Problem 50 - Chemistry The Central Science In Si Units

Question

The compound \(\mathrm{XCl}_{4}\) contains \(75.0 \% \mathrm{Cl}\) by mass. What is the element X?

Step-by-Step Solution

Answer

The element X is titanium (Ti).

Step 1: Define Variables

Let the atomic mass of chlorine (Cl) be denoted as \(M_{Cl} = 35.5\, \text{g/mol}\). Suppose the atomic mass of element X is \(M_X\, \text{g/mol}\). The molar mass of the compound \(\text{XCl}_4\) is \(M_X + 4 \times M_{Cl}\).

Step 2: Express Mass Percentage of Chlorine

Use the given mass percentage of chlorine in the compound, which is 75%. Therefore, the mass of chlorine in \(\text{XCl}_4\) is \((4 \times M_{Cl})\) grams and the expression for the mass percentage is: \[ \frac{4 \times M_{Cl}}{M_X + 4 \times M_{Cl}} = 0.75 \]

Step 3: Substitute Known Values

Substitute \(M_{Cl} = 35.5\) into the mass percentage equation: \[ \frac{4 \times 35.5}{M_X + 4 \times 35.5} = 0.75 \]

Step 4: Solve for X's Atomic Mass

Simplify the equation to solve for \(M_X\): \[ 4 \times 35.5 = 0.75(M_X + 4 \times 35.5) \]\[ 142 = 0.75(M_X + 142) \]Eliminate decimals by isolating \(M_X\):\[ 142 = 0.75M_X + 106.5 \]Subtract 106.5 from both sides:\[ 35.5 = 0.75M_X \]Divide by 0.75 to find \(M_X\):\[ M_X = \frac{35.5}{0.75} = 47.33 \]

Step 5: Determine Element X

Round 47.33 to the nearest whole number, which gives 47. Refer to the periodic table to identify the element with an atomic mass of approximately 47. This element is titanium (Ti).

Key Concepts

Atomic Mass

Atomic mass is a fundamental concept in chemistry that refers to the mass of an individual atom of an element. It's often expressed in atomic mass units (amu) or grams per mole (g/mol). The atomic mass is essentially the sum of the masses of the protons and neutrons in the nucleus.
Each element on the periodic table has a characteristic atomic mass. For example:

• Hydrogen's atomic mass is approximately 1 amu.
• Chlorine's atomic mass is approximately 35.5 amu.

The atomic mass helps chemists determine the mass of a certain number of atoms, or the molar mass of compounds. It's crucial for calculating and comparing the proportions of elements within compounds, as we explored in the problem above.

Mass Percentage

Mass percentage is a way to express the concentration of a component in a mixture or a compound. It tells us what fraction of the total mass is attributable to a specific element. Here is how mass percentage is calculated:

• Mass percentage is defined as: \( \text{Mass Percentage} = \frac{\text{Mass of Element in Compound}}{\text{Total Mass of Compound}} \times 100 \% \)

Mass percentage is an important metric when analyzing chemical compositions. In our example, chlorine constitutes 75% of the mass in \( \text{XCl}_4 \). This helps us determine the mass of other elements in the compound, like element X.
Once you determine the mass percentages, you can also backtrack to find out the empirical formula of the compound. This is essential in chemistry to reveal the makeup of unknown substances.

Molar Mass

Molar mass is the weight of one mole (approximately 6.022 x \(10^{23}\) particles) of a given substance. It's expressed in grams per mole (g/mol). Molar mass is incredibly useful for converting between the mass of a substance and the number of moles, which is crucial for stoichiometric calculations in chemical reactions.
For elements, the molar mass is numerically equal to the atomic mass shown on the periodic table. For compounds, it's the sum of the atomic masses of all the atoms in the molecule.
Using our compound \( \text{XCl}_4 \) as an example:

• The molar mass can be calculated by adding together the molar mass of 'X' and four times the molar mass of Cl.
• This aids in determining composition and reacting masses.

Understanding molar mass allows you to predict how substances will react and how much of each reactant will be required or produced in reactions.

Periodic Table

The periodic table is a comprehensive chart that organizes all known chemical elements. It's laid out in order of increasing atomic number, which represents the number of protons in the nucleus of each element's atom. This logical framework is vital in predicting elemental properties.
The periodic table's structure allows chemists to infer relationships between elements:

• Elements in the same column (group) generally exhibit similar chemical behaviors.
• Atomic mass increases across periods from left to right.
• The later sections of each row contain metals, while non-metals populate the table's central areas.

For tasks like the one in our example, the periodic table was used to identify the unknown element X based on its atomic mass. Finding the atomic mass closest to an element on the table helped us identify it as titanium (Ti). The periodic table remains an indispensable tool in solving chemistry problems, understanding elemental properties, and organizing chemical information efficiently.