Problem 69 - Chemistry and Chemical Reactivity

Question

Decide whether each of the following statements is true or false. If false, change the wording to make it true. (a) The magnitude of the equilibrium constant is always independent of temperature. (b) When two chemical equations are added to give a net equation, the equilibrium constant for the net equation is the product of the equilibrium constants of the summed equations. (c) The equilibrium constant for a reaction has the same value as \(K\) for the reverse reaction. (d) Only the concentration of \(\mathrm{CO}_{2}\) appears in the equilibrium constant expression for the reaction \(\mathrm{CaCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g})\) (e) For the reaction \(\mathrm{CaCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{CaO}(\mathrm{s})+\) \(\mathrm{CO}_{2}(\mathrm{g}),\) the value of \(K\) is numerically the same, whether the amount of \(\mathrm{CO}_{2}\) is expressed as moles/liter or as gas pressure.

Step-by-Step Solution

Answer

(a) False, (b) True, (c) False, (d) True, (e) True.

Step 1: Assess Statement (a)

Statement (a) claims that the magnitude of the equilibrium constant is always independent of temperature. This statement is **false** because the equilibrium constant is dependent on temperature according to the van 't Hoff equation, which relates changes in temperature to changes in the equilibrium constant due to enthalpy changes. Thus, the equilibrium constant is not temperature-independent.

Step 2: Evaluate Statement (b)

Statement (b) asserts that when two chemical equations are added to give a net equation, the equilibrium constant for the net equation is the product of the equilibrium constants of the summed equations. This statement is **true**. When two reactions are added, their respective equilibrium constants multiply to give the equilibrium constant for the overall reaction.

Step 3: Consider Statement (c)

Statement (c) states that the equilibrium constant for a reaction has the same value as \(K\) for the reverse reaction. This statement is **false**. For the reverse reaction, the equilibrium constant is the reciprocal of the equilibrium constant of the forward reaction.

Step 4: Analyze Statement (d)

In Statement (d), it is claimed that only the concentration of \(\mathrm{CO}_{2}\) appears in the equilibrium constant expression for the reaction \(\mathrm{CaCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g})\). This statement is **true**. Equilibrium expressions do not include solids or liquids, only gases and aqueous species, hence only \(\mathrm{CO}_{2}\) appears.

Step 5: Analyze Statement (e)

Statement (e) claims that for the reaction \(\mathrm{CaCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g})\), the value of \(K\) is the same if \(\mathrm{CO}_{2}\) is expressed as concentration (moles/liter) or pressure. This statement is **true**, but with a clarification: the "concentration" used in the expression for gases in terms of \(K\) is typically partial pressure (related to concentration via the ideal gas law).

Key Concepts

Temperature Dependence of Equilibrium Constants

Equilibrium constants, usually denoted as \( K \), express the ratio of the concentrations of products to reactants at equilibrium. These constants are sensitive to temperature. When temperature changes, it can alter the enthalpy (\( \Delta H \)) of the reaction, affecting the equilibrium position. This causes \( K \) to vary, meaning the equilibrium constant is not independent of temperature.

A rise in temperature generally favors endothermic reactions (positive \( \Delta H \)), increasing \( K \). Conversely, exothermic reactions (negative \( \Delta H \)) typically have a decreased \( K \) at higher temperatures. Understanding how temperature affects these constants is crucial in predicting how reactions shift in different environments.

Understanding the van 't Hoff Equation

The van 't Hoff equation provides a quantitative relationship between equilibrium constants and temperature changes. It is represented as: \[ \ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]where:

• \( K_1 \) and \( K_2 \) are the equilibrium constants at temperatures \( T_1 \) and \( T_2 \).
• \( \Delta H \) is the change in enthalpy.
• \( R \) is the universal gas constant.

This equation shows how an increase or decrease in temperature will impact \( K \), based on the enthalpy change of the reaction. Practically, it can be used to estimate new equilibrium constants when a system experiences a temperature shift.

Reciprocal Equilibrium Constants

Every chemical reaction can be reversed. The equilibrium constant for the reverse reaction is the reciprocal of that for the forward reaction. Mathematically, if the equilibrium constant for a forward reaction is \( K \), then for the reverse, it is \( \frac{1}{K} \).

This concept helps predict reaction direction. A higher \( K \) for forward reactions indicates a favorable formation of products. However, a small \( K \) suggests the reaction favors reactants and might proceed better in the reverse. This reciprocal relationship underscores how dynamic and context-dependent chemical equilibria can be.

Solids in Equilibrium Expressions

In equilibrium expressions, the focus is typically on gaseous and aqueous reactants and products. Solids and liquids are excluded from these expressions due to their constant densities. As a result, their concentrations do not change with reaction progress.

For example, in the decomposition of calcium carbonate \( \text{CaCO}_3(s) \leftrightarrow \text{CaO}(s) + \text{CO}_2(g) \), only \( \text{CO}_2 \) appears in the equilibrium constant expression since both \( \text{CaCO}_3\) and \( \text{CaO} \) are solids. This simplification is critical for accurately calculating \( K \) based solely on variable properties, namely gases and aqueous solutions.

Partial Pressure vs. Concentration

When evaluating gaseous equilibria, partial pressures are often used instead of concentrations. This choice is because gases do not have fixed volumes like solids or liquids, and their concentrations can be easily correlated with their partial pressures using the ideal gas law: \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is moles of gas, \( R \) is the gas constant, and \( T \) is temperature.

For reactions involving gases, the equilibrium constant \( K \) can be expressed in terms of pressure, called \( K_p \), or in terms of concentration, \( K_c \). These parameters, although related, offer different views of the same system. Typically, chemical problems clarify how \( K \) should be expressed, but knowing the relationship helps switch between pressure and concentration effortlessly.