Question

Consider the following reaction at equilibrium: $$\mathrm{A}(g) \rightleftarrows 2 \mathrm{~B}(g)$$ From the data shown here, calculate the equilibrium constant (both \(K_{P}\) and \(K_{\mathrm{c}}\) ) at each temperature. Is the reaction endothermic or exothermic? $$ \begin{array}{clc} \text { Temperature }\left({ }^{\circ} \mathbf{C}\right) & {[\mathbf{A}](\boldsymbol{M})} & {[\mathbf{B}](\boldsymbol{M})} \\ \hline 200 & 0.0125 & 0.843 \\ 300 & 0.171 & 0.764 \\ 400 & 0.250 & 0.724 \end{array} $$

Short answer

\(K_c\) decreases with temperature; thus, the reaction is exothermic.

Step-by-step solution

Understand the Reaction

The reaction given is \(\mathrm{A}(g) \rightleftarrows 2 \mathrm{B}(g)\). For this reaction, we'll use the concentrations of A and B to calculate the equilibrium constants \(K_c\) and \(K_P\).

Calculate Equilibrium Constant \(K_c\)

We know the equilibrium constant for concentration (\(K_c\)) is given by: \[ K_c = \frac{[B]^2}{[A]} \] Calculate \(K_c\) for each temperature using the concentrations provided. For 200°C, \[ K_c = \frac{(0.843)^2}{0.0125} = 56.8 \] For 300°C, \[ K_c = \frac{(0.764)^2}{0.171} = 3.41 \] For 400°C, \[ K_c = \frac{(0.724)^2}{0.25} = 2.1 \]

Calculate Equilibrium Constant \(K_P\)

The equilibrium constant for pressure \(K_P\) is related to \(K_c\) by the relation: \[ K_P = K_c(RT)^{\Delta n} \] where \(\Delta n\) is the change in moles of gas (products \(-\) reactants), \(R\) is the gas constant \(0.0821 \frac{L \cdot atm}{mol \cdot K}\), and \(T\) is the temperature in Kelvin. Here, \(\Delta n = 1\). Convert each temperature from Celsius to Kelvin by adding 273.15. For 200°C (473.15 K): \[ K_P = 56.8 \times (0.0821 \times 473.15)^1 = 2198.7 \] For 300°C (573.15 K): \[ K_P = 3.41 \times (0.0821 \times 573.15)^1 = 160.2 \] For 400°C (673.15 K): \[ K_P = 2.1 \times (0.0821 \times 673.15)^1 = 116.8 \]

Determine Reaction Endothermic or Exothermic

Examine how \(K_c\) changes with temperature to determine whether the reaction is endothermic or exothermic. \(K_c\) decreases with increasing temperature, suggesting the reaction is exothermic since increasing temperature drives the reaction towards reactants.

Key concepts

Understanding Endothermic and Exothermic Reactions

In chemical reactions, the terms endothermic and exothermic refer to how energy, mainly in the form of heat, is involved in a reaction process. These differences are key to predicting how reactions behave under various conditions.
Endothermic reactions absorb energy from the surroundings, meaning they need energy to proceed. As a result, the products of an endothermic reaction have more energy than the reactants. A common everyday example is the melting of ice, which requires heat to convert it into water.
An exothermic reaction, on the other hand, releases energy in the form of heat. This release occurs because the products contain less energy than the reactants. Combustion is a typical exothermic process, where burning wood releases heat and light energy.

• For endothermic reactions: Heat is absorbed; products > reactants in energy.
• For exothermic reactions: Heat is released; products < reactants in energy.

This distinction not only helps describe reactions but also aids in predicting how temperature changes influence reaction direction, as seen in the given exercise.

Overview of Reaction Equilibrium

In chemistry, equilibrium refers to a state where both the forward and reverse reactions occur at the same rates, resulting in no net change in the concentrations of reactants and products. At this point, the reaction mixture is said to be at dynamic equilibrium.
For the given reaction \[\mathrm{A}(g) \rightleftarrows 2 \mathrm{B}(g)\]we can express the equilibrium condition using the equilibrium constant, commonly represented by \(K_c\) for concentration. This constant is derived from the ratio of the concentrations of products to reactants at equilibrium.

• Equilibrium is reversible: forward and backward reaction rates equalize.
• Constant concentration ratios characterize the equilibrium state.

Analyzing the equilibrium constant provides insight into the relative amounts of products and reactants at equilibrium, which further implies how favorably a reaction proceeds.

Influence of Temperature on Reaction Equilibrium

The effect of temperature on equilibrium is a fascinating subject that reveals much about the nature of the reaction. According to Le Chatelier's Principle, when a system at equilibrium experiences a change in temperature, the equilibrium position shifts in a direction that counteracts the change.
In the case of exothermic reactions, like the one in our exercise, increasing temperature typically causes the equilibrium to favor the formation of reactants. Therefore, the equilibrium constant \(K_c\) decreases with rising temperature. This shift occurs because adding heat to an exothermic reaction provides energy that encourages the reverse reaction.
Conversely, in endothermic reactions, an increase in temperature shifts the equilibrium toward the products as the system absorbs additional heat to promote the forward reaction.

• Exothermic: Increase in temperature – equilibrium favors reactants.
• Endothermic: Increase in temperature – equilibrium favors products.

Understanding these shifts helps predict how temperature variations impact product and reactant concentrations, which is crucial for practical applications in industrial processes.