Problem 137 - Chemistry The Science in Context

Question

You are given a \(0.6240 \mathrm{g}\) sample of a substance with the generic formula MCl \(_{2}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2} .\) After complete drying of the sample (which means removing the 2 mol of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{MCl}_{2}\) ), the sample has a mass of \(0.5471 \mathrm{g} .\) What is the identity of element M?

Step-by-Step Solution

Answer

Answer: The identity of element M in the given substance is gold (Au).

Step 1: Determine the mass of the water removed

To find the mass of the water removed, we can subtract the mass of the dried sample from the initial mass of the sample: Initial mass of the sample: 0.6240 g Mass of the dried sample: 0.5471 g Mass of the water removed: 0.6240 g - 0.5471 g = 0.0769 g

Step 2: Determine the moles of water removed

Since we know the mass of the water removed and the molar mass of water (H\(_2\)O) is 18.015 g/mol, we can find the number of moles of water removed: Moles of water removed = \(\frac{0.0769 \text{ g}}{18.015\text{ g/mol}} = 0.00427 \text{ mol}\)

Step 3: Determine the moles of MCl\(_2\) in the sample

The formula MCl\(_2\)(H\(_2\)O)\(_2\) indicates that there are two moles of water for each mole of MCl\(_2\). Thus, by dividing the moles of water removed by 2, we can determine the moles of MCl\(_2\) in the sample: Moles of MCl\(_2\) = \(\frac{0.00427 \text{ mol}}{2} = 0.002135 \text{ mol}\)

Step 4: Calculate the molar mass of MCl\(_2\)

To find the molar mass of MCl\(_2\), we can use the mass of the dried sample (which is the mass of MCl\(_2\)) and the moles of MCl\(_2\) that we have already determined: Molar mass of MCl\(_2\) = \(\frac{0.5471 \text{ g}}{0.002135\text{ mol}} = 256.34\text{ g/mol}\)

Step 5: Determine the molar mass of element M

We can calculate the molar mass of M in MCl\(_2\) by subtracting the molar mass of two moles of Cl from the molar mass of MCl\(_2\): Molar mass of 2 moles of Cl: 2 * (35.45 g/mol) = 70.90 g/mol Molar mass of M: 256.34 g/mol - 70.90 g/mol = 185.44 g/mol

Step 6: Identify the element M

Now that we have the molar mass of M (185.44 g/mol), let's compare it with the molar masses of elements in the periodic table. The element with a molar mass close to 185.44 g/mol is gold (Au) with a molar mass of 196.97 g/mol. Therefore, the identity of element M is gold (Au). The formula of the substance can be written as AuCl\(_2\) (H\(_2\)O)\(_2\).

Key Concepts

Moles Determination

To determine the number of moles of a substance, you need to know its mass and molar mass. In the given problem, you are tasked with finding the moles of water and later moles of MCl_{2} in the sample. Start by calculating the moles of water removed. You have 0.0769 grams of water, and the molar mass of water is 18.015 g/mol. The formula for calculating moles is:
\[\text{Moles of water} = \frac{\text{mass of water}}{\text{molar mass of water}} = \frac{0.0769\, \text{g}}{18.015\, \text{g/mol}}\]After you find the moles of water (0.00427 mol), use this information to find the moles of MCl_{2}. Because each formula unit of MCl_{2}(H_2O)_2 contains two moles of water, simply divide the moles of water by 2 to get the moles of MCl_{2}:
\[\text{Moles of MCl}_2 = \frac{0.00427\, \text{mol}}{2} = 0.002135\, \text{mol}\]This systematic approach helps you unravel the mysteries behind chemical samples by knowing how substances are composed.

Mass of the Dried Sample

In chemical analysis, it is often necessary to determine the mass of a dried sample, especially when the compound contains water of crystallization. In our exercise, a sample containing water is dried to calculate the mass of MCl_{2} after water is removed. First, we calculate the mass of the water lost:

• Initial mass of sample = 0.6240 g
• Mass of the dried sample = 0.5471 g

Subtract the mass of the dried sample from the initial mass to find the mass of the water:
\[\text{Mass of water removed} = 0.6240\, \text{g} - 0.5471\, \text{g} = 0.0769\, \text{g}\]Understanding the mass of the dried sample helps in executing precise chemical calculations and better understanding the nature of hydrated compounds.

Element Identification

Once you know the molar mass of a compound, it's possible to identify the unknown element by examining its composition. In our case, after calculating the molar mass of MCl_{2} as 256.34 g/mol, subtract the known mass of the chlorine components (2 * 35.45 g/mol = 70.90 g/mol) to find the molar mass of the unknown element M:
\[\text{Molar mass of M} = 256.34\, \text{g/mol} - 70.90\, \text{g/mol} = 185.44\, \text{g/mol}\]By comparing this calculated value with the periodic table, you determine that element M is gold (Au), which has a close molar mass of 196.97 g/mol. This step validates the identity of M in the compound, which can be written as AuCl_{2}(H_2O)_2.

• Make sure to check for close matches on the periodic table.
• Note that slight differences can occur due to rounding.

Chemical Compound Analysis

Chemical compound analysis involves determining the structure and composition of a chemical substance. Understanding the makeup of MCl _{2}(H_2O)_2 provides insight into its properties and potential applications. In the given exercise, we accounted for components such as chlorine and water, ultimately identifying the central element, gold (Au).
Compound analysis steps include:

• Determining the empirical formula of the compound based on the given data.
• Calculating the molar mass from the sample mass after drying.
• Deciphering the identity of each component part of the compound, such as calculating moles of individual elements.

This detailed analysis allows chemists to understand not just what the compound is made of, but also how its components interact. Mastery of these techniques can provide foundational knowledge necessary for more advanced chemical explorations.