Question

In the uncatalyzed reaction $$ \mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g) $$ at \(100^{\circ} \mathrm{C}\) the pressures of the gases at equilibrium are \(P_{\mathrm{N}_{2} \mathrm{O}_{4}}=0.377 \mathrm{~atm}\) and \(P_{\mathrm{NO}_{2}}=1.56 \mathrm{~atm} .\) What would happen to these pressures if a catalyst were present?

Short answer

Introduction of a catalyst won't affect the equilibrium pressures. So, the pressures at equilibrium remain \(P_{\mathrm{N}_{2} \mathrm{O}_{4}}=0.377 \mathrm{~atm}\) and \(P_{\mathrm{NO}_{2}}=1.56 \mathrm{~atm}\).

Step-by-step solution

Understanding Catalysts

Catalysts are substances that increase the rate of a reaction by lowering the activation energy, but are not consumed in the reaction. They work by providing an alternative reaction pathway with a lower activation energy. However, catalysts do not affect the equilibrium state of the reaction.

Analyzing Initial Conditions

The initial pressures of the gases at equilibrium are given by \(P_{\mathrm{N}_{2} \mathrm{O}_{4}}=0.377 \mathrm{~atm}\) and \(P_{\mathrm{NO}_{2}}=1.56 \mathrm{~atm} \). At this point, the rates of forward and reverse reactions are equal.

Implication of a Catalyst on Pressures

As mentioned in Step 1, a catalyst will speed up the rate of both the forward and reverse reactions equally, bringing the system to equilibrium more swiftly. However, it will not change the equilibrium state itself. Therefore, the pressures at equilibrium will remain the same, hence \(P_{\mathrm{N}_{2} \mathrm{O}_{4}}=0.377 \mathrm{~atm}\) and \(P_{\mathrm{NO}_{2}}=1.56 \mathrm{~atm} \) after adding a catalyst.

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs in a reversible reaction when the rate of the forward reaction equals the rate of the reverse reaction. At this point, the concentrations or pressures of the reactants and products remain constant over time.

Understanding chemical equilibrium is crucial because it gives insight into how a reaction proceeds and when it 'settles down' to a state of balance. For instance, in the reaction \( \mathrm{N_2O_4}(g) \rightleftharpoons 2 \mathrm{NO_2}(g) \) at equilibrium, the pressures of the reactants and products are stationary, which means the conversion of \( \mathrm{N_2O_4} \) to \( \mathrm{NO_2} \) occurs at the same rate as the reverse process.

It's essential to realize that this dynamic balance does not mean that the chemicals stop reacting, but rather that their concentrations remain unchanged due to the equal rates of both reactions.

Catalysis in Chemical Reactions

Catalysis plays a pivotal role in modifying reaction rates without altering the equilibrium of the chemical system. A catalyst is a substance that increases the rate at which a reaction reaches equilibrium by providing an alternative pathway with a lower activation energy.

It's a common misconception that catalysts affect the position of equilibrium. In reality, a catalyst does not have any effect on the equilibrium constants or pressures. It simply ensures that equilibrium is achieved more rapidly. This is particularly useful in industrial processes where time is a critical factor for productivity. Whether the reaction involves gases, liquids, or solids, a catalyst provides a unique advantage by hastening the reaction rate while remaining chemically unchanged itself.

Equilibrium Constant and Pressure

The equilibrium constant (\( K \) or \( K_{eq} \) for gases) is a measure that describes the ratio of products to reactants at chemical equilibrium for a particular temperature. For the reaction mentioned, \( K_{eq} \) would be calculated using the partial pressures of the gases involved.

However, it's crucial to understand that while a change in pressure will influence the rates at which equilibrium is achieved, it does not affect the value of the equilibrium constant, as long as the temperature remains constant. Any shifts in pressure will initially disturb the system, causing it to adjust and re-establish equilibrium through Le Chatelier's Principle, without altering the \( K_{eq} \) value. To put it plainly, altering the pressure of the system can lead to temporary changes in the reactant and product concentrations, but at equilibrium, the pressures will settle to values that yield the same \( K_{eq} \) as before.

Activation Energy

Activation energy is the minimum amount of energy required to initiate a chemical reaction. It is a barrier that reactants must overcome to transform into products.

The significance of activation energy lies in its ability to determine the rate at which a reaction will proceed. A higher activation energy means fewer molecules have enough energy to react when they collide, resulting in a slower reaction. Conversely, a lower activation energy, such as one provided by the presence of a catalyst, allows more reactants to achieve the transition state, thereby speeding up the reaction.

While catalysts lower the activation energy, it is vital to note that their presence does not alter the energy of the reactants or the products. Hence, the overall energy change for the reaction remains the same, ensuring the equilibrium position is unaffected by the catalyst.