Question

A certain metal \(\mathrm{M}\) forms a bromide containing 53.79 percent Br by mass. What is the chemical formula of the compound?

Short answer

The chemical formula of the compound is \( \text{ZnBr}_2 \).

Step-by-step solution

Understand the Problem

We have a metal bromide compound, which consists of a metal element M and bromine (Br), and we know that 53.79% of the compound's mass is from bromine. We need to find the chemical formula of this compound by determining the ratio of metal M to bromine atoms in the formula.

Determine the Mass Ratio of Bromine

Since Bromine makes up 53.79% of the compound, the remaining percentage must belong to the metal M. Calculate the percentage of the metal as follows: 100% - 53.79% = 46.21%. So, 46.21% of the compound is metal M.

Calculate Moles of Bromine

We assume 100 g of the compound for easier calculation. This means there are 53.79 grams of Br. The molar mass of Br (bromine) is approximately 79.9 g/mol. Calculate the moles of Br:\[\text{Moles of Br} = \frac{53.79\,g}{79.9\,g/mol} = 0.67345\,mol\approx 0.673\,mol\]

Calculate Moles of Metal M

Similarly, there are 46.21 grams of the metal M in 100 g of the compound. Let the atomic mass of metal M be \(x\) g/mol. Thus, the moles of metal M are:\[\text{Moles of M} = \frac{46.21}{x}\]

Establish the Ratio of M to Br

Assume the simplest whole number ratio of moles between M and Br is 1:1 (since there are no additional details). Thus:\[0.673 \approx \frac{46.21}{x}\]Solve for \(x\):\[x = \frac{46.21}{0.673} \approx 68.67\ g/mol\]

Identify the Metal

Look up the periodic table to find the element with an atomic mass close to 68.67 g/mol. Zinc (Zn) has an atomic mass of about 65.38 g/mol, which is the closest. This suggests that metal M is likely zinc (Zn).

Write the Chemical Formula

Given the atomic mass corresponds to zinc and the simplest ratio is 1:1, the chemical formula of the compound is \( \text{ZnBr}_2 \), as this satisfies the mass percentage and ratio requirements.

Key concepts

Mass Percentage

Mass percentage is a way to express how much of a particular element is present within a compound relative to the total mass of the compound. It is calculated by taking the mass of the element in question and dividing it by the total mass of the compound, then multiplying by 100 to convert to a percentage.

In the context of our problem, bromine (Br) accounts for 53.79% of the mass in the metal bromide compound. This percentage helps determine how the compound's mass is distributed between the metal and bromine.

Understanding mass percentage is crucial for breaking down the composition of compounds and eventually figuring out molecular or empirical formulas. Often, by knowing the mass percentage, you can surmise not only which elements are in a compound but also how they are arrayed in terms of quantity.

Molar Mass Calculation

Molar mass calculation refers to finding the mass of one mole of a substance. It is typically expressed in grams per mole (g/mol). This information is critical in chemistry for converting mass of a substance to the number of moles, which aids in solving stoichiometric problems.

In our scenario, we began calculating molar mass using the known amount of bromine and its molar mass of 79.9 g/mol. This allows us to compute the moles of bromine in the compound: 53.79 grams of Br converts to approximately 0.673 moles.

Calculating molar mass also involves determining the unknown element's mass by establishing it as a variable (denoted as \(x\)) and solving the equation derived from the known mass percentage and moles. This step is pivotal in finding not only the identity but also the stoichiometric makeup of unknown compounds.

Metal Identification

Identifying a metal in a compound when only partial information is available revolves around understanding its atomic mass. In the exercise, the metal's atomic mass is derived through calculations from given data points such as mass percentage and the formula weight of known elements like bromine.

From the calculated mass percentage and moles, the mass of metal \(M\) is computed as 46.21% of an assumed sample mass of 100 grams. Solving for the atomic mass using the relationship \(\frac{46.21}{x} = 0.673\), where \(x\) is the molar mass of the metal, gives us an estimated atomic mass of approximately 68.67 g/mol.

Cross-referencing with the periodic table helps isolate possible metals, leading us to identify the metal \(M\) as zinc (Zn), whose atomic mass closely matches our calculated value. Accurate metal identification relies heavily on this combination of problem-solving skills and known periodic elements.

Empirical Formula

The empirical formula is the simplest whole-number ratio of elements present in a compound, distilled to its most basic form without detailing repeated units or specific molecular configurations.

To determine the empirical formula from percentage composition, convert each element's mass into moles, then find the smallest whole-number ratio between them. In our exercise, converting the grams of bromine and the unknown metal to moles guided us to the formula.

For both elements in the compound, we calculated:

• Bromine moles: 0.673
• Metal moles: inferred from the same relationship since a 1:1 ratio is proposed

These values suggest a 1:1 ratio, matching our stoichiometric calculations leading us to propose the formula as \(\text{ZnBr}_2\), indicating two bromine atoms per zinc atom. This empirical composition satisfies all calculated and known constraints from the mass percentages and the stoichiometric patterns. The empirical formula helps clarify the fundamental composition, crucial for understanding chemical behavior in reactions.