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 alkali metal hydroxide is approximately 102.59 g/mol. From the calculated molar mass, the identity of the alkali metal cation is Rb+ (rubidium).

Step 1: Calculate moles of hydrochloric acid

First, we need to calculate the moles of HCl added to the solution. To do this, we will use the formula: moles of HCl = concentration of HCl × volume of HCl The volume of HCl must be converted to liters: \(17.0\,\mathrm{mL}\times \frac{1\,\mathrm{L}}{1000\,\mathrm{mL}} = 0.017\,\mathrm{L}\) Now, we can calculate the moles of HCl: \(moles\,of\,HCl=2.50\,\mathrm{M}\times 0.017\,\mathrm{L}=0.0425\,\mathrm{mol}\)

Step 2: Calculate moles of alkali metal hydroxide

Since we have reached the equivalence point, we know that the moles of H+ ions are equal to the moles of OH- ions. The balanced chemical equation for the reaction between the metal hydroxide (M(OH)x) and hydrochloric acid (HCl) is: \(M(OH)_x + x\,HCl \rightarrow M^{(x+)} + x\,Cl^- + x\,H_2O\) From the balanced equation, we can see that x moles of HCl react with 1 mole of M(OH)x. Therefore, we can calculate the moles of alkali metal hydroxide (M(OH)x) using stoichiometry: moles of M(OH)x = moles of HCl ÷ x Since the alkali metal hydroxide can be LiOH, NaOH, KOH, RbOH, or CsOH, x can be 1. moles of M(OH)x = 0.0425 ÷ 1 = 0.0425 mol

Step 3: Calculate molar mass of the metal hydroxide

Now that we have the moles of the alkali metal hydroxide, we can find its molar mass by using the formula: molar mass of M(OH)x = mass of M(OH)x ÷ moles of M(OH)x molar mass of M(OH)x = 4.36 g ÷ 0.0425 mol = 102.59 g/mol

Step 4: Identify the alkali metal cation

The molar mass of the metal hydroxide is approximately 102.59 g/mol. Comparing this value to the molar masses of the alkali metal cations: - LiOH: 23.95 g/mol - NaOH: 39.99 g/mol - KOH: 56.11 g/mol - RbOH: 102.48 g/mol - CsOH: 135.93 g/mol We can conclude that the identity of the alkali metal cation is Rb+ (rubidium) since the molar mass of RbOH is closest to the calculated molar mass of our unknown metal hydroxide.

Key Concepts

Stoichiometry

Stoichiometry is a critical concept in chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It serves as the mathematical method behind the recipe for reactions. When applied to an acid-base titration, stoichiometry allows us to determine the amount of acid necessary to neutralize a base, or vice versa.

Understanding the mole-to-mole ratios, evident in the balanced chemical equations, is essential for solving titration problems. In our example, the stoichiometric relationship between HCl and the alkali metal hydroxide is 1:1, as indicated by the reaction's coefficients. By knowing the amount of HCl used to reach the equivalence point, we can determine the amount of metal hydroxide originally present in the solution. This information can prove invaluable in making calculations related to the molar mass and in identifying the unknown alkali metal.

Molar Mass

Molar mass, the weight of one mole of a substance, is a key concept in chemical calculations, especially when dealing with reactions and titrations. It represents the mass in grams of 1 mole of atoms, molecules, or ions of a substance and is expressed in grams per mole (g/mol).

For an unknown compound, such as the alkali metal hydroxide in our exercise, molar mass is calculated by dividing the mass of the sample by the number of moles of the substance. The significance of molar mass is pronounced when identifying substances in a titration. By comparing the calculated molar mass with known molar masses of potential candidates, one can deduce the identity of the unknown substance, as we did to identify the unknown alkali metal hydroxide as RbOH.

Equivalence Point

The equivalence point in a titration is the exact point at which the amount of titrant added is stoichiometrically equivalent to the amount of substance in the sample being titrated. In the case of an acid-base titration, this occurs when the number of moles of hydrogen ions (H+) equals the number of moles of hydroxide ions (OH-).

Indicators are typically used to signal the approach to the equivalence point by changes in color. In a calculation scenario, knowing the volume and concentration of the titrant (HCl in our case) allows us to calculate the exact moles of titrant used to reach the equivalence point. These moles are then equivalent to the moles of the substance being titrated, highlighting the close interrelationship between stoichiometry, molar mass, and the equivalence point in titrations.

Alkali Metal Hydroxides

Alkali metal hydroxides are a group of chemical compounds formed by alkali metals, such as lithium (Li), sodium (Na), potassium (K), rubidium (Rb), and cesium (Cs), combined with hydroxide ions (OH-). These bases are highly soluble in water and are characterized by their strong alkalinity and reactivity.

One mole of any alkali metal hydroxide will typically neutralize one mole of a strong acid, like HCl, forming water and an alkali metal chloride as the products of the reaction. Knowing the properties of these hydroxides aids in understanding their behavior in acid-base titrations. In practical applications, they're frequently involved in neutralization reactions, which can be harnessed for identifying the cation in an unknown hydroxide sample through calculating its molar mass and comparing it to standard values, just as was demonstrated in solving our textbook problem.