Question

Carbon dioxide gas, in the dry state, may be produced by heating calcium carbonate. $$ \mathrm{CaCO}_{3}(s) \rightarrow \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) $$ What volume of \(\mathrm{CO}_{2},\) collected dry at \(55 \quad \mathrm{C}\) and a pressure of 774 torr, is produced by complete thermal decomposition of \(10.0 \mathrm{~g}\) of \(\mathrm{CaCO}_{3} ?\)

Short answer

The volume of CO2 gas produced by the complete thermal decomposition of 10.0 g of CaCO3 at 55 °C and a pressure of 774 Torr is 2.65 liters.

Step-by-step solution

Convert mass of CaCO3 to moles

To convert the mass of CaCO3 to moles, we will use its molar mass. The molar mass of calcium carbonate is 100.09 g/mol (40.08 g/mol for Ca, 12.01 g/mol for C, and 16.00 g/mol for each O). Using the formula: moles = mass / molar mass moles of CaCO3 = 10.0 g / 100.09 g/mol = 0.0999 mol

Calculate moles of CO2 produced

The balanced chemical equation provides the stoichiometry of the decomposition reaction: CaCO3(s) → CaO(s) + CO2(g) From the equation, 1 mole of CaCO3 produces 1 mole of CO2. Therefore, the moles of CO2 produced are equal to the moles of CaCO3. moles of CO2 = moles of CaCO3 = 0.0999 mol

Convert moles of CO2 to volume using the Ideal Gas Law

The Ideal Gas Law is given by: PV = nRT where P is pressure, V is volume, n is moles, R is the ideal gas constant, and T is the temperature in Kelvin. We are given the pressure in Torr (774 Torr), which we need to convert to atmospheres by dividing by 760: P = 774 Torr / 760 Torr/atm = 1.0184 atm The temperature in Celsius (55 °C) should be converted to Kelvin: T = 55 °C + 273.15 = 328.15 K We will use the value of the ideal gas constant R in the units L·atm·mol⁻¹·K⁻¹: R = 0.0821 L·atm·mol⁻¹·K⁻¹ Now we can solve the Ideal Gas Law equation for V: V = nRT / P V = (0.0999 mol)×(0.0821 L·atm·mol⁻¹·K⁻¹)×(328.15 K) / (1.0184 atm) = 2.65 L

Report final answer

The volume of CO2 gas produced by the complete thermal decomposition of 10.0 g of CaCO3 at 55 °C and a pressure of 774 Torr is 2.65 liters.

Key concepts

Stoichiometry

Stoichiometry is the backbone of chemical equations and involves the calculation of reactants and products in chemical reactions. It plays a crucial role when we are interested in understanding how much of a substance is consumed or produced during a reaction.

Consider the decomposition of calcium carbonate (CaCO₃), which produces calcium oxide (CaO) and carbon dioxide (CO₂). The balanced chemical equation is written as:
\[ \mathrm{CaCO}_{3}(s) \rightarrow \mathrm{CaO}(s) + \mathrm{CO}_{2}(g) \]
The equation tells us that 1 mole of calcium carbonate yields exactly 1 mole of carbon dioxide. When we apply stoichiometry, beginning with the mass of calcium carbonate, we first determine the moles of calcium carbonate present and then, through the coefficients provided by the balanced equation, ascertain the moles of carbon dioxide produced.

Using stoichiometric coefficients, the number of moles and the law of conservation of mass, we ensure that matter is neither created nor destroyed; so, if we start with 10.0 grams of calcium carbonate, we would theoretically produce an equivalent amount of product, which in our case, is carbon dioxide. The step by step solution included also is an example of applying stoichiometry principles.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas. It is commonly expressed as:
\[PV = nRT\]
Here, R is the gas constant. This law is pivotal when determining the volume of a gas, such as carbon dioxide in our example, under specific conditions of temperature and pressure.

To apply the Ideal Gas Law, one must ensure that the pressure is in atmospheres (atm) and the temperature is in Kelvin (K). Any given values in other units should be converted accordingly - this ensures accuracy in the final calculations. For instance, in the provided solution, the pressure was initially in Torr and had to be converted to atmospheres, while the temperature was given in Celsius and had to be converted to Kelvin.

The law is applied best when gases behave ideally, which means they follow certain assumptions, including no interactions between gas particles and that the particles occupy no volume. While these assumptions are not perfectly true for real gases, the Ideal Gas Law provides a good approximation for many conditions, especially those involving higher temperatures and lower pressures, like in our example where CO₂ is being produced at a fairly high temperature.

Chemical Reaction

Chemical reactions involve the transformation of substances through the breaking of old bonds and the formation of new ones. These can be described through chemical equations that illustrate what reactants are used and what products are formed.

Thermal decomposition, such as when heating calcium carbonate to yield calcium oxide and carbon dioxide, is a specific type of chemical reaction where a compound breaks down into simpler ones upon heating. This process is represented in the equation:
\[\mathrm{CaCO}_{3}(s) \xrightarrow{\Delta} \mathrm{CaO}(s) + \mathrm{CO}_{2}(g)\]
The triangle above the arrow signifies that heat is applied for the reaction to occur. It is essential to know that not all compounds will decompose with heat alone, and different substances will require different amounts of energy (heat) for decomposition to take place.

The exercise provided is a practical application of understanding chemical reactions. It requires comprehension of how substances interact under specific conditions, predicting the outcomes, and then using other chemistry concepts like stoichiometry and the ideal gas law to quantify the products of the reaction.