Le Châtelier's principle and industrial processes

In the catalysis stage of the industrial contact process for obtaining sulfuric acid, the following reversible reaction takes place:

$$2\,\mathrm{SO_2(g)} + \mathrm{O_2(g)} \rightleftharpoons 2\,\mathrm{SO_3(g)} \quad \Delta H^\circ = -198\,\text{kJ}\times\text{mol}^{-1}$$

Explain the effect of the following independent changes on the position of the equilibrium at constant temperature:

1. a) The addition of divanadium pentoxide, $\mathrm{V_2O_5(s)}$, used as a solid catalyst in the reactor.
2. b) The continuous and selective removal of $\mathrm{SO_3(g)}$ from the reactor as it forms.

The process of calcination of limestone in a closed furnace at $1073\,\text{K}$ corresponds to the heterogeneous equilibrium:

$$\mathrm{CaCO_3(s)} \rightleftharpoons \mathrm{CaO(s)} + \mathrm{CO_2(g)} \quad \Delta H^\circ = +178\,\text{kJ}\times\text{mol}^{-1}$$

Explain with reasoning how each of the following independent actions at $1073\,\text{K}$ affects the equilibrium:

1. a) The introduction of an additional mass of $50\,\text{g}$ of pure $\mathrm{CaCO_3(s)}$ into the vessel, keeping the volume and temperature unchanged.
2. b) An isothermal expansion of the vessel that doubles the total volume available to the gases. Assume that the two pure solid phases remain present.

Phosphorus pentachloride is introduced into a closed reactor and heated to $523\,\text{K}$, establishing the following gaseous equilibrium:

$$\mathrm{PCl_5(g)} \rightleftharpoons \mathrm{PCl_3(g)} + \mathrm{Cl_2(g)}$$

Analyze, with justification, the effect on the amount of $\mathrm{PCl_5(g)}$ of introducing an amount of inert gas (argon, $\mathrm{Ar}$) under the following different thermodynamic conditions:

1. a) The noble gas is injected while the volume of the vessel and the temperature are kept strictly constant.
2. b) The noble gas is injected while the vessel is allowed to expand so that the total pressure of the mixture and the temperature remain constant.

The synthesis of ammonia by the Haber-Bosch process is based on the exothermic reaction:

$$\mathrm{N_2(g)} + 3\,\mathrm{H_2(g)} \rightleftharpoons 2\,\mathrm{NH_3(g)} \quad \Delta H^\circ = -92.2\,\text{kJ}\times\text{mol}^{-1}$$

Industrially, the process is operated neither at room temperature nor at unlimitedly high pressures, but under conditions of thermal and mechanical compromise ($T \approx 400\text{--}450\,^\circ\text{C}$ and $P \approx 150\text{--}200\,\text{atm}$) in the presence of promoted iron catalysts. Justify these industrial conditions by considering:

1. a) The choice of high working pressures in light of the cost and safety considerations of the plant.
2. b) The need to use an intermediate temperature ($450\,^\circ\text{C}$) in light of the analysis of equilibrium yield versus reaction rate.

In an industrial Haber-Bosch plant, ammonia is synthesized at $700\,\text{K}$ and $200\,\text{atm}$ total according to the equilibrium:

$$\mathrm{N_2(g)} + 3\,\mathrm{H_2(g)} \rightleftharpoons 2\,\mathrm{NH_3(g)} \quad \Delta H^\circ = -92.2\,\text{kJ}\times\text{mol}^{-1}$$

The feed gas contains impurities of methane and argon from natural gas and air, which do not react. To avoid losses of reactants, the unreacted exit gases are continuously recycled to the reactor after condensing the $\mathrm{NH_3}$.

1. a) Deduce how the progressive accumulation of these inert gases in the closed recycling loop affects the partial pressure of the reactants and the molar yield of formation of $\mathrm{NH_3}$, assuming that the total mechanical pressure ($200\,\text{atm}$) and the temperature are kept fixed. Explain why it is essential to install a continuous gas purge.
2. b) The synthesis reaction releases a large amount of heat energy. If the reactor did not have an efficient cooling system and were allowed to operate adiabatically, explain how the temperature of the catalyst bed, the constant $K_p$ and the final conversion of reactants would vary in a coupled way.

In the gas-phase synthesis of methanol according to the equilibrium:

$$\mathrm{CO(g)} + 2\,\mathrm{H_2(g)} \rightleftharpoons \mathrm{CH_3OH(g)} \quad \Delta H^\circ = -90.7\,\text{kJ}\times\text{mol}^{-1}$$

a chemical engineering team monitors the methanol yield in a pilot reactor. In a first experiment at $500\,\text{K}$ and $50\,\text{atm}$, the mole fraction of $\mathrm{CH_3OH}$ at equilibrium is $x_1 = 0.22$.

Subsequently, the state of the system is modified simultaneously, operating at a new temperature $T_2 = 560\,\text{K}$ and a new total pressure $P_2 = 120\,\text{atm}$.

1. a) Analyze the opposing effects of the change in temperature and in pressure on the position of the equilibrium, stating in which direction each of these changes acts separately.
2. b) From the experimental value obtained under the second set of conditions, $x_2 = 0.18$, deduce which of the two effects has prevailed in the overall response of the system and justify formally whether the equilibrium constant $K_p$ at $560\,\text{K}$ is greater than, less than, or equal to that at $500\,\text{K}$.