Exercises: Dynamic equilibrium, K and Q

Solid calcium carbonate is placed in a closed reactor, and the following heterogeneous equilibrium is established at $1073\text{ K}$: $\mathrm{CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)}$.

1. a) Write the expressions for the equilibrium constants $K_c$ and $K_p$ for this reaction, explaining with reasoning why pure solid species are not included.
2. b) Explain the physical meaning of "dynamic equilibrium" as opposed to a static equilibrium, stating whether the decomposition of $\mathrm{CaCO_3(s)}$ stops once this state is reached. Assume that the two pure solid phases remain present.

For the synthesis of ammonia according to the equation $\mathrm{N_2(g) + 3\,H_2(g) \rightleftharpoons 2\,NH_3(g)}$, the value of the equilibrium constant $K_{c,1}$ at $673\text{ K}$ is equal to $0.50$.

1. a) Determine the value of the equilibrium constant $K_{c,2}$ at the same temperature for the reverse reaction: $2\,\mathrm{NH_3(g) \rightleftharpoons N_2(g) + 3\,H_2(g)}$.
2. b) Calculate the value of the constant $K_{c,3}$ at $673\text{ K}$ corresponding to the decomposition of one mole of ammonia: $\mathrm{NH_3(g) \rightleftharpoons \frac{1}{2}\,N_2(g) + \frac{3}{2}\,H_2(g)}$. For the calculations, use the ideal model and the teaching values provided.

In a closed flask of capacity $2.00\text{ L}$, at a fixed temperature of $700\text{ K}$, a gaseous mixture is introduced consisting of $0.40\text{ mol}$ of $\mathrm{H_2(g)}$, $0.20\text{ mol}$ of $\mathrm{I_2(g)}$ and $0.80\text{ mol}$ of $\mathrm{HI(g)}$. The equilibrium constant at that temperature for the process $\mathrm{H_2(g) + I_2(g) \rightleftharpoons 2\,HI(g)}$ is $K_c = 54.0$.

1. a) Calculate the initial reaction quotient $Q_c$.
2. b) Deduce, with justification, whether the system is at equilibrium and, if not, in which direction it will evolve spontaneously to reach it. For the calculations, use the ideal model and the teaching values provided.

The reversible elementary reaction $\mathrm{A(g) \rightleftharpoons B(g)}$ is studied kinetically in the gas phase in a closed isothermal vessel of constant volume at $300\text{ K}$, starting only from the reactant $\mathrm{A}$ at an initial concentration of $1.00\text{ mol}\times\text{L}^{-1}$. The attached graph monitors the variation with time of the rate of the forward reaction ($v_d$) and of the reverse reaction ($v_i$).

Look at the following representative figure:

1. a) Explain, using the graph, why the final state reached after a sufficiently long time, within experimental precision, corresponds to a dynamic equilibrium and not to a static cessation of reactivity.
2. b) If the forward rate constant is $k_d = 0.030\text{ s}^{-1}$ and at equilibrium the common rate shown is $v_{\mathrm{eq}} = 0.012\text{ mol}\times\text{L}^{-1}\times\text{s}^{-1}$, calculate the equilibrium concentration of $\mathrm{A}$, the equilibrium concentration of $\mathrm{B}$ and the value of the equilibrium constant $K_c$. For the calculations, use the ideal model and the teaching values provided.

At a temperature of $500\text{ K}$, the equilibrium constant for the synthesis of sulfur trioxide according to the equation $\mathrm{2\,SO_2(g) + O_2(g) \rightleftharpoons 2\,SO_3(g)}$ is $K_c = 400$.

Data: gas constant $R = 0.082\text{ atm}\times\text{L}\times\text{mol}^{-1}\times\text{K}^{-1}$.

