Question

A 25.0-g sample of ammonium carbamate, \(\mathrm{NH}_{4}\left(\mathrm{NH}_{2} \mathrm{CO}_{2}\right)\), was placed in an evacuated \(0.250\) - L flask and kept at \(25^{\circ} \mathrm{C}\). At equilibrium, \(17.4 \mathrm{mg}\) of \(\mathrm{CO}_{2}\) was present. What is the value of \(K_{c}\) for the decomposition of ammonium carbamate into ammonia and carbon dioxide? The reaction is \(\mathrm{NH}_{4}\left(\mathrm{NH}_{2} \mathrm{CO}_{2}\right)(\mathrm{s}) \rightleftharpoons\) \(2 \mathrm{NH}_{3}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g}) .\)

Short answer

Kc is the value obtained by substituting the equilibrium concentrations of NH3 and CO2 in the Kc expression, given that the concentration of solid NH4(NH2CO2) is omitted.

Step-by-step solution

Calculate the moles of CO2 at equilibrium

To find the number of moles of CO2 at equilibrium, convert the mass of CO2 to moles using its molar mass (Molar mass of CO2 = 44.01 g/mol). Use the formula: number of moles = mass (g) / molar mass (g/mol).

Calculate the moles of NH3 at equilibrium

From the balanced chemical equation, the molar ratio of NH4(NH2CO2) to NH3 is 1:2 and to CO2 is 1:1. Therefore, the moles of NH3 produced will be twice the moles of CO2 at equilibrium.

Calculate the initial moles of NH4(NH2CO2)

Convert the initial mass of NH4(NH2CO2) to moles using its molar mass (Molar mass of NH4(NH2CO2) = 78.11 g/mol).

Calculate the moles of NH4(NH2CO2) at equilibrium

Subtract the moles of CO2 at equilibrium from the initial moles of NH4(NH2CO2) to calculate the moles of NH4(NH2CO2) remaining at equilibrium.

Calculate the equilibrium concentrations

Use the moles of NH3, CO2, and NH4(NH2CO2) at equilibrium and the volume of the flask to calculate their equilibrium concentrations (mol/L).

Write the expression for Kc

Set up the expression for the equilibrium constant Kc based on the balanced chemical equation. Kc = [NH3]^2[CO2] / [NH4(NH2CO2)]. Since NH4(NH2CO2) is a solid, its concentration is omitted from the expression.

Calculate the value of Kc

Substitute the equilibrium concentrations into the Kc expression and solve for Kc.

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is crucial when studying reactions that can occur in both the forward and reverse directions. At equilibrium, the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. However, it’s important to note that being at equilibrium does not mean the reactants and products are present in equal amounts; rather, it indicates a stable ratio between the concentrations of the two.

In the process of reaching equilibrium, the system may shift to favor the formation of either reactants or products, depending on the conditions. This is described by Le Chatelier’s Principle, which states that a system at equilibrium will respond to a disturbance by countering the change and re-establishing equilibrium. This principle can be applied to changes in concentration, pressure, volume, or temperature.

Molar Mass Calculation

The molar mass of a substance, expressed in grams per mole (g/mol), is a vital concept in chemistry that corresponds to the mass of one mole of a compound. Molar mass is calculated by summing up the atomic masses of all atoms in a molecule as given by the periodic table. For instance, CO2 has a molar mass of 44.01 g/mol, which is derived from the atomic mass of carbon (12.01 g/mol) and two oxygens (2 × 16.00 g/mol).

Understanding molar mass is essential in stoichiometric calculations as it allows converting between the mass of a substance and the number of moles, which is the bridge between the macroscopic scale we measure and the microscopic scale of molecules and atoms.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It involves using the balanced chemical equation to determine the ratios at which substances react and are produced. Because chemical reactions obey the law of conservation of mass, stoichiometry allows us to predict the amounts of products formed from given quantities of reactants and vice versa.

The coefficients in the balanced equation represent the molar ratio of reactants and products. For the decomposition of ammonium carbamate, the balanced equation indicates that one mole of the compound yields two moles of NH3 and one mole of CO2. Stoichiometry is vital for calculating the moles of each substance at equilibrium, a step that is indispensable in determining the equilibrium constant.

Equilibrium Concentrations

Equilibrium concentrations refer to the molarity of reactants and products present when a chemical system reaches equilibrium. These are expressed in moles per liter (mol/L) and are essential in quantifying the equilibrium state. In the lab, after calculating the moles of each substance at equilibrium using stoichiometry, the next step is to divide these moles by the volume of the system to find their concentrations.

To give a concrete example, if we know the volume of the flask and the moles of CO2 at equilibrium, we can calculate the concentration of CO2 by dividing the moles of CO2 by the volume in liters. Accurate measurement of these concentrations is necessary for determining the equilibrium constant, which quantifies the extent of the reaction at equilibrium.

Equilibrium Constant Expression

The equilibrium constant expression (Kc) is a mathematical representation of the ratio of product concentrations to reactant concentrations at chemical equilibrium, each raised to the power of their respective stoichiometric coefficients. For the reaction involving the decomposition of ammonium carbamate, we derive the expression Kc = [NH3]^2[CO2] as given in the chemical equation, excluding any pure solids or liquids, as their concentration changes are negligible.

Calculating the value of the equilibrium constant involves substituting the equilibrium concentrations into this expression. Kc values indicate the position of equilibrium—the larger the value, the more product-favored the reaction. By calculating the Kc, scientists can predict the extent of a reaction and compare the relative amounts of reactants and products at equilibrium for different systems.