Question

For the reaction below, \(K_{\mathrm{p}}=1.16\) at \(800 .{ }^{\circ} \mathrm{C}\). $$ \mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) $$ If a \(20.0-\mathrm{g}\) sample of \(\mathrm{CaCO}_{3}\) is put into a \(10.0\) - \(\mathrm{L}\) container and heated to \(800 .{ }^{\circ} \mathrm{C}\), what percentage by mass of the \(\mathrm{CaCO}_{3}\) will react to reach equilibrium?

Short answer

82.35% by mass of CaCO3 will react to reach equilibrium at 800°C in a 10.0 L container.

Step-by-step solution

Calculate the initial moles of CaCO3

First, find the number of moles of CaCO3 in the initial 20.0 g sample. You can do this by dividing the mass of the sample by the molar mass of CaCO3. Molar mass of CaCO3 = 40.08(Ca) + 12.01(C) + 3 * 16.00(O) = 100.09 g/mol Initial moles of CaCO3 = Initial mass / Molar mass = \( \frac{20.0}{100.09} \) mol

Write the expression for Kp

We are given that Kp = 1.16 for the reaction at 800°C. The balanced reaction is: CaCO3(s) ⇌ CaO(s) + CO2(g) Since solids are not included in the equilibrium constant expression, the expression for Kp will only include the partial pressure of CO2(g) at equilibrium. Kp = PCO2

Create an equation relating moles of CO2 to initial moles of CaCO3

At equilibrium, let x mol of CaCO3 react: CaCO3(s) - x ⇌ (CaO) + (CO2) The initial moles of CaCO3 is \( \frac{20.0}{100.09} \) mol. At equilibrium, x mol of CaCO3 has reacted, producing x mol of CO2. Partial pressure of CO2 at equilibrium: PCO2 = \( \frac{x \times R \times T}{V} \) Here, R is the gas constant = 0.0821 L atm/(mol K) T is the temperature in Kelvin = 800 + 273.15 = 1073.15 K V is the volume of the container = 10.0 L

Solve for x

Now, we can substitute the expression for PCO2 into the equation for Kp: Kp = PCO2 = \( \frac{x \times R \times T}{V} \) 1.16 = \( \frac{x \times 0.0821 \times 1073.15}{10.0} \) Solve for x: x = \( \frac{1.16 \times 10.0}{0.0821 \times 1073.15} \) = 0.1646 mol

Calculate the percentage of CaCO3 that reacted

Now, we can find the mass of CaCO3 that reacted at equilibrium: Mass of CaCO3 reacted = x * Molar mass of CaCO3 = 0.1646 * 100.09 = 16.47 g Calculate the percentage of CaCO3 that reacted: Percentage = \( \frac{Mass \, of \, CaCO3 \, reacted}{Initial \, mass \, of \, CaCO3} \) × 100 Percentage = \( \frac{16.47}{20.0} \) × 100 = 82.35% So, 82.35% by mass of CaCO3 will react to reach equilibrium at 800°C in a 10.0 L container.

Key concepts

Equilibrium Constant (Kp)

The equilibrium constant, represented as Kp, is a numerical value that represents the ratio of the concentration of products to the concentration of reactants, raised to their respective coefficients, once a chemical reaction has reached equilibrium. However, Kp specifically refers to equilibrium involving gases and is expressed in terms of partial pressure. For example, in the decomposition of calcium carbonate, Kp relates to the partial pressure of carbon dioxide gas at equilibrium. It's given by the simple expression Kp = PCO2, since there are no gas reactants in this reaction and the solid calcium oxide is omitted from the expression. Understanding Kp helps predict how far the reaction progresses toward the products or reactants as conditions change. In practice, the constant can be used to calculate unknown concentrations or, as in our exercise, the extent of a reaction.

Le Chatelier's Principle

Le Chatelier's principle is a fundamental concept in chemical equilibrium which states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. This principle helps us predict how an equilibrium will shift when subject to alterations in pressure, temperature, volume, or concentration. For instance, increasing the pressure of a gas involved in equilibrium typically shifts the reaction toward the side with fewer moles of gas. In the case of calcium carbonate decomposition, if the pressure above the reaction mixture is increased, the principle suggests that the reaction would shift toward the production of more solid reactants to reduce the pressure.

Calcium Carbonate Decomposition

Calcium carbonate decomposition is a chemical process where calcium carbonate (CaCO3) breaks down into calcium oxide (CaO) and carbon dioxide (CO2) when heated. This decomposition is an example of a thermal decomposition reaction, which is endothermic, meaning it requires heat to proceed. The reaction is reversible, leading to a dynamic equilibrium between the solid reactants and products at a given temperature. The balance between the solid and gas phases at this equilibrium is captured by the equilibrium constant, Kp, which in this instance is only influenced by the pressure of CO2 as solids don't have pressure.

Molar Mass Calculation

Calculating molar mass is essential in stoichiometry to convert between the mass of a substance and the amount in moles. The molar mass is the weight of one mole of a substance, typically expressed in grams per mole (g/mol). It is calculated by summing the weights of the individual atoms that constitute the molecule, based on the atomic weights found on the periodic table. For calcium carbonate (CaCO3), this is done by adding the molar mass of calcium, carbon, and three oxygen atoms. Correctly determining molar mass is crucial for accurately calculating the moles of a substance, as seen in our exercise to determine the initial moles of CaCO3, to further understand how much of it decomposes at equilibrium.

Partial Pressure

Partial pressure is a concept in gases that refers to the pressure a single gas in a mixture would exert if it alone occupied the entire volume of the mixture at the same temperature. It is a way to understand the behavior of individual gases within a mixture, such as in a chemical reaction at equilibrium. The partial pressure of a gas can be calculated using the ideal gas law equation, where it's a function of the number of moles of the gas, the temperature, the volume of the container, and the gas constant R. In the step-by-step solution provided, the partial pressure of CO2 generated by the decomposition of CaCO3 at equilibrium plays a key role in using the ideal gas law to relate the number of moles of CO2 at equilibrium to the known equilibrium constant Kp.