Question

A gaseous mixture consisting of \(1.1 \mathrm{mmol} \mathrm{SO}_{2}\) and \(2.2 \mathrm{mmol}\) \(\mathrm{O}_{2}\) in a \(250-\mathrm{mL}\) container was heated to \(500 . \mathrm{K}\) and allowed to reach equilibrium. Will more sulfur trioxide be formed if that equilibrium mixture is cooled to \(298 \mathrm{~K}\) ? For the reaction \(2 \mathrm{SO}_{2}(\mathrm{~g})+\) \(\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g}), K=2.5 \times 10^{10}\) at \(500 . \mathrm{K}\) and \(4.0 \times 10^{24}\) at \(298 \mathrm{~K}\).

Short answer

Yes, more sulfur trioxide will be formed if the equilibrium mixture is cooled to 298 K, because the equilibrium constant at 298 K is significantly larger, indicating a shift to the right in the reaction, favoring the formation of SO3.

Step-by-step solution

- Identify the Reaction Type

Examine the balanced chemical equation to identify the type of reaction. For the reaction\(\[\begin{equation}2 \text{SO}_{2}(g) + \text{O}_{2}(g) \rightleftharpoons 2 \text{SO}_{3}(g) \end{equation}\]\), it is a synthesis reaction where sulfur dioxide and oxygen combine to form sulfur trioxide.

- Determine the Direction of Equilibrium Shift

To determine the direction in which the equilibrium will shift when the temperature is changed, consider Le Chatelier's Principle. According to the principle, if a system at equilibrium is disturbed, the system will adjust to counteract the disturbance. In this case, cooling the system will remove heat. If the reaction is exothermic (releases heat), the system will shift to the right (forward direction) to produce more heat, which means producing more \(SO_3\). If the reaction is endothermic (absorbs heat), the system will shift to the left (reverse direction) upon cooling.

- Analyze the Equilibrium Constants

Compare the equilibrium constants (\(K\)) at both temperatures. A larger \(K\) value indicates a higher concentration of products at equilibrium. Here, the value of \(K\) at 298 K (\(4.0 \times 10^{24}\)) is much larger than at 500 K (\(2.5 \times 10^{10}\)). This indicates that at 298 K, the equilibrium position favors the formation of more \(SO_3\) compared to at 500 K.

- Conclude the Effect of Cooling

Since cooling the system leads to a higher equilibrium constant at 298 K, it can be concluded that more sulfur trioxide (\(SO_3\)) will be formed when the temperature is decreased to 298 K. The system shifts to the right, favoring the formation of products.

Key concepts

Le Chatelier's Principle

When studying chemical reactions, one of the most fundamental concepts is Le Chatelier's Principle. This principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. It's an essential tool for predicting how different changes in conditions such as pressure, temperature, and concentration will affect the outcome of a reaction at equilibrium.

Imagine a seesaw in balance; Le Chatelier's principle is akin to predicting how the seesaw will tilt if extra weight is added or removed from one side. For reactions, it means if you increase the concentration of reactants, the system shifts to make more products. Conversely, if you remove products, the system will also produce more to balance the loss. In the context of temperature, for a reaction that releases heat (exothermic), cooling the system would lead to the formation of more products, as the system seeks to generate the heat that was removed. On the other hand, for a reaction that absorbs heat (endothermic), an increase in temperature would cause the system to produce more products.

• If pressure is increased, the system shifts towards the side with fewer moles of gas.
• If a catalyst is added, the equilibrium is not affected; instead, it helps the system reach equilibrium faster.
• When a reactant or product is added or removed, the system will shift to rebalance the equilibrium concentration.

This principle is incredibly useful when trying to manipulate the yields of industrial chemical reactions.

Equilibrium Constant

The equilibrium constant (often abbreviated as K) is a number that expresses the ratio of the concentrations of products to reactants at equilibrium for a particular reaction at a constant temperature. The expression for the equilibrium constant depends on the stoichiometry of the reaction and is a way of quantifying the position of equilibrium.

Mathematically, for a generic reaction where \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant expression is \(K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\), where the concentrations \( [A], [B], [C], [D] \) are those at equilibrium, and \(^a, ^b, ^c, ^d\) are the reaction stoichiometries.

The value of \(K\) offers insight into the reaction mixture at equilibrium:

• If \(K >> 1\), the reaction favors the formation of products.
• If \(K << 1\), the reaction favors the reactants.
• If \(K \approx 1\), neither reactants nor products are favored, and the reaction mixture contains comparable amounts of both.

The equilibrium constant is temperature-dependent, meaning that it will change if the temperature of the system changes. This is a direct reflection of the reaction's sensitivity to temperature and has practical implications in understanding how different temperatures affect the composition of the system at equilibrium.

Synthesis Reaction

Within the realm of chemical reactions, a synthesis reaction is a type of reaction where multiple reactants combine to form a single product. This kind of reaction is also frequently called a direct combination reaction because it involves substances directly combining to make something new. The general form of a synthesis reaction is \(A + B \rightarrow AB\).

These reactions are among the simplest and most straightforward in chemistry, as they often involve only a few reactants and one main product. In synthesis reactions, atoms or molecules bond together, which requires energy to initiate the reaction. The bonding can involve a variety of interactions including ionic, covalent, or metallic bonds depending on the nature of the reactants.

Examples of synthesis reactions include the formation of water from hydrogen and oxygen gas \(2H_2 + O_2 \rightarrow 2H_2O\), and the formation of sodium chloride (table salt) from sodium and chlorine gas \(2Na + Cl_2 \rightarrow 2NaCl\). These reactions are fundamentally important, not just in laboratory chemistry, but also in industrial processes that synthesize products ranging from pharmaceuticals to construction materials. Understanding the mechanism of these reactions is crucial for optimizing product yields and designing effective chemical processing methods.