Problem 88 - Chemistry and Chemical Reactivity

Question

Group 2 A metal carbonates are decomposed to the metal oxide and \(\mathrm{CO}_{2}\) on heating: $$ \mathrm{MCO}_{3}(\mathrm{s}) \longrightarrow \mathrm{MO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g}) $$ You heat \(0.158 \mathrm{g}\) of a white, solid carbonate of a Group \(2 \mathrm{A}\) metal (M) and find that the evolved \(\mathrm{CO}_{2}\) has a pressure of \(69.8 \mathrm{mm} \mathrm{Hg}\) in a \(285-\mathrm{mL}\). Hask at \(25^{\circ} \mathrm{C} .\) Identify \(\mathrm{M}\).

Step-by-Step Solution

Answer

The metal is Strontium (Sr).

Step 1: Write Down the Balanced Equation

The decomposition of a Group 2A metal carbonate (MCO_3) results in the formation of the metal oxide (MO) and carbon dioxide (CO_2) gas: \[ \mathrm{MCO}_{3}(\mathrm{s}) \longrightarrow \mathrm{MO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g}) \] This equation will help us understand the stoichiometry of the reaction.

Step 2: Convert Pressure to Atmospheres

We need to convert the pressure of CO_2 from mm Hg to atmospheres using the conversion factor (1 atm = 760 mm Hg). Thus, \( P = \frac{69.8 \text{ mm Hg}}{760 \text{ mm Hg/atm}} \approx 0.09184 \text{ atm} \).

Step 3: Convert Temperature to Kelvin

To use the ideal gas law, convert the temperature from Celsius to Kelvin. The formula is: \[ T = 25^\circ C + 273.15 = 298.15 \text{ K} \].

Step 4: Calculate Moles of CO_2 using Ideal Gas Law

Apply the ideal gas law \( PV = nRT \) where R is the ideal gas constant 0.0821 L atm/mol K. Solve for n (moles of CO_2): \[ n = \frac{PV}{RT} = \frac{(0.09184 \text{ atm})(0.285 \text{ L})}{(0.0821 \text{ L atm/mol K})(298.15 \text{ K})} \approx 0.00107 \text{ mol} \].

Step 5: Calculate Molar Mass of MCO_3

The reaction shows a 1:1 molar ratio between MCO_3 and CO_2. Thus, moles of MCO_3 also equal 0.00107 mol. Calculate molar mass: \[ \text{Molar Mass of MCO}_3 = \frac{0.158 \text{ g}}{0.00107 \text{ mol}} \approx 147.66 \text{ g/mol} \].

Step 6: Identify the Group 2A Metal, M

Subtract the molar mass of CO_3 (roughly 60 g/mol) from the calculated molar mass of MCO_3 to get the molar mass of M. \[ \text{Molar Mass of M} = 147.66 \text{ g/mol} - 60 \text{ g/mol} \approx 87.66 \text{ g/mol} \] This molar mass is closest to the molar mass of Strontium (Sr), which is 87.62 g/mol.

Key Concepts

Chemical Decomposition

Chemical decomposition is a type of chemical reaction where a single compound breaks down into two or more simpler substances. This process is often initiated by the addition of heat and is seen commonly in reactions like the decomposition of metal carbonates. In this exercise, a Group 2A metal carbonate breaks down into a metal oxide and carbon dioxide gas:

• The general formula for this decomposition reaction is: \(\mathrm{MCO}_{3} \rightarrow \mathrm{MO} + \mathrm{CO}_{2}\)
• "M" denotes a Group 2A metal (such as Mg, Ca, Sr, or Ba), leading to different metal oxides and carbon dioxide.

This reaction is significant because it shows the stoichiometric relationship between the metal carbonate, metal oxide, and carbon dioxide. It is useful in stoichiometry problems as it helps calculate the quantities of the substances involved, such as how much carbon dioxide is produced from a given mass of the metal carbonate.

Ideal Gas Law

The Ideal Gas Law is a powerful equation used to relate different properties of a gas. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure in atmospheres
• \( V \) is the volume in liters
• \( n \) is the number of moles
• \( R \) is the ideal gas constant (0.0821 L atm/mol K)
• and \( T \) is the temperature in Kelvin.

When dealing with gases, converting measurements properly is crucial. For instance, we convert the pressure in mm Hg to atmospheres with the conversion factor of 1 atm = 760 mm Hg. Also, the temperature must be converted from Celsius to Kelvin to fit the requirements of the Ideal Gas Law. In this exercise, using these conversions and the Ideal Gas Law formula allowed us to calculate the number of moles of carbon dioxide released upon the decomposition of the metal carbonate.

Molar Mass Calculation

Molar mass is a fundamental concept used to find the mass of a substance per mole. It is especially important when dealing with chemical equations because it allows you to convert between the mass of a substance and the amount in moles, making stoichiometry calculations possible. In the given exercise, we calculated the molar mass of the Group 2A metal carbonate \( \mathrm{MCO}_3 \) by using the relationship: \[ \text{Molar Mass} = \frac{\text{mass}}{\text{moles}} \]Here's how it was done:

• The measured mass of the metal carbonate was used along with the moles calculated from the Ideal Gas Law.
• The result yielded the molar mass of the carbonate, which is an essential step to identify the metal \( M \).

This value was crucial in determining the identity of the metal \( M \) because each metal in Group 2A has a unique molar mass.

Stoichiometry

Stoichiometry is the section of chemistry that involves calculating the relationships between the reactants and products in a chemical reaction. It relies on the balanced equation to determine these relationships and involve concepts like molar ratios and conservation of mass.In this problem:

• The balanced equation \( \mathrm{MCO}_{3} \rightarrow \mathrm{MO} + \mathrm{CO}_{2} \) indicates a 1:1 molar ratio between the decomposing carbonate and the carbon dioxide produced.
• Knowing this ratio allows the moles of carbon dioxide calculated from the Ideal Gas Law to be directly linked to the moles of the carbonate that decomposed.

Using stoichiometry, we can trace the path from the initial mass of the carbonate to the identity of the metal \( M \), by deriving from the stoichiometric balances involved in the decomposition process. These calculations underscore the importance of stoichiometry in solving real-world chemical problems accurately.