Problem 118 - Chemistry

Question

Analysis of a metal chloride \(\mathrm{XCl}_{3}\) show that it contains 67.2 percent \(\mathrm{Cl}\) by mass. Calculate the molar mass of \(\mathrm{X}\) and identify the element.

Step-by-Step Solution

Answer

The element X in the molecule \(XCl_3\) is Chromium (Cr) with a calculated molar mass of approximately 52.687 g/mol.

Step 1: Calculate the mass fraction of Chlorine and the unknown element in the compound

Given that the compound \(XCl_3\) contains 67.2 percent Chlorine by mass, it means the rest is the unknown element. Therefore the mass percent of X (the unknown element) is \(100\% - 67.2\% = 32.8\%\)

Step 2: Calculate the molar mass of Chlorine

The atomic mass of Chlorine (Cl) is 35.453 g/mol. In the molecule \(XCl_{3}\), we have 3 atoms of Chlorine, so the total mass contributed by Chlorine in this molecule is \(3 \times 35.453g/mol = 106.359 g/mol\)

Step 3: Calculate the molar mass of the unknown element

Given that the mass ratio is 32.8:67.2, the mass of X (unknown element) can be calculated from the mass of Cl derived in the step before in the following way: \((32.8/67.2) \times 106.359 g/mol = 52.687 g/mol\) This is the molar mass of the unknown element X.

Step 4: Identify the unknown element

Now, comparing this molar mass with the periodic table or a list of atomic masses, we find that the element with this molar mass closest to 52.687 g/mol is Chromium (Cr) - its atomic weight is 52.00 g/mol. So, the unknown element X in \(XCl_3\) is Chromium (Cr).

Key Concepts

Metal Chloride Analysis

When analyzing a compound like metal chloride, specifically with the chemical formula \( XCl_3 \), it's important to understand its composition. In this type of molecule, there is a metal element (\( X \)) bonded to three chlorine atoms. Analyzing such compounds typically involves determining the proportion of each type of atom in the compound by mass.

In the given exercise, we know the mass percentage of chlorine is 67.2%. Since the compound must account for 100% of the total mass, the unknown metal element makes up the remaining percentage. Therefore, the mass percent of the metal is simply calculated by subtracting the chlorine's mass percent from the total:

• 100% - 67.2% = 32.8%

By knowing these percentages and applying them to the solution, we can take a deeper dive into the component analysis of the compound.

Unknown Element Identification

To identify an unknown element in a compound like \( XCl_3 \), it's essential to infer its molar mass from the compound's overall makeup. Each component of the compound, in this case, chlorine and the unknown element \( X \), contributes to the total molar mass.

The given percentage of chlorine by mass helps to deduce the mass contribution from chlorine. Since we know chlorine's atomic mass (35.453 g/mol), and there are three atoms in the compound, the total chlorine mass is found by calculating:

• \(3 \times 35.453 \text{ g/mol} = 106.359 \text{ g/mol}\)

Using the percentage compositions, you can calculate the mass of the unknown element by applying cross-multiplication based on these mass ratios.

Once this molar mass is calculated for element \( X \), compare it against known values from the periodic table. In this instance, the closest molar mass to our calculated 52.687 g/mol is Chromium (Cr), with a known atomic mass of 52.00 g/mol. This process underscores how elemental identification involves recognizing these approximations and understanding periodic trends.

Elemental Composition Calculation

Elemental composition calculation is crucial when you want to determine the specific elements within a compound. Given mass percentages are used in conjunction with atomic masses to establish the molar proportions within a compound like \( XCl_3 \).

For example, once we establish the mass percent of chlorine (67.2%) and the unknown element (32.8%), we use these proportions to calculate individual molar masses. By knowing Chlorine's atomic mass (35.453 g/mol), the entire chlorine contribution in the compound is computed by multiplying by the number of chlorine atoms:

• \(3 \times 35.453 \text{ g/mol} = 106.359 \text{ g/mol}\)

With the unknown element, the mass can be found using its percentage ratio relative to chlorine's mass:

• \((32.8/67.2) \times 106.359 \text{ g/mol} = 52.687 \text{ g/mol}\)

This calculated molar mass then hints at the identity of element \( X \) as Chromium (Cr), whose atomic mass closely matches what was obtained. This calculation method exhibits how molecular formula, atomic masses, and percentage compositions converge to enable precise elemental determination within compounds.