Problem 65 - Chemistry and Chemical Reactivity

Question

In an experiment \(1.056 \mathrm{g}\) of a metal carbonate, containing an unknown metal \(\mathrm{M}\), is heated to give the metal oxide and \(0.376 \mathrm{g} \mathrm{CO}_{2}\) $$\mathrm{MCO}_{3}(\mathrm{s})+\text { heat } \longrightarrow \mathrm{MO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g})$$ What is the identity of the metal M? (a) \(\mathrm{M}=\mathrm{Ni}\) (c) \(\mathbf{M}=\mathbf{Z} \mathbf{n}\) (b) \(\mathrm{M}=\mathrm{Cu}\) (d) \(\mathrm{M}=\mathrm{Ba}\)

Step-by-Step Solution

Answer

M is Copper (Cu).

Step 1: Determine the Molar Mass of CO2

First, calculate the molar mass of carbon dioxide \(\mathrm{CO}_2\). The atomic mass of carbon \(\mathrm{C}\) is approximately \(12.01\, \mathrm{g/mol}\), and for oxygen \(\mathrm{O}\), it is \(16.00\, \mathrm{g/mol}\). Therefore, the molar mass of \(\mathrm{CO}_2\) is approximately \(12.01 + 2 \times 16.00 = 44.01\, \mathrm{g/mol}\).

Step 2: Calculate Moles of CO2

Using the mass of \(\mathrm{CO}_2\) produced, which is \(0.376\, \mathrm{g}\), we can calculate the moles of \(\mathrm{CO}_2\) using the formula: \(\text{Moles of } \mathrm{CO}_2 = \frac{0.376}{44.01} \approx 0.00855\, \mathrm{mol}\).

Step 3: Apply Stoichiometry of the Decomposition Reaction

The balanced reaction \(\mathrm{MCO}_{3} \rightarrow \mathrm{MO} + \mathrm{CO}_{2}\) indicates a 1:1 molar ratio between \(\mathrm{MCO}_{3}\) and \(\mathrm{CO}_2\). Thus, the moles of \(\mathrm{MCO}_{3}\) are also \(0.00855\, \mathrm{mol}\).

Step 4: Calculate Molar Mass of MCO3

Given that \(1.056\, \mathrm{g}\) of \(\mathrm{MCO}_{3}\) produces \(0.00855\, \mathrm{mol}\), the molar mass of \(\mathrm{MCO}_3\) can be calculated as: \(\text{Molar mass of } \mathrm{MCO}_3 = \frac{1.056}{0.00855} \approx 123.68\, \mathrm{g/mol}\).

Step 5: Determine Molar Mass of M

We know the components of \(\mathrm{MCO}_3\) include \(\mathrm{M}\), \(\mathrm{C}\), and \(\mathrm{3O}\). Totaling the known weights: \(\mathrm{C} = 12.01\, \mathrm{g/mol}\) and \(3\mathrm{O} = 3 \times 16.00 = 48.00\, \mathrm{g/mol}\). Thus, the molar mass of \(\mathrm{M}\) is \(123.68 - 12.01 - 48.00 = 63.67\, \mathrm{g/mol}\).

Step 6: Identify the Metal M

Compare the calculated molar mass of \(\mathrm{M}\) to known atomic masses: Nickel (Ni) has a molar mass of about \(58.69\, \mathrm{g/mol}\), Copper (Cu) is \(63.55\, \mathrm{g/mol}\), Zinc (Zn) is \(65.38\, \mathrm{g/mol}\), and Barium (Ba) is \(137.33\, \mathrm{g/mol}\). The closest match is Copper (Cu) with \(63.55\, \mathrm{g/mol}\).

Key Concepts

Molar Mass Calculation

Molar mass calculation is an essential skill in stoichiometry and helps us understand how much of each element is present in a compound or reaction. At its core, the molar mass of a compound reflects the sum of the atomic masses of all atoms in its molecular formula.
To calculate the molar mass, we sum up the masses of each element or group of elements involved:

• First, identify each element in the compound.
• Then, multiply the atomic mass of each element by the number of times it appears in the formula.
• Finally, add these values to obtain the total molar mass.

The molar mass is crucial for converting between grams and moles, which is a standard practice in chemistry to relate mass to the amount of substance present. For example, in the calculation of carbon dioxide's molar mass in this exercise, we combined the atomic masses of carbon (12.01 g/mol) and oxygen (16.00 g/mol) multiplied by its subscript, leading us to the molar mass of CO₂ being approximately 44.01 g/mol.
This calculation helps bridge the gap between the microscopic world of atoms and the macroscopic world of grams, making chemistry tangible.

Metal Carbonates

Metal carbonates, such as the one examined in this problem, are compounds that contain a metal cation and the carbonate anion \(\mathrm{CO}_3^{2-}\). When heated, these compounds typically decompose into metal oxides and carbon dioxide gas.
The general reaction of metal carbonates decomposing can be written as:\[\mathrm{MCO}_{3}(\text{s}) \rightarrow \mathrm{MO}(\text{s}) + \mathrm{CO}_{2}(\text{g})\]

• "M" represents the metal ion in the compound.
• This decomposition is an example of a thermal decomposition reaction.
• The metal carbonate loses carbon dioxide, which is released as a gas, leaving a solid metal oxide behind.

Understanding metal carbonates and their reactions is essential for applications in geology, environmental science, and materials science, where such reactions are prevalent.
In the exercise, identifying the unknown metal involves using the stoichiometry of the reaction and the mass of the components to deduce the metal's identity through its molar mass.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products, often with a change in physical properties such as mass, color, or gas evolution.
The stoichiometric principles applied in these reactions allow us to predict amounts of products and reactants consumed and formed. In this example, the decomposition of a metal carbonate into a metal oxide and carbon dioxide illustrates the process clearly.
Key points about chemical reactions:

• The law of conservation of mass dictates that the mass of reactants equals the mass of products in a chemical reaction.
• Balanced chemical equations represent the stoichiometry of reactions, showing the proportions in which substances react and are produced.
• In a decomposition reaction like the one here, a single compound breaks down into two or more simpler substances.

The balanced equation provided:\[\mathrm{MCO}_{3} \rightarrow \mathrm{MO} + \mathrm{CO}_{2}\]indicates a one-to-one molar conversion, which simplifies identifying the amount of metal carbonate decomposed and linking it to the unknown metal. By understanding these aspects, we can better determine the outcomes of reactions and apply this in industrial and laboratory settings.