Question

How many milliliters of carbon dioxide gas at STP are produced from the decomposition of \(1.59 \mathrm{~g}\) of lithium hydrogen carbonate? $$2 \mathrm{LiHCO}_{3}(s) \longrightarrow \mathrm{Li}_{2} \mathrm{CO}_{3}(s)+\mathrm{H}_{2} \mathrm{O(l)+\mathrm{CO}_{2}(g)$$

Short answer

262.08 mL of \(\text{CO}_2\) are produced.

Step-by-step solution

Write Down the Chemical Equation

The balanced chemical equation for the decomposition is: \( 2 \text{LiHCO}_3 (s) \rightarrow \text{Li}_2 \text{CO}_3 (s) + \text{H}_2\text{O} (l) + \text{CO}_2 (g) \). This tells us that 2 moles of \(\text{LiHCO}_3 \) produce 1 mole of \( \text{CO}_2 \).

Calculate the Molar Mass of LiHCO3

To proceed, we must calculate the molar mass of \( \text{LiHCO}_3 \). Li: 6.94 g/mol, H: 1.01 g/mol, C: 12.01 g/mol, O: 16.00 g/mol. Calculate: \(6.94 + 1.01 + 12.01 + (3 \times 16) = 67.95\) g/mol.

Determine Moles of LiHCO3

Using the molar mass, calculate the moles of \( \text{LiHCO}_3 \) from the mass given: \( \text{Moles} = \frac{1.59 \text{ g}}{67.95 \text{ g/mol}} \approx 0.0234 \) moles.

Relate Moles of LiHCO3 to CO2 Produced

According to the equation, 2 moles of \( \text{LiHCO}_3 \) yield 1 mole of \( \text{CO}_2 \). Thus, \(0.0234\) moles of \( \text{LiHCO}_3 \) will yield \( \frac{0.0234}{2} = 0.0117 \) moles of \( \text{CO}_2 \).

Convert Moles of CO2 to Milliliters

At standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 L. Therefore, \(0.0117\) moles of \( \text{CO}_2 \) will occupy \(0.0117 \times 22,400 = 262.08\) mL of \( \text{CO}_2 \).

Key concepts

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It involves using a balanced chemical equation to determine the number of moles of each substance involved. The key skill is to interpret the equation correctly. In the given problem, stoichiometry helps to deduce how many moles of carbon dioxide (\(\text{CO}_2\)) are produced from the decomposition of lithium hydrogen carbonate (\(\text{LiHCO}_3\)).

From the balanced equation:

• 2 moles of \(\text{LiHCO}_3\) decompose to produce 1 mole of \(\text{CO}_2\).
• Therefore, this relationship allows us to calculate the moles of \(\text{CO}_2\) produced from a known amount of \(\text{LiHCO}_3\).

By doing so, stoichiometry underpins the conversion of reactant measurement to product measurement in a scalable way.

Molar Mass Calculation

Calculating molar mass is a critical step when transitioning from mass to moles. Molar mass is defined as the mass of one mole of a substance, typically expressed in grams per mole (g/mol). In our case, we calculate the molar mass of lithium hydrogen carbonate (\(\text{LiHCO}_3\)). This allows us to convert 1.59 grams of \(\text{LiHCO}_3\) into moles, which is necessary for further stoichiometric calculations.

The calculation involves summing the atomic masses of all atoms in the compound:

• Lithium (Li): 6.94 g/mol
• Hydrogen (H): 1.01 g/mol
• Carbon (C): 12.01 g/mol
• Oxygen (O): 16.00 g/mol (times 3 due to three oxygen atoms)

Adding these masses together, we obtain a molar mass of 67.95 g/mol for \(\text{LiHCO}_3\). This reveals how much one mole of \(\text{LiHCO}_3\) weighs, crucial for moles calculation.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a standard set of conditions for measuring the volume of gases. These conditions are defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm. At STP, one mole of any gas occupies 22.4 liters. This provides a convenient way to work between moles and volume for gaseous substances.

In our problem, understanding STP is crucial when we convert moles of \(\text{CO}_2\) to volume. For stoichiometric calculations involving gases, STP conditions simplify the conversion since 1 mole = 22.4 liters. Recognizing this standardized measurement allows for consistent communication and computation of volumes in diverse contexts.

Gas Volume Conversion

Gas volume conversion is the step where the amount of gas in moles is translated to a measurable volume. This is particularly relevant for scenarios like our exercise, where we have to find out the volume of carbon dioxide gas produced.

The conversion uses the relationship that under STP conditions, 1 mole of gas equals 22.4 liters. Once the moles of \(\text{CO}_2\) are determined from stoichiometry and molar mass calculation, this factor allows easy conversion into liters or milliliters.

• For example, if 0.0117 moles of \(\text{CO}_2\) are produced, this can be converted to a volume of 262.08 milliliters by multiplying the number of moles by 22,400 mL/mol.

This final conversion step translates theoretical moles into practical, observable volumes, which is often required in quantitative experiments or process analysis.