Question

A solution of scandium chloride was treated with silver nitrate. The chlorine in the scandium compound was converted to silver chloride, AgCl. A 58.9-mg sample of scandium chloride gave \(167.4 \mathrm{mg}\) of silver chloride. What are the mass percentages of \(\mathrm{Sc}\) and \(\mathrm{Cl}\) in scandium chloride? What is its empirical formula?

Short answer

The empirical formula is ScCl3, with 29.7% Sc and 70.3% Cl.

Step-by-step solution

Determine Moles of AgCl Formed

First, calculate the moles of AgCl formed from the given mass of silver chloride. Use the molar mass of AgCl \( \text{AgCl} \ = \ \text{Ag}\ (107.87 \ \text{g/mol}) + \text{Cl}\ (35.45 \ \text{g/mol}) = 143.32 \ \text{g/mol} \).\ Moles of AgCl = \( \frac{167.4 \ \text{mg}}{143.32 \ \text{mg/mol}} \) = \(1.168 \times 10^{-3} \ \text{mol} \).

Calculate Moles of Cl

The amount of chlorine in the silver chloride is equivalent to the moles of AgCl, as there is 1 Cl atom per AgCl molecule. Therefore, the moles of Cl is also \(1.168 \times 10^{-3} \ \text{mol} \).

Find Mass of Cl in Scandium Chloride

To find the mass of chlorine, use the calculated moles and the molar mass of chlorine:\[ \text{Mass of Cl} = 1.168 \times 10^{-3} \ \text{mol} \times 35.45 \ \text{g/mol} = 0.0414 \ \text{g} = 41.4 \ \text{mg} \].

Calculate Mass of Sc in Scandium Chloride

Subtract the mass of chlorine from the mass of the initial sample of scandium chloride to find the mass of scandium: \[ \text{Mass of Sc} = 58.9 \ \text{mg} - 41.4 \ \text{mg} = 17.5 \ \text{mg} \].

Calculate Mass Percentages of Sc and Cl

Find the mass percentages by dividing the mass of each element by the total mass of the sample and converting to percentages:\[ \text{Mass Percent of Sc} = \frac{17.5 \ \text{mg}}{58.9 \ \text{mg}} \times 100 \approx 29.7\% \]\[ \text{Mass Percent of Cl} = \frac{41.4 \ \text{mg}}{58.9 \ \text{mg}} \times 100 \approx 70.3\% \].

Determine Empirical Formula

Using the moles calculated:- Moles of Sc: \( \frac{17.5 \ \text{mg}}{44.96 \ \text{g/mol}} = 3.89 \times 10^{-4} \ \text{mol} \). - Moles of Cl: \(1.168 \times 10^{-3} \ \text{mol} \).Calculate the ratio: - \( \frac{\text{Moles of Cl}}{\text{Moles of Sc}} = \frac{1.168 \times 10^{-3}}{3.89 \times 10^{-4}} = 3\).Thus, the empirical formula is \( \text{ScCl}_3 \).

Key concepts

Mass Percentage

Mass percentage is a way to express the concentration of an element in a compound. To calculate the mass percentage of an element, you compare the mass of the element to the total mass of the compound, then convert this ratio into a percentage by multiplying by 100.
For example, in the exercise, we have scandium chloride. We determined the mass of chlorine (Cl) and scandium (Sc) separately from our calculations. Here's how you calculate the mass percentages:

• For chlorine (Cl), divide the mass of Cl by the total mass of scandium chloride, then multiply by 100.
  In our solution, it was: \( \text{Mass Percent of Cl} = \frac{41.4\, \text{mg}}{58.9\, \text{mg}} \times 100 \approx 70.3\% \).
• For scandium (Sc), do the same: \( \text{Mass Percent of Sc} = \frac{17.5\, \text{mg}}{58.9\, \text{mg}} \times 100 \approx 29.7\% \).

This helps you understand how much of each element is present in the compound, relative to its total mass.

Moles Calculation

Calculating moles is an essential part of stoichiometric calculations in chemistry. In simple terms, a mole is a quantity that represents a specific number of molecules or atoms, which is Avogadro's number \(6.022 \times 10^{23}\). It helps chemists convert between the mass of a substance and the number of atoms or molecules.
In the exercise, we started by determining the moles of silver chloride (AgCl) formed from its given mass. This is done by using the molar mass of AgCl. Here's how you calculate it:

• First, identify the molar mass by adding the atomic masses of silver (Ag) and chlorine (Cl):
  \( \text{AgCl} = 107.87\, \text{g/mol} + 35.45\, \text{g/mol} = 143.32\, \text{g/mol} \).
• Next, calculate the moles using the formula: \( \text{Moles of AgCl} = \frac{167.4\, \text{mg}}{143.32\, \text{mg/mol}} = 1.168 \times 10^{-3}\, \text{mol} \).

The same calculation principle applies to find the moles of chlorine (Cl) in AgCl. Since each AgCl contains one Cl atom, the moles of Cl are equal to the moles of AgCl. Understanding moles helps in balancing chemical equations and determining quantities involved in chemical reactions.

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a given substance. It is expressed in g/mol and is crucial for converting between the mass of a substance and the moles, which represent a count of molecules or atoms.
To determine the molar mass, look at the periodic table and sum up the atomic masses of all atoms present in a molecule. Here's how it was applied in our silver chloride (AgCl) example:

• The molar mass of AgCl is found by adding the atomic mass of silver (Ag) with chlorine (Cl):
  Ag has an atomic mass of 107.87 g/mol, and Cl has an atomic mass of 35.45 g/mol.
• Therefore, the molar mass of AgCl is: \( 107.87\, \text{g/mol} + 35.45\, \text{g/mol} = 143.32\, \text{g/mol} \).

This value is vital in calculations where you convert mass to moles, allowing you to use the proportions of a chemical reaction accurately. A deep understanding of molar mass enables accurate stoichiometry in all chemical calculations.