Problem 15 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Calcium carbonate \(\left(\mathrm{CaCO}_{3}\right)\) decomposes to form \(\mathrm{CaO}\) and \(\mathrm{CO}_{2}\) with \(\Delta_{\mathrm{r}} \mathrm{H}_{29 \mathrm{g}}=+178 \mathrm{kJ} \mathrm{mol}^{-1}\) and \(\Delta_{r} \mathrm{S}_{2 \mathrm{SQ}}=+161 \mathrm{JK}^{-1} \mathrm{mol}^{-1}\) Estimate the temperature at which the decomposition becomes spontaneous. (Section \(14.6)\)

Step-by-Step Solution

Answer

The decomposition becomes spontaneous at approximately 1107 K.

Step 1: Understand the Gibbs Free Energy Formula

To determine the temperature at which the reaction becomes spontaneous, we use the Gibbs Free Energy formula: \[ \Delta G = \Delta H - T \Delta S \]For spontaneity, \( \Delta G \) should be less than zero. We aim to find the temperature \( T \) such that \( \Delta G = 0 \). This translates to the equation:\[ 0 = \Delta H - T \Delta S \]

Step 2: Set Up the Spontaneity Equation

For the reaction to become spontaneous, set \( \Delta G = 0 \) in the Gibbs Free Energy equation:\[ 0 = \Delta H - T \Delta S \]Rearrange this equation to find the temperature \( T \):\[ T = \frac{\Delta H}{\Delta S} \]

Step 3: Substitute Known Values Into the Equation

Substitute the given values for \( \Delta H \) and \( \Delta S \) into the rearranged equation:\[ T = \frac{178 \text{ kJ mol}^{-1}}{161 \text{ J K}^{-1} \text{ mol}^{-1}} \]Note that \( \Delta H \) should be in the same units as \( \Delta S \), so convert \( 178 \text{ kJ mol}^{-1}\) to \( 178000 \text{ J mol}^{-1} \).

Step 4: Calculate the Temperature

Calculate the temperature \( T \) by dividing the converted \( \Delta H \) by \( \Delta S \):\[ T = \frac{178000 \text{ J mol}^{-1}}{161 \text{ J K}^{-1} \text{ mol}^{-1}} \approx 1106.8 \text{ K} \]

Step 5: Conclusion

The decomposition of calcium carbonate becomes spontaneous at approximately 1107 K. This is the temperature above which the reaction will proceed without external energy input.

Key Concepts

Spontaneity

In the world of chemistry, understanding whether a reaction will occur on its own is crucial. This concept is known as spontaneity. When we talk about spontaneity, we're asking whether a chemical reaction will proceed without needing an outside push, such as heat or energy.
A spontaneous reaction is one where the Gibbs Free Energy change (9 G) is less than zero. This means the reaction can happen naturally. To find the temperature at which a reaction like the decomposition of calcium carbonate becomes spontaneous, we set 9 G to zero, as it represents the point of transition to spontaneity.
This involves solving for the temperature (9 G = 9 H - T9 S = 0). Understanding this relationship helps us predict when and how a reaction will occur based on thermodynamic data, such as changes in enthalpy (9 H) and entropy (9 S).

Thermodynamics

Thermodynamics is the study of energy transformations in reactions. It helps us understand how energy changes influence the way reactions occur. In terms of spontaneous reactions, the change in Gibbs Free Energy is key.
Thermodynamics involves two important aspects: enthalpy (9 H) and entropy (9 S). Enthalpy measures the total heat content in a system. It indicates if heat is absorbed or released. Entropy, on the other hand, is a measure of randomness or disorder within a system.
In the context of our reaction, 9 H is positive, meaning heat is absorbed (endothermic). Since 9 S is also positive, the system becomes more disordered. The balance between these two factors, temperature, and their contribution to Gibbs Free Energy (9 G = 9 H - T9 S) informs us about the reaction's spontaneity at different temperatures.

Calcium Carbonate Decomposition

Calcium carbonate (9 CaCO_{3}9) decomposition is a well-studied reaction in chemistry. It breaks down into calcium oxide (9 CaO9) and carbon dioxide (9 CO_{2}9).
This process is generally endothermic, requiring heat to proceed. During decomposition, the system absorbs heat (9 H = +178 kJ/mol), while the entropy of the system increases (9 S = +161 J/K/mol).
To determine the temperature at which this reaction becomes spontaneous, we consider the relationship between these thermodynamic quantities. Using the Gibbs Free Energy formula, we find the temperature where 9 G equals zero. In this situation, the temperature is approximately 1107 K.
Above this temperature, the decomposition of calcium carbonate occurs naturally without added energy, illustrating the role of temperature in chemical reactions.