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 theequivalence 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 alkalimetal cation: \(\mathrm{Li}^{+}, \mathrm{Na}^{+}, \mathrm{K}^{+}, \mathrm{Rb}^{+},\) or \(\mathrm{Cs}^{+} ?\)

Short answer

The molar mass of the unknown alkali metal hydroxide (MOH) is approximately 40 g/mol. Based on the comparison with the molar masses of the known alkali metal hydroxides, the identity of the alkali metal cation is Na⁺ (sodium).

Step-by-step solution

Find the moles of HCl

: We are given the concentration (2.50 M) and volume (17.0 mL) of the HCl solution. To find the moles of HCl, we can use the formula: moles = concentration × volume Since the volume is given in milliliters, we need to convert it to liters by dividing by 1000. \(moles_{HCl} = (2.50 \, \mathrm{M}) \times (17.0 \, \mathrm{mL} \times \frac{1 \, \mathrm{L}}{1000 \, \mathrm{mL}}) \)

Find the moles of MOH

: At the equivalence point, the moles of HCl and MOH are equal (1:1 ratio). Therefore, \(moles_{MOH} = moles_{HCl}\).

Calculate the molar mass of MOH

: We are given the mass of MOH (4.36 g), and we found the moles of MOH in step 2. To calculate the molar mass, use the formula: \(molar \, mass_{MOH} = \frac{mass_{MOH}}{moles_{MOH}}\). Substitute the known values in the formula and calculate the molar mass.

Identify the alkali metal cation

: Compare the calculated molar mass of MOH with the molar masses of known alkali metal hydroxides (LiOH, NaOH, KOH, RbOH, and CsOH). The cation with the closest molar mass to the calculated molar mass will be the identity of the alkali metal cation in the unknown hydroxide.

Key concepts

Molar Mass Calculation

Understanding the molar mass of a substance is crucial for various chemical calculations. Molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). It is the sum of the atomic masses of all atoms in a molecule. To calculate the molar mass, you will use the formula:
\[\begin{equation}\text{Molar Mass} = \frac{\text{Total Mass of Compound}}{\text{Number of Moles}}\text{.}\end{equation}\]In an acid-base titration scenario, after finding the number of moles of the titrant (in this case, HCl), you can then compute the molar mass of the unknown substance by dividing the mass of the unknown by the number of moles of the titrant, assuming a 1:1 ratio at the equivalence point.

When the substance is an unknown alkali metal hydroxide, the process includes the additional step of obtaining the substance's mass first, which then enables you to proceed with the calculation. It is important to convert the volume of solutions to liters and note that all substances involved in calculations are pure, without impurities, for accurate molar mass determination.

Alkali Metal Hydroxide Identification

In the context of acid-base titrations, identifying an unknown alkali metal hydroxide involves comparing the calculated molar mass of the metal hydroxide with the known molar masses of common alkali metal hydroxides: LiOH, NaOH, KOH, RbOH, and CsOH. Each alkali metal has a distinctive molar mass.

Once you have the molar mass from the titration data, you check which known alkali metal hydroxide's molar mass closely matches your calculated value. Careful measurement and calculation are vital in this step to avoid misidentification. An alkali metal hydroxide is composed of an alkali metal cation and the hydroxide anion, and their combined atomic/ionic masses give the molar mass of the hydroxide.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. In a titration, stoichiometry helps you understand the proportions in which reactants combine. In the case of acid-base titrations, the reaction typically occurs with a 1:1 mole ratio. Therefore, the moles of acid (HCl, in our example) will equal the moles of the base (MOH) at the equivalence point.

Understanding stoichiometry allows you to make predictions about the outcome of a reaction and calculate unknown concentrations, which is particularly important when working with solutions in a titration experiment. It provides the foundation for understanding the equivalence point and is especially significant in determining the concentration of an unknown solution through titration.

Equivalence Point

The equivalence point in a titration is the moment when the quantity of titrant added is stoichiometrically equivalent to the quantity of substance present in the sample. For acid-base titrations, this is when moles of acid equal moles of base. A pH indicator or a pH meter can signal when the equivalence point is reached through a change in color or a specific pH measurement, respectively.

The significance of the equivalence point extends beyond indicating when stoichiometric amounts of reactants have been mixed. It enables chemists to calculate unknown concentrations, as seen in the process of identifying an unknown alkali metal hydroxide. Correctly interpreting the equivalence point is critical for accurate titrimetric analysis and subsequent calculations, such as molar mass determination and substance identification.