Problem 32 - Chemistry in 30 days

Question

In the conversion of lime stone to lime, \(\mathrm{CaCO}_{3}(\mathrm{~s}) \rightarrow \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{~g})\) The values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are \(+179.1 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(160.2 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\), respectively at 298 \(\mathrm{K}\) and 1 bar. Assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not change with temperature; temperature above which conversion of lime stone to lime will be spontaneous is (a) \(1118 \mathrm{~K}\) (b) \(1008 \mathrm{~K}\) (c) \(1200 \mathrm{~K}\) (d) \(845 \mathrm{~K}\)

Step-by-Step Solution

Answer

The reaction is spontaneous above 1118 K, which is option (a).

Step 1: Understanding the Reaction

The given reaction is the decomposition of limestone \( \text{CaCO}_3(s) \) into lime \( \text{CaO}(s) \) and carbon dioxide gas \( \text{CO}_2(g) \). We are asked to determine the temperature at which this reaction becomes spontaneous.

Step 2: Understanding Spontaneity

A reaction becomes spontaneous when the Gibbs free energy change \( \Delta G \) is negative. The Gibbs free energy change is given by the equation: \[ \Delta G = \Delta H - T\Delta S \] where \( \Delta H \) is enthalpy change, \( T \) is temperature in Kelvin, and \( \Delta S \) is entropy change.

Step 3: Solving for Temperature

To find the temperature at which the reaction is spontaneous, set \( \Delta G < 0 \), thus: \[ 0 > \Delta H - T\Delta S \] Solving for \( T \): \[ T > \frac{\Delta H}{\Delta S} \] Plug in the given values for \( \Delta H = 179.1 \text{ kJ mol}^{-1} \) and \( \Delta S = 160.2 \text{ J K}^{-1} \text{ mol}^{-1} \). Note that \( \Delta H \) should be converted to joules: \( 179.1 \text{ kJ mol}^{-1} = 179100 \text{ J mol}^{-1} \).

Step 4: Calculating the Temperature

Use the formula: \[ T > \frac{179100}{160.2} \] Calculating this gives: \[ T > 1118.1 \text{ K} \] Thus, the reaction becomes spontaneous above 1118 K.

Step 5: Conclusion

Comparing the calculation with provided options, the temperature above which the conversion of limestone to lime is spontaneous is 1118 K, which corresponds to option (a).

Key Concepts

Gibbs Free Energy

The concept of Gibbs Free Energy is crucial to understanding chemical reactions and their spontaneity. Gibbs Free Energy, denoted as \( \Delta G \), combines the system's enthalpy, temperature, and entropy into one equation: \[ \Delta G = \Delta H - T \Delta S \].

• \( \Delta G \) provides insights into whether a reaction will occur without external input.
• If \( \Delta G \) is negative, the reaction is spontaneous and can proceed on its own.
• If \( \Delta G \) is positive, the reaction requires energy input to take place.
• When \( \Delta G \) is zero, the system is at equilibrium and no net reaction occurs.

Therefore, calculating \( \Delta G \) allows us to predict the feasibility of chemical processes under specific conditions.

Enthalpy Change

Enthalpy change, symbolized as \( \Delta H \), represents the heat absorbed or released during a reaction at constant pressure. It provides insights into the nature of chemical reactions:

• Positive \( \Delta H \) indicates an endothermic reaction, where heat is absorbed from the surroundings.
• Negative \( \Delta H \) signifies an exothermic reaction, where heat is released into the surroundings.

In the limestone to lime reaction, \( \Delta H = +179.1 \text{ kJ mol}^{-1} \) shows it is endothermic. This heat absorption is critical to driving the reaction forward, requiring higher temperatures to promote spontaneity.

Entropy Change

Entropy change, denoted as \( \Delta S \), is a measure of disorder or randomness in a system. Every system tends to move towards increased entropy, influencing reaction spontaneity.

• A positive \( \Delta S \) indicates increased disorder, favoring the reaction's spontaneity.
• A negative \( \Delta S \) suggests decreased disorder, possibly counteracting reaction progression.

For the conversion of limestone to lime, \( \Delta S = 160.2 \text{ J K}^{-1} \text{ mol}^{-1} \) is positive, as the gas formation increases disorder. This factor works alongside the enthalpy in determining the spontaneity at specific temperatures.

Spontaneity of Reaction

The spontaneity of a reaction is dictated by the Gibbs Free Energy change (\( \Delta G \)). For a reaction to be spontaneous, it must result in a decrease in free energy, i.e., \( \Delta G < 0 \).By rearranging the Gibbs Free Energy equation: \[ \Delta G = \Delta H - T \Delta S \],

• Set \( \Delta G < 0 \) to predict spontaneity.
• This means: \[ 0 > \Delta H - T \Delta S \]
• Solving for temperature gives the condition: \[ T > \frac{\Delta H}{\Delta S} \]

Applying this to our reaction helps us determine the temperature above which the reaction becomes spontaneous. In this case, the calculation identified 1118 K as the threshold, showing that higher temperatures are essential for spontaneity in endothermic reactions like the limestone to lime conversion.