Problem 166 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

The temperature above 1118 K makes the conversion spontaneous, so the answer is (c) 1118 K.

Step 1: Understand the Reaction and Given Data

The given reaction is the decomposition of calcium carbonate (\(\mathrm{CaCO}_3\)) into calcium oxide (\(\mathrm{CaO}\)) and carbon dioxide (\(\mathrm{CO}_2\)). You are given \(\Delta \mathrm{H}^{\circ} = +179.1 \, \mathrm{kJ} \, \mathrm{mol}^{-1}\) and \(\Delta \mathrm{S}^{\circ} = 160.2 \, \mathrm{J} \, / \, \mathrm{K}\). Note that \(\Delta \mathrm{S}^{\circ}\) should be converted into kJ by dividing by 1000.

Step 2: Convert \(\Delta S^{\circ}\) to kJ

\(\Delta \mathrm{S}^{\circ} = 160.2 \, \mathrm{J} \, / \, \mathrm{K} = 0.1602 \, \mathrm{kJ} \, / \, \mathrm{K}\). This conversion is crucial since \(\Delta \mathrm{H}^{\circ}\) is given in kJ.

Step 3: Apply the Gibbs Free Energy Equation

For the reaction to be spontaneous, \(\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} < 0\). We need to find the temperature \(T\) such that \(\Delta G^{\circ}\) becomes negative.

Step 4: Solve for Temperature \(T\)

Set up the equation: \(\Delta G^{\circ} = 179.1 - T \times 0.1602 = 0\). Solve for \(T\) to find \(T = \frac{179.1}{0.1602}\) to find the temperature above which the reaction starts becoming spontaneous.

Step 5: Calculate the Critical Temperature

Calculate \(T = \frac{179.1}{0.1602} \approx 1118 \, \mathrm{K}\). This is the temperature above which the conversion from limestone to lime becomes spontaneous.

Key Concepts

Gibbs Free Energy

Gibbs Free Energy is a central concept in understanding chemical reactions and their spontaneity. It is represented as \( \Delta G \), which indicates the change in free energy during a reaction.
The equation to calculate Gibbs Free Energy is:

• \( \Delta G = \Delta H - T \Delta S \)

Here, \( \Delta H \) represents the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy. For a reaction to occur spontaneously, the Gibbs Free Energy change must be negative, meaning \( \Delta G < 0 \).
Interpretation of \( \Delta G \) is straightforward:

• If \( \Delta G < 0 \), the process is spontaneous and can proceed without external energy.
• If \( \Delta G = 0 \), the system is in equilibrium.
• If \( \Delta G > 0 \), the reaction is non-spontaneous.

Gibbs Free Energy is especially useful for predicting the spontaneity of reactions under constant temperature and pressure. It can also provide insights into the balance between enthalpy and entropy changes for a given reaction.

Spontaneous Reaction

A spontaneous reaction is one that occurs naturally without the need for external energy. The key factor that determines spontaneity is the Gibbs Free Energy change \( \Delta G \).
When describing a spontaneous process,

• \( \Delta G < 0 \) indicates that the reaction proceeds in the forward direction.
• Spontaneity can change with conditions like temperature and pressure.

In the context of the limestone reaction, we want to find the temperature where the reaction becomes spontaneous. This means calculating if \( \Delta G \) becomes less than zero when temperature is increased. Above this critical temperature, the decomposition of limestone occurs naturally, converting it to lime and carbon dioxide.

Entropy Change

Entropy, denoted as \( \Delta S \), measures the degree of disorder or randomness in a system. It plays a crucial role in determining whether a reaction is spontaneous. The second law of thermodynamics tells us that in an isolated system, the total entropy will always increase over time.
Here are some key points about entropy:

• When \( \Delta S > 0 \), the disorder of the system increases, often favoring spontaneity.
• Entropy can be affected by changes in state, such as solid to gas, which greatly increases disorder.

Within our reaction of converting limestone to lime, the production of \( \text{CO}_2 \) gas increases the entropy. This increase aids in making the reaction spontaneous above a certain temperature. Entropy ensures that reactions may proceed even if they are endothermic, given a high enough value that counters the enthalpy.

Enthalpy Change

Enthalpy change, represented as \( \Delta H \), is the measure of heat absorbed or released in a reaction. This thermodynamic concept informs us about the energy stability of reactants and products.

• \( \Delta H > 0 \) indicates an endothermic process, which absorbs heat.
• \( \Delta H < 0 \) indicates an exothermic process, which releases heat.

In our stone-to-lime conversion, \( \Delta H \) is positive, meaning it's endothermic. This tells us that the reaction requires the input of energy. However, coupled with a large entropy change, the reaction can become spontaneous at high temperatures. When assessing a reaction, balancing between \( \Delta H \) and \( \Delta S \) alongside \( T \) helps to understand whether a process will naturally occur. It helps in predicting reaction conditions favoring product formation.