Question

The volume of carbon dioxide gas evolved at STP by heating \(7.3 \mathrm{~g}\) of \(\mathrm{Mg}\left(\mathrm{HCO}_{3}\right)_{2}\) will be: (a) \(1100 \mathrm{~mL}\) (b) \(1120 \mathrm{~mL}\) (c) \(2230 \mathrm{~mL}\) (d) \(3240 \mathrm{~mL}\)

Short answer

The volume of CO2 evolved is closest to (c) 2230 mL.

Step-by-step solution

Determine the Balanced Chemical Equation

When \( \mathrm{Mg(HCO_3)_2} \) is heated, it decomposes into \( \mathrm{MgO} \), \( \mathrm{CO_2} \), and \( \mathrm{H_2O} \). The balanced equation is: \[ \mathrm{Mg(HCO_3)_2 \rightarrow MgO + 2CO_2 + 2H_2O} \]. This equation shows that 1 mole of \( \mathrm{Mg(HCO_3)_2} \) produces 2 moles of \( \mathrm{CO_2} \).

Determine the Molar Mass of Mg(HCO3)2

Calculate the molar mass of \( \mathrm{Mg(HCO_3)_2} \). The atomic masses are approximately: Mg = 24, H = 1, C = 12, and O = 16. Therefore, the molar mass: \[ 24 + 2 \times (1+12+3 \times 16) = 146 \text{ g/mol} \].

Calculate the Moles of Mg(HCO3)2

Use the given mass (7.3 g) to find the moles of \( \mathrm{Mg(HCO_3)_2} \): \[ \text{Moles} = \frac{7.3 \text{ g}}{146 \text{ g/mol}} \approx 0.05 \text{ moles} \].

Calculate the Moles of CO2 Produced

According to the balanced equation, 1 mole of \( \mathrm{Mg(HCO_3)_2} \) produces 2 moles of \( \mathrm{CO_2} \). Hence, 0.05 moles of \( \mathrm{Mg(HCO_3)_2} \) will produce: \[ 0.05 \times 2 = 0.1 \text{ moles of } \mathrm{CO_2} \].

Convert Moles of CO2 to Volume at STP

At STP (Standard Temperature and Pressure), 1 mole of any gas occupies 22.4 liters. Thus, the volume of 0.1 moles of \( \mathrm{CO_2} \) is: \[ 0.1 \times 22.4 \text{ L} = 2.24 \text{ L} = 2240 \text{ mL} \].

Determine the Closest Answer Choice

The calculated volume of \( \mathrm{CO_2} \) is 2240 mL. However, the closest provided option is (c) 2230 mL.

Key concepts

Decomposition Reaction

A decomposition reaction is a type of chemical reaction where one compound breaks down into two or more simpler substances. It's like dismantling a complex molecule into its building blocks.
These reactions generally require heat, light, or electricity to break the chemical bonds.
In the given exercise, when magnesium bicarbonate a compound with the formula \( \mathrm{Mg(HCO_3)_2} \) is heated, it undergoes decomposition.
It breaks down into magnesium oxide \( \mathrm{MgO} \), carbon dioxide \( \mathrm{CO_2} \), and water \( \mathrm{H_2O} \).

• Balanced Reaction: \( \mathrm{Mg(HCO_3)_2 \rightarrow MgO + 2CO_2 + 2H_2O} \)
• The balanced equation reveals that from one mole of magnesium bicarbonate, two moles of carbon dioxide are evolved.
• This is a characteristic of decomposition reactions: a single compound yielding multiple products.

Understanding this fundamental concept is essential in predicting the amount of product formed when a given mass of a compound is decomposed.

STP Conditions

Standard Temperature and Pressure, commonly known as STP, refers to a standard set of conditions for measuring gases.
These conditions provide a baseline for scientists to compare gases under a consistent frame of reference.
STP is defined as a temperature of 0 degrees Celsius, which is 273.15 Kelvin and a pressure of 1 atmosphere (atm).

• Under these settings, gases behave predictably as described by gas laws.
• Any gas measured under STP will have properties that can be directly compared to those of other gases.
• Using STP conditions allows chemists to standardize calculations and results, simplifying the comparison of different gas-scenarios.

In the exercise provided, calculating the volume of \( \mathrm{CO_2} \) at STP allows us to use the molar volume concept seamlessly.

Molar Volume

Molar volume is an important concept in chemistry, especially when dealing with gases.
It refers to the volume that one mole of a gas occupies at a given set of temperature and pressure conditions.

• At STP, one mole of any ideal gas occupies 22.4 liters.
• This value is derived from the ideal gas law and is crucial for calculating the volume of gases formed or consumed in chemical reactions.

In the problem, you'd utilize the molar volume concept to find out the volume of \( \mathrm{CO_2} \) gas produced.
For instance, if 0.1 moles of \( \mathrm{CO_2} \) are produced, it occupies: \[ 0.1 \times 22.4 \text{ L} = 2.24 \text{ L (or 2240 mL)} \].
This value is then used to find the appropriate answer from the provided choices.

Gas Laws

Gas laws are equations that describe how the volume, temperature, pressure, and number of gas particles interact.
These laws are the framework for understanding gas behavior in various conditions. They include concepts such as Boyle's Law, Charles's Law, and Avogadro's Law.

• Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume.
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature.
• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles.

In the problem at hand, the ideal gas law, which combines these individual laws, is often used.
This law states that for one mole of gas at STP, the volume is 22.4 liters. This correlation, derived from these traditional gas laws, makes it essential for solving problems related to gaseous reactions and volume calculations.