Problem 116 - Chemistry and Chemical Reactivity

Question

A Calcium and magnesium carbonates occur together in the mineral dolomite. Suppose you heat a sample of the mineral to obtain the oxides, \(\mathrm{CaO}\) and \(\mathrm{MgO}\), and then treat the oxide sample with hydrochloric acid. If \(7.695 \mathrm{g}\) of the oxide sample requires \(125 \mathrm{mL}\) of \(2.55 \mathrm{M} \mathrm{HCl}\) \(\mathrm{CaO}(\mathrm{s})+2 \mathrm{HCl}(\mathrm{aq}) \longrightarrow \mathrm{CaCl}_{2}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)\) \(\mathrm{MgO}(\mathrm{s})+2 \mathrm{HCl}(\mathrm{aq}) \longrightarrow \mathrm{MgCl}_{2}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)\) What is the weight percent of each oxide (CaO and \(\mathrm{MgO}\) ) in the sample?

Step-by-Step Solution

Answer

The weight percent of CaO is 54.7% and the weight percent of MgO is 45.3%.

Step 1: Calculate Moles of HCl Used

To find the moles of HCl, use the molarity (M) and volume (L) of HCl: \[ \text{Moles of HCl} = 0.125 \text{ L} \times 2.55 \text{ M} = 0.31875 \text{ moles} \]

Step 2: Determine Moles of Each Oxide Reacted

According to the reactions, each mole of \[\mathrm{CaO}\]and\[\mathrm{MgO}\]reacts with 2 moles of HCl:Let the moles of \[\mathrm{CaO}\] be \[x\] and the moles of \[\mathrm{MgO}\] be \[y\].Then, \[ \text{x + y} = 0.31875 / 2 = 0.159375 \text{ moles} \] due to the 1:2 ratio in both equations.

Step 3: Set Up Mass Equations

Given the total mass of oxides is 7.695 g, relate moles to mass:- Molar mass of \(\mathrm{CaO} = 56.08 \text{ g/mol}\)- Molar mass of \(\mathrm{MgO} = 40.30 \text{ g/mol}\)Set up the equation: \[56.08x + 40.30y = 7.695 \text{ g} \] where \(x\) is moles of \(\mathrm{CaO}\) and \(y\) is moles of \(\mathrm{MgO}\).

Step 4: Solve Simultaneous Equations

Solve the equations:1. \(x + y = 0.159375\)2. \(56.08x + 40.30y = 7.695\)Using substitution or elimination methods to find:\[x = 0.075 \text{ moles (CaO)} \] and \[y = 0.084375 \text{ moles (MgO)}\].

Step 5: Calculate Mass of Each Oxide

Use molar masses to find the mass of each oxide:\[ \text{Mass of } \mathrm{CaO} = 0.075 \times 56.08 = 4.206 \text{ g} \]\[ \text{Mass of } \mathrm{MgO} = 0.084375 \times 40.30 = 3.489 \text{ g} \]

Step 6: Calculate Weight Percent

Find the weight percent of each component:\[ \text{Weight percent of } \mathrm{CaO} = \left(\frac{4.206}{7.695}\right) \times 100\% = 54.7\% \]\[ \text{Weight percent of } \mathrm{MgO} = \left(\frac{3.489}{7.695}\right) \times 100\% = 45.3\% \]

Key Concepts

Weight Percent Calculation

Weight percent helps us understand the composition of a mixture by expressing the concentration of a component as a percentage of the total mass. Here's how you can break down this process:

• To calculate the weight percent, first determine the exact mass of the component of interest.
• Next, divide the mass of the individual component by the total mass of the mixture or sample.
• Finally, multiply the result by 100 to convert it into a percentage.

For instance, if we have asked to find the weight percent of CaO in dolomite, we would divide the mass of CaO by the combined mass of CaO and MgO, then multiply by 100 as follows: \[\text{Weight percent of } \mathrm{CaO} = \left(\frac{\text{mass of CaO}}{\text{total mass of sample}}\right) \times 100\%\]
Understanding weight percent is fundamental in chemistry for quantifying the distribution of components within a compound.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical equation. It allows us to predict how much of each substance is involved in a reaction.

In an ideal chemical reaction, stoichiometry assumes that:

• The reaction goes to completion, meaning all reactants are transformed into products.
• Moles of substances combine according to the ratios given by the balanced chemical equation.

In the context of dolomite mineral analysis, when CaO and MgO react with hydrochloric acid (HCl), each oxide reacts with HCl in a 1:2 molar ratio. This ratio helps us calculate the number of moles of oxide that reacted, based on the amount of HCl used (as shown by the equation):
- CaO + 2HCl → CaCl₂ + H₂O
- MgO + 2HCl → MgCl₂ + H₂O

From this stoichiometric relationship, you can set up equations and determine how the reactants are consumed, leading to determining the masses needed for weight percent calculations.

Molar Mass

Molar mass is a crucial concept in chemistry that allows for the conversion between mass and moles of a substance. It is the mass of one mole of a chemical element or compound, typically expressed in grams per mole (g/mol).

Knowing the molar mass is essential for:

• Converting the mass of a substance to moles, using the formula: \[\text{Number of moles} = \frac{\text{mass}}{\text{molar mass}}\]
• Determining the mass composition of compounds in mixtures.
• Calculating the individual masses of elements in a compound.

For example, in this dolomite analysis, the molar masses were:

• CaO, with a molar mass of 56.08 g/mol.
• MgO, with a molar mass of 40.30 g/mol.

Using these values, you can easily compute the exact amount of oxides formed in reactions and their roles in the total composition of the sample given the stoichiometry.