Problem 104 - Chemistry The Molecular Science

Question

For the reaction cis 2 -butene \(\rightleftharpoons\) trans-2-butene \(K_{\mathrm{c}}\) is 1.65 at \(500 . \mathrm{K}, 1.47\) at \(600 . \mathrm{K},\) and 1.36 at \(700 . \mathrm{K}\) Predict whether the conversion from the cis to the trans isomer of 2 -butene is exothermic or endothermic.

Step-by-Step Solution

Answer

The conversion is exothermic.

Step 1: Definition of Equalibrium Constant and Dependence on Temperature

The equilibrium constant \(K_{\mathrm{c}}\) is a measure of the extent of a reaction at equilibrium. Its value depends on the temperature. Generally, if \(K_{\mathrm{c}}\) decreases with an increase in temperature, the reaction is exothermic. Conversely, if \(K_{\mathrm{c}}\) increases with an increase in temperature, the reaction is endothermic.

Step 2: Analysis of Given \(K_{\mathrm{c}}\) Values

We are given \(K_{\mathrm{c}} = 1.65\) at \(500\ \mathrm{K}\), \(K_{\mathrm{c}} = 1.47\) at \(600\ \mathrm{K}\), and \(K_{\mathrm{c}} = 1.36\) at \(700\ \mathrm{K}\). Observing these values, \(K_{\mathrm{c}}\) decreases as the temperature increases.

Step 3: Conclusion Based on \(K_{\mathrm{c}}\) and Temperature Change

Since the equilibrium constant \(K_{\mathrm{c}}\) decreases with an increase in temperature, it indicates that the formation of the trans isomer from the cis isomer is exothermic. The decrease in \(K_{\mathrm{c}}\) suggests that the reaction releases heat.

Key Concepts

Equilibrium Constant

The equilibrium constant, often symbolized as \(K_c\), is a crucial number that tells us how far a reaction goes toward completion before equilibrium is achieved. It provides a snapshot of the ratio of product concentrations to reactant concentrations, each raised to the power of their coefficients in the balanced chemical equation, when the reaction has reached equilibrium.
In simpler terms, the value of \(K_c\) helps us understand how much of either side of the reaction predominates when equilibrium is established. A large \(K_c\) means more products are present, while a small \(K_c\) means more reactants remain. For reversible reactions, changes in the conditions such as temperature can lead to changes in \(K_c\), which reflects how the position of equilibrium shifts.
The relationship between \(K_c\) and temperature can reveal whether a reaction is endothermic or exothermic. This relationship is based on the Le Chatelier's Principle, where a system at equilibrium will adjust to counteract disturbances such as changes in temperature.

Endothermic Reaction

An endothermic reaction is one that absorbs heat from its surroundings. Essentially, in these reactions, energy is required to convert reactants to products. This uptake of energy can usually be observed as a cooling effect in the surroundings, as the system draws in heat.
With endothermic reactions, an increase in temperature tends to increase the equilibrium constant \(K_c\). This is because adding heat shifts the equilibrium position in favor of the product formation, effectively making the reaction proceed more to the right. This aligns with Le Chatelier's Principle, as the reaction compensates for the extra heat by absorbing more of it, thus favoring the forward reaction.
Common examples of endothermic reactions include photosynthesis in plants and the thermal decomposition of calcium carbonate. In both cases, heat is an essential ingredient for the reaction to proceed.

Exothermic Reaction

An exothermic reaction, in contrast, releases heat into its surroundings. These reactions often occur spontaneously and result in a temperature rise of the system's environment due to the release of energy.
For exothermic reactions, when the temperature increases, the equilibrium constant \(K_c\) typically decreases. According to Le Chatelier's Principle, immediately when extra heat is added to the system, the reaction will shift to diminish the impact of the added heat. Thus, the equilibrium favors the reactant side, and as a result, \(K_c\) will lower, suggesting a reduction in the formation of products.
A common exothermic reaction example is the combustion of a candle, where heat and light are released as the wax burns. Therefore, understanding these temperature effects on equilibrium constants allows us to anticipate how chemical reactions behave under varying conditions.