1. a) Deduce the general mathematical relationship between $K_p$ and $K_c$ for a mixture of ideal gases and calculate the numerical value of $K_p$ for this reaction at $500\text{ K}$.
2. b) If, in a closed flask at that temperature, the partial pressures at equilibrium are $p_{\mathrm{SO_2}} = 0.10\text{ atm}$ and $p_{\mathrm{O_2}} = 0.20\text{ atm}$, calculate the partial pressure of $\mathrm{SO_3}$ at equilibrium and the total pressure of the mixture. For the calculations, use the ideal model and the teaching values provided.

In a previously evacuated flask of $5.00\text{ L}$, $1.50\text{ mol}$ of carbon monoxide, $\mathrm{CO(g)}$, and $1.50\text{ mol}$ of water vapour, $\mathrm{H_2O(g)}$, are mixed at a constant temperature of $900\text{ K}$, and the following reaction takes place:

$\mathrm{CO(g) + H_2O(g) \rightleftharpoons CO_2(g) + H_2(g)}$, whose constant is $K_c = 4.00$.

1. a) Construct a table with the initial amounts of substance, the change undergone and the amounts at equilibrium, expressed in moles.
2. b) Determine the molar concentration of each species at equilibrium and calculate the mole fraction of carbon dioxide in the gas phase. For the calculations, use the ideal model and the teaching values provided.

You can do this on paper and send a photo with your answer.

Dinitrogen tetroxide dissociates thermally into nitrogen dioxide according to the equilibrium $\mathrm{N_2O_4(g) \rightleftharpoons 2\,NO_2(g)}$. In a closed vessel of fixed volume $V = 10.0\text{ L}$ at $320\text{ K}$, a mass of $18.4\text{ g}$ of pure $\mathrm{N_2O_4}$ is introduced. Once chemical equilibrium has been established at that temperature, the total pressure recorded on the manometer is $0.650\text{ atm}$.

Data: relative atomic masses: $\mathrm{N} = 14.0$; $\mathrm{O} = 16.0$. Constant $R = 0.082\text{ atm}\times\text{L}\times\text{mol}^{-1}\times\text{K}^{-1}$.

1. a) Calculate the degree of dissociation $\alpha$ of $\mathrm{N_2O_4(g)}$ under these conditions.
2. b) Deduce the expression for $K_p$ in terms of $\alpha$ and the total pressure $P_t$, and determine its numerical value at $320\text{ K}$. For the calculations, use the ideal model and the teaching values provided.

In a rigid reactor of $4.00\text{ L}$ at $720\text{ K}$, $1.20\text{ mol}$ of carbon monoxide, $\mathrm{CO(g)}$, and $3.60\text{ mol}$ of water vapour, $\mathrm{H_2O(g)}$, are introduced. At that temperature the water-gas shift reaction takes place:

$\mathrm{CO(g) + H_2O(g) \rightleftharpoons CO_2(g) + H_2(g)}$, with $K_c = 1.60$.

1. a) Set up the quadratic equation corresponding to the extent of reaction $x$ at equilibrium and rigorously justify discarding the root that has no physical meaning.
2. b) Calculate the molar composition of each gas at equilibrium and the percentage yield of the process with respect to the limiting reactant. For the calculations, use the ideal model and the teaching values provided.

In a flask of $2.00\text{ L}$ at $523\text{ K}$, a mixture of $0.200\text{ mol}$ of $\mathrm{PCl_5(g)}$ and $0.100\text{ mol}$ of $\mathrm{Cl_2(g)}$ is initially introduced. At that temperature the decomposition equilibrium is established:

$\mathrm{PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)}$, with $K_c = 0.0400$.

Data: $R = 0.082\text{ atm}\times\text{L}\times\text{mol}^{-1}\times\text{K}^{-1}$.

1. a) Show, by means of an equilibrium table with the extent of reaction in moles as the unknown, that the molar concentration of $\mathrm{PCl_3}$ at equilibrium requires solving a quadratic equation, and calculate the concentrations of all species at equilibrium.
2. b) Determine the total pressure inside the flask at equilibrium and calculate the value of the constant $K_p$ at that temperature. For the calculations, use the ideal model and the teaching values provided.