Question

Aceticacid can be manufactured by combining methanol with carbon monoxide, an example of a carbonylation reaction: $$ \mathrm{CH}_{3} \mathrm{OH}(l)+\mathrm{CO}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{COOH}(l) $$ (a) Calculate the equilibrium constant for the reaction at \(25^{\circ} \mathrm{C}\). (b) Industrially, this reaction is run at temperatures above \(25^{\circ} \mathrm{C}\). Will an increase in temperature produce an increase or decrease in the mole fraction of acetic acid at equilibrium? Why are elevated temperatures used? (c) At what temperature will this reaction have an equilibrium constant equal to \(1 ?\) (You may assume that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are temperature independent, and you may ignore any phase changes that might occur.)

Short answer

(a) Calculate the equilibrium constant at \(25^{\circ} \mathrm{C}\) using the given \(\Delta H^{\circ}_{298}\) and \(\Delta S^{\circ}_{298}\) values: \(K_{298}=e^{-\frac{\Delta G^{\circ}_{298}}{RT}}\) (b) If the \(\Delta H^{\circ}\) is positive (endothermic reaction), an increase in temperature will increase the equilibrium constant, leading to a higher mole fraction of acetic acid at equilibrium. Elevated temperatures are used because they provide the energy needed for the reaction to proceed and increase the product yield. (c) To find the temperature at which the equilibrium constant equals 1, use the van't Hoff equation: \(T=\frac{\Delta H^{\circ}}{\Delta S^{\circ}}\) Convert temperature to degrees Celsius if necessary.

Step-by-step solution

(a) Calculate the equilibrium constant at \(25^{\circ} \mathrm{C}\).

With the values of \(\Delta H^{\circ}_{298}\) and \(\Delta S^{\circ}_{298}\), you can calculate the equilibrium constant by plugging them into the equations above and solving for \(K_{298}\): \(K_{298}=e^{-\frac{\Delta G^{\circ}_{298}}{RT}}\) Step 3: Analyze the effect of an increase in temperature on the equilibrium constant. The van't Hoff equation relates the change in equilibrium constant with temperature: \[ \frac{d \ln K}{d T}=\frac{\Delta H^{\circ}}{R T^{2}} \] To determine how increasing the temperature affects the equilibrium constant, we can analyze the van't Hoff equation's sign by considering the sign of \(\Delta H^{\circ}\).

(b) Determine the effect of increased temperatures

If the \(\Delta H^{\circ}\) is negative (exothermic reaction), then an increase in temperature will decrease the equilibrium constant. If the \(\Delta H^{\circ}\) is positive (endothermic reaction), then an increase in temperature will increase the equilibrium constant. Elevated temperatures can provide the necessary energy for the reaction to proceed and increase the yield of the product. Step 4: Calculate the temperature at which the equilibrium constant equals 1. We can use the van't Hoff equation to find the temperature at which the equilibrium constant is equal to 1: \[ \ln K=\frac{-\Delta H^{\circ}}{RT}+\frac{\Delta S^{\circ}}{R} \]

(c) Calculate the temperature for \(K=1\)

To find the temperature at which the equilibrium constant equals 1, set \(\ln K = 0\) and solve for \(T\): \(T=\frac{\Delta H^{\circ}}{\Delta S^{\circ}}\) Make sure to convert the temperature back to degrees Celsius if necessary.

Key concepts

Understanding Carbonylation Reactions

Carbonylation reactions are a class of organic chemistry processes where a carbon monoxide (CO) molecule is added to a substrate. These reactions are not simply a matter of mixing two reactants; they often require specific conditions to proceed effectively and are catalyzed by metals to ensure a higher yield and rate.

In the scenario of producing acetic acid (CH_3COOH), methanol (CH_3OH) reacts with carbon monoxide in a carbonylation reaction. This particular process is widely used in industry due to its efficiency and the high importance of acetic acid as a chemical building block. The reaction proceeds as follows:
\[ \text{CH}_3 \text{OH}(l) + \text{CO}(g) \rightarrow \text{CH}_3\text{COOH}(l) \]
The equilibrium constant (K) at a certain temperature quantifies the ratio of the product's concentration to the reactants'. For the reaction to be productive in an industrial setting, understanding and optimising the conditions that favour the formation of the desired acetic acid is essential.

The van't Hoff Equation and its Role

The van't Hoff equation forms the core of understanding how temperature influences the equilibrium constant (K) of a chemical reaction. Given by the expression:
\[ \frac{d \text{ln} K}{d T}=\frac{\text{Δ} H^{\text{∘}}}{R T^{2}} \] it showcases the relationship between the temperature (T), the equilibrium constant (K), and the enthalpy change of the reaction (\text{Δ} H^{\text{∘}}), R being the ideal gas constant.

In practical terms, if a reaction is exothermic (\text{Δ} H^{\text{∘}} < 0), increasing the temperature typically leads to a decrease in the equilibrium constant; the system adjusts by favouring the reactants to absorb the added heat. Conversely, for endothermic reactions (\text{Δ} H^{\text{∘}} > 0), a temperature rise will generally lead to an increase in K, favouring formation of more products. This relationship is crucial in adjusting the conditions under which a reaction is carried out to optimize yield, especially in industrial processes like the carbonylation of methanol.

Le Chatelier's Principle in Action

Le Chatelier's principle is a guiding concept in chemistry that helps predict the effect of a change in conditions on a chemical equilibrium. It states that if a dynamic equilibrium is disturbed by changing the conditions, such as temperature, pressure, or concentration, the position of equilibrium will shift to counteract the change.

Applying Le Chatelier's principle to the carbonylation of methanol, if the temperature is increased, the principle suggests that the equilibrium will shift in the direction that absorbs heat, which, for exothermic reactions, is towards the reactants. This explains why, despite a higher temperature often leading to decreased equilibrium constant for exothermic reactions, it is still used industrially: the rate of reaction increases, leading to faster production, even if the position of equilibrium is less favourable.

In summary, using Le Chatelier's principle, chemists and engineers are able to fine-tune reaction conditions to obtain the best possible yield of desired products, utilizing shifts in equilibrium to their advantage.