Question

For the dissociation of \(\mathrm{CaCO}_{3}(\mathrm{s})\) at \(25^{\circ} \mathrm{C}, \mathrm{CaCO}_{3}(\mathrm{s})\) \(\rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g}) \Delta G^{\circ}=+131 \mathrm{kJ} \mathrm{mol}^{-1} .\) A sample of pure \(\mathrm{CaCO}_{3}(\mathrm{s})\) is placed in a flask and connected to an ultrahigh vacuum system capable of reducing the pressure to \(10^{-9} \mathrm{mmHg}\) (a) Would \(\mathrm{CO}_{2}(\mathrm{g})\) produced by the decomposition of \(\mathrm{CaCO}_{3}(\mathrm{s})\) at \(25^{\circ} \mathrm{C}\) be detectable in the vacuum system at \(25^{\circ} \mathrm{C} ?\) (b) What additional information do you need to determine \(P_{\mathrm{CO}_{2}}\) as a function of temperature? (c) With necessary data from Appendix D, determine the minimum temperature to which \(\mathrm{CaCO}_{3}(\mathrm{s})\) would have to be heated for \(\mathrm{CO}_{2}(\mathrm{g})\) to become detectable in the vacuum system.

Short answer

Detectability of CO2 from CaCO3 dissociation in the vacuum system at 25°C is unlikely due to positive change in Gibbs free energy. For determining the pressure of CO2 as a function of temperature, the enthalpy change (ΔH°) and entropy change (ΔS°) of the reaction are needed. The minimum temperature for detectable CO2 cannot be calculated without these values.

Step-by-step solution

Determine Whether CO2 would be Detectable

We first need to determine whether the CO2 produced by the decomposition of CaCO3 would be detectable in the vacuum system at 25°C. The change in Gibbs free energy under standard conditions, ΔG° (+131 kJ/mol), is positive, indicating that the reaction does not favor the formation of products at 25°C. Thus, the amount of CO2 formed at this temperature would be minimal and likely would not be detectable in the vacuum system.

Identify the Additional Information Needed

To find the partial pressure of CO2 as a function of temperature, we need the ΔH° and ΔS° values for the reaction. Using the ΔH° and ΔS° values, we can set up the equation ΔG = ΔH - TΔS and solve for the equilibrium constant K. We can then use the equation \n\[ p_{CO2} = K \] \n where p_{CO2} is the partial pressure of CO2, and K is the equilibrium constant.

Determine the Minimum Temperature

Given the ΔH° and ΔS° values from the Appendix D, we can solve for the temperature (T) that would produce an equilibrium constant of 1 using the equation ΔG = ΔH - TΔS, therefore we solve for T which gives \(T = \frac{\Delta H}{\Delta S}\). This would represent the minimum temperature at which CO2 would become detectable in the vacuum system. However, since we do not know the ΔH° and ΔS° values in this specific case, we cannot calculate the exact temperature.

Key concepts

Understanding Gibbs Free Energy

Gibbs free energy, represented by the symbol \( G \), plays a pivotal role in predicting the behavior of chemical reactions under constant temperature and pressure conditions. It is a thermodynamic quantity that combines the system's enthalpy (\( H \)), entropy (\( S \)), and temperature (\( T \)) to assess the spontaneity of a process.

The formula for Gibbs free energy is \( \Delta G = \Delta H - T\Delta S \), where \( \Delta G \) is the change in Gibbs free energy, \( \Delta H \) is the change in enthalpy, and \( \Delta S \) is the change in entropy. A negative value of \( \Delta G \) indicates that a reaction is spontaneous, which means it can occur without any external input of energy. Conversely, a positive \( \Delta G \) suggests that the reaction is non-spontaneous and requires additional energy to proceed.

For our textbook exercise discussing the dissociation of \( \text{CaCO}_{3}(\text{s}) \), the positive \( \Delta G \) value of +131 kJ/mol means that, at 25°C (room temperature), the reaction does not occur spontaneously and \( \text{CO}_{2}(\text{g}) \) would not be produced in significant amounts. Understanding this concept is essential for interpreting reaction feasibility and the extent of reaction completion in chemical processes.

Deciphering the Equilibrium Constant

The equilibrium constant, denoted as \( K \), is a dimensionless value that quantifies the ratio of concentrations of products to reactants at equilibrium for a particular reaction at a given temperature. It provides deep insights into the position of equilibrium and the extent of a reaction.

Mathematically, if a generic reaction is written as \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant \( K \) can be expressed as: \[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \] where the square brackets denote the molar concentrations of the reactants \( A \) and \( B \), and products \( C \) and \( D \), while \( a \), \( b \), \( c \), and \( d \) are the stoichiometric coefficients.

For reactions involving gases, partial pressures are used instead of concentrations, as seen in the solved problem where \( p_{CO2} \) represents the partial pressure of carbon dioxide. By using the equilibrium constant, we can predict the direction of a reaction and the possible concentrations or pressures of reactants and products at equilibrium. In thermodynamics, the equilibrium constant is directly related to the change in Gibbs free energy for the reaction through the relationship: \[ \Delta G = -RT \ln(K) \] where \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. This equation helps to understand the interplay between \( \Delta G \) and \( K \), as they both contribute to whether a reaction will proceed or not.

Exploring Thermodynamics of Reactions

Thermodynamics of reactions refers to the study of energy changes that accompany chemical processes. The central laws of thermodynamics govern these changes, outlining how the energy is transformed and transferred during a reaction.

Three primary thermodynamic quantities are often discussed in the context of chemical reactions: enthalpy (\( H \)), entropy (\( S \)), and Gibbs free energy (\( G \)). Enthalpy represents the heat content of a system, and changes in enthalpy (\( \Delta H \)) reflect the heat absorbed or released during a reaction. Entropy is a measure of disorder or randomness in a system, with increases in entropy (\( \Delta S \)) often associated with the spreading out of energy and matter.

Understanding how these quantities change during a reaction is crucial for predicting its course and outcome. For instance, knowing the values of \( \Delta H \) and \( \Delta S \) can help us solve for \( \Delta G \) and, in turn, determine the spontaneity of the reaction, as seen in step 2 of the textbook solution where we require these values to predict the partial pressure of \( CO_2 \) over temperature. When both are known, we can calculate the equilibrium constant, which tells us the extent to which a reaction will go under certain conditions, and in step 3, we could determine the minimum temperature for the detectability of \( CO_2 \) if we knew these values. This integrative approach highlights the importance of thermodynamics in understanding and controlling chemical reactions.