unknown alkali metal hydroxide #

Step 2: Using the total mass and number of moles of the alkali metal hydroxide: \( molar\ mass_{MOH} = \frac{mass_{MOH}}{moles_{MOH}} \) \( molar\ mass_{MOH} = \frac{4.36\ g}{0.0425\ mol} \) \( molar\ mass_{MOH} = 102.6\ g/mol \) #Step 4: Identify the alkali metal cation #

To do this, subtract the molar mass of the hydroxide ion (OH-) from the molar mass of the entire metal hydroxide: \( molar\ mass_{M^{+}} = molar\ mass_{MOH} - molar\ mass_{OH^{-}} \) \( molar\ mass_{M^{+}} = 102.6\ g/mol - 17.0\ g/mol \) \( molar\ mass_{M^{+}} = 85.6\ g/mol \) By comparing the molar mass of the alkali metal cation (85.6 g/mol) with those of the given cations, we can conclude that the alkali metal cation is K+ (potassium), as its molar mass is approximately 39.1 g/mol.

Key Concepts

Acid-Base Titration

Acid-base titration is a laboratory method used to determine the concentration of an unknown acid or base by reacting it with a standard solution of the opposite type. The titration process involves adding a titrant (a solution of known concentration) from a burette to a solution of the unknown substance, which contains an indicator that changes color at a certain pH level. In our exercise, hydrochloric acid (HCl) was the titrant and the unknown alkali metal hydroxide (MOH) was the substance being analyzed. An indicator was used to visually signal when the reaction reached its end point, also known as the equivalence point. This method relies on the stoichiometric relationship between the acid and base, which in this case is a 1:1 molar ratio, allowing us to determine the number of moles of the alkali metal hydroxide based on the volume and molarity of HCl used.

Molar Mass Calculation

The molar mass of a substance is the mass in grams of one mole of that substance. In chemical reactions and titrations, calculating molar mass is an essential step for identifying an unknown compound or for converting between mass and moles of a substance. To calculate the molar mass of the alkali metal hydroxide in our exercise, the mass of the sample was divided by the moles of hydroxide. The total mass given was 4.36 g, and the moles of hydroxide, which are equivalent to the moles of HCl at the equivalence point, were found to be 0.0425 mol, leading us to find the molar mass of the hydroxide compound. Understanding molar mass calculation is crucial for any chemical analysis and is particularly important when identifying unknown samples, as demonstrated in the exercise.

Equivalence Point

The equivalence point in a titration is the moment when the number of moles of titrant added equals the number of moles of substance present in the solution. It is a critical juncture where the chemical reaction between the acid and the base is complete. Determining the equivalence point accurately is vital since it allows chemists to calculate the concentration of the unknown solution. In our case, the equivalence point was identified by the color change of the indicator introduced into the mixture of the unknown hydroxide and the HCl. The volume of the acid used at this point was 17.0 mL, which gave us enough information to calculate the number of moles of both the acid and the base, assuming a 1:1 stoichiometric ratio in the reaction. This concept is fundamental in understanding how to execute a titration and interpret its results.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It is based on the conservation of mass and the concept of moles, allowing chemists to predict the outcomes of reactions and to calculate various quantities of interest. In the context of our exercise, stoichiometry was applied to connect the volume and molarity of the HCl titrant to the number of moles of HCl, and by extension, to the number of moles of alkali metal hydroxide. The stoichiometric ratio between HCl and MOH was 1:1, indicating that for every mole of HCl used, one mole of MOH reacted. This straightforward relationship is the essence of stoichiometry, enabling the calculation of the molar mass of the unknown hydroxide and subsequently identifying the metal cation involved.