Problem 69 - Chemistry

Question

Consider the following reaction at equilibrium: $$\mathrm{A}(g) \rightleftharpoons 2 \mathrm{~B}(g)$$ 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? $$ \begin{array}{clr} \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & {[\mathrm{A}](M)} & {[\mathrm{B}](M)} \\ 200 & 0.0125 & 0.843 \\ 300 & 0.171 & 0.764 \\ 400 & 0.250 & 0.724 \end{array} $$

Step-by-Step Solution

Answer

After calculating the equilibrium constants, observe their temperature dependence to determine if the reaction is endothermic or exothermic. In an endothermic reaction, \( K_{\mathrm{c}} \) increases as the temperature increases, while in an exothermic reaction, \( K_{\mathrm{c}} \) decreases as the temperature increases.

Step 1: Understanding the Equilibrium Constants

For any chemical reaction, the equilibrium constant refers to the ratio of concentrations of the products to the reactants, each raised to the power equals to the stoichiometric coefficients at equilibrium. The equilibrium constant can be calculated using concentrations, denoted as \(K_{\mathrm{c}}\), or partial pressures, denoted as \(K_{P}\). In this exercise, the given reaction is: \(\mathrm{A}(g) \rightleftharpoons 2 \mathrm{~B}(g)\), with each compound's concentrations given at different temperatures.

Step 2: Calculate the Equilibrium Constants

Starting with the equilibrium constant in terms of concentration, we have \[ K_{\mathrm{c}} = \frac{[\mathrm{B}]^2}{[\mathrm{A}]} \] We substitute the concentrations at each given temperature into the equation and calculate \( K_{\mathrm{c}} \) for each temperature. Equilibrium constant in terms of pressure, \( K_{\mathrm{P}} \), is not relevant in this case as we deal with concentrations and not pressures.

Step 3: Determine if the Reaction is Endothermic or Exothermic

From the equilibrium constants calculated, if \( K_{\mathrm{c}} \) increases with the temperature, then the reaction is endothermic, since adding heat shifts the equilibrium to the right (to the products), resulting in a larger value of \( K_{\mathrm{c}} \). If the \( K_{\mathrm{c}} \) value decreases with temperature, then the reaction is exothermic, because adding heat shifts the equilibrium to the left (the reactants), leading to a smaller value of \( K_{\mathrm{c}} \).

Key Concepts

Endothermic and Exothermic Reactions

Let's dive into the fascinating world of chemical reactions and explore how we can pick out endothermic from exothermic reactions. These terms describe how energy is transferred in a reaction.

• Endothermic Reactions: These absorb heat from the surroundings. For the reaction in the question, if the equilibrium constant ( K_{c} ) increases as the temperature rises, it means that the system is absorbing energy for reactants to transform into products. This hints towards an endothermic process.
• Exothermic Reactions: These release heat into the surroundings. In this case, a decrease in the equilibrium constant with increasing temperature would suggest the release of heat and a shift towards the reactants, indicating an exothermic reaction.

Understanding whether a reaction is endothermic or exothermic helps to explain reaction conditions and can guide us in controlling reactions for industrial processes or laboratory experiments.

Chemical Equilibrium

Chemical equilibrium is a crucial concept in chemistry that describes the state of a reversible reaction where the rates of the forward and reverse reactions are equal. This means the concentrations of reactants and products remain constant over time.
For our reaction A(g) \rightleftharpoons 2B(g), equilibrium is depicted by calculating the equilibrium constant (K_{c}). This value is derived from the ratio of product concentrations to reactant concentrations, raised to their stoichiometric coefficients. At equilibrium:
\[ K_{c} = \frac{[B]^2}{[A]} \]
We plug in the concentration values at varied temperatures to find the equilibrium constant at those temperatures. By assessing these values, we can determine how shifts in conditions such as temperature affect the system, helping us predict the behavior of the reaction under new conditions.

Reaction Kinetics

Reaction kinetics focuses on the rate at which reactants are converted into products. It is essential to understand how fast a reaction proceeds and what influences its rate. Even with equilibrium being a state of equal forward and reverse reaction rates, knowing how quickly equilibrium is reached is vital.
Factors influencing reaction kinetics include:

• Temperature: Generally, increasing temperature speeds up reactions since molecules have more energy and collide more frequently. However, for equilibrium constants, the effect of temperature varies based on whether a reaction is endothermic or exothermic.
• Concentration: The concentrations of reactants can affect how quickly products are formed until equilibrium is reached.
• Presence of Catalysts: Catalysts can lower the activation energy required for a reaction, increasing its rate without altering the equilibrium position.

Understanding kinetics helps chemists to maximize efficiency and control the speed of reactions, which is pivotal in both research and industrial processes.