Problem 45 - Chemistry The Molecular Science

Question

Metal carbonates decompose to the metal oxide and \(\mathrm{CO}_{2}\) on heating according to this general equation. $$\mathrm{M}_{x}\left(\mathrm{CO}_{3}\right)_{y}(\mathrm{~s}) \longrightarrow \mathrm{M}_{x} \mathrm{O}_{y}(\mathrm{~s})+y \mathrm{CO}_{2}(\mathrm{~g})$$ You heat \(0.158 \mathrm{~g}\) of a white, solid carbonate of a Group 2A metal and find that the evolved \(\mathrm{CO}_{2}\) has a pressure of \(69.8 \mathrm{mmHg}\) in a \(285-\mathrm{mL}\) flask at \(25^{\circ} \mathrm{C}\). Determine the molar mass of the metal carbonate.

Step-by-Step Solution

Answer

The molar mass of the metal carbonate is approximately 149.06 g/mol, which suggests the metal is Strontium.

Step 1: Convert Pressure to Atmospheres

The pressure of the CO₂ is given as 69.8 mmHg. Convert this pressure to atmospheres using the conversion factor 1 atm = 760 mmHg: \[ P = \frac{69.8 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.0918 \text{ atm} \]

Step 2: Convert Volume to Liters

The volume of the flask is given as 285 mL. Convert this volume to liters: \[ V = \frac{285 \text{ mL}}{1000 \text{ mL/L}} = 0.285 \text{ L} \]

Step 3: Use the Ideal Gas Law to Find Moles of CO2

Using the ideal gas law \( PV = nRT \), where \( R = 0.0821 \text{ L atm } \text{K}^{-1} \text{mol}^{-1} \) and \( T = 25^\circ C = 298 \text{ K} \), calculate the moles of CO₂:\[ n = \frac{PV}{RT} = \frac{0.0918 \text{ atm} \times 0.285 \text{ L}}{0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \times 298 \text{ K}} = 0.00106 \text{ mol} \]

Step 4: Calculate Grams of CO2

Using the moles of CO₂ calculated and the molar mass of CO₂ (44.01 g/mol), find the mass of CO₂:\[ \text{mass of CO}_2 = 0.00106 \text{ mol} \times 44.01 \text{ g/mol} = 0.0467 \text{ g} \]

Step 5: Find the Mass of the Metal Oxide

Calculate the mass of the metal oxide produced by subtracting the mass of CO₂ from the initial mass of the carbonate:\[ \text{mass of metal oxide} = 0.158 \text{ g} - 0.0467 \text{ g} = 0.1113 \text{ g} \]

Step 6: Calculate Total Molar Mass of the Carbonate

The carbonate is composed of the metal and the carbonate ion CO₃²⁻. Using stoichiometry and known mass, the moles of CO₂ also represent the moles of metal carbonate. The molar mass can also be derived using the masses of the pair: \[ M_{total} = \frac{0.158 \text{ g}}{0.00106 \text{ mol}} = 149.06 \text{ g/mol} \]

Step 7: Determine the Metal in Group 2A

Since Group 2A metals tend to form metal carbonates in the form MCO₃, subtract the molar mass of CO₃²⁻ (60.01 g/mol) from the total molar mass of the carbonate calculated:\[ M_{metal} = 149.06 \text{ g/mol} - 60.01 \text{ g/mol} \approx 89.05 \text{ g/mol} \] The molar mass of the metal suggests that the metal is likely Strontium, as it closely matches the known atomic mass of Sr (87.62 g/mol).

Step 8: Conclusion

The metal in the carbonate is Sr, making the compound SrCO₃ with an approximate molar mass of 147.63 g/mol.

Key Concepts

Metal Carbonates

Metal carbonates are compounds that consist of a metal ion combined with the carbonate ion, \(\text{CO}_3^{2-}\). They are typically solid at room temperature and often appear as white powders. When heated, metal carbonates undergo thermal decomposition, a process where the compound breaks down into simpler substances due to the application of heat.
For example, the equation for decomposition is:

• \( \text{M}_x(\text{CO}_3)_y(\text{s}) \rightarrow \text{M}_x\text{O}_y(\text{s}) + y \text{CO}_2(\text{g}) \)

Here, a metal carbonate \( \text{M}_x(\text{CO}_3)_y \) decomposes to form metal oxide \( \text{M}_x\text{O}_y \) and carbon dioxide gas (\( \text{CO}_2 \)). The evolution of carbon dioxide gas is often a telltale sign of this reaction.
The exact metal oxide and the amount of carbon dioxide produced depend on the specific metal involved. This is where the structure of the metal carbonate plays a role in determining its properties and decomposition behavior.

Gas Laws

Gas laws are a set of principles that describe the behavior of gases in relation to pressure, volume, temperature, and moles. The Ideal Gas Law, represented by the equation \( PV = nRT \), is one of the most fundamental equations when studying gases.
In this equation:

• \( P \) is the pressure of the gas (in atmospheres).
• \( V \) is the volume that the gas occupies (in liters).
• \( n \) is the amount of gas in moles.
• \( R \) is the ideal gas constant (0.0821 L atm K\(^{-1}\) mol\(^{-1}\)).
• \( T \) is the temperature of the gas (in Kelvin).

When calculating or predicting gaseous behavior, these variables are often interrelated, meaning that changes in one can cause changes in the others. The ideal gas law allows us to solve for any one of these variables if the others are known. By using this equation, we can calculate the amount of gas produced during the thermal decomposition of metal carbonates.

Group 2A Elements

Group 2A elements, also known as the alkaline earth metals, include Beryllium, Magnesium, Calcium, Strontium, Barium, and Radium. They are located in the second column of the periodic table and are known for their shiny, silvery-white appearance.
Some characteristics of Group 2A elements include:

• They have two valence electrons.
• These metals are highly reactive, especially with water, though less so than Group 1A metals.
• They typically form divalent cations, meaning they tend to lose two electrons to form \( \text{M}^{2+} \) ions.
• Compounds they form, such as metal carbonates, are often insoluble in water.

In reactions, especially decompositions, Group 2A elements tend to release \( \text{CO}_2 \) when in the form of carbonates. Identifying which particular metal is present in a given compound often relies on knowing the properties and molar masses of the Group 2A metals.

Molar Mass Calculation

Molar mass is a measure of the mass of one mole of a substance, typically expressed in grams per mole (g/mol). Calculating molar mass is vital in determining the composition of compounds or the identity of an unknown metal.
Here’s how we perform molar mass calculations:

• First, calculate the moles of a known component, such as \( \text{CO}_2 \), using the ideal gas law \( n = \frac{PV}{RT} \).
• Next, using the stoichiometry from the decomposition equation, infer the moles of other components involved, like the initial metal carbonate.
• Finally, use the mass of the initial sample and the number of moles to determine molar mass: \( M = \frac{\text{mass}}{\text{moles}} \).

This process allows one to deduce not only the molar mass of the entire compound but also if necessary, identify the unknown metal by comparing calculated values to known atomic masses. This methodology is essential in analytical chemistry and helps to identify materials based on their decomposition products and properties.