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 [2007] (a) \(1200 \mathrm{~K}\) (b) \(845 \mathrm{~K}\) (c) \(1118 \mathrm{~K}\) (d) \(1008 \mathrm{~K}\)

Step-by-Step Solution

Answer

The temperature above which the conversion is spontaneous is 1118 K, option (c).

Step 1: Understand the question

We are asked to find the temperature above which the conversion of limestone to lime becomes spontaneous. This involves using the Gibbs free energy equation: \( \Delta G = \Delta H - T \Delta S \). The process becomes spontaneous when \( \Delta G < 0 \).

Step 2: Set up the equation for Gibbs free energy

Since the reaction becomes spontaneous when \( \Delta G \) is less than zero, we set up the inequality: \( \Delta H - T \Delta S < 0 \). We substitute the given values: \( \Delta H^\circ = 179.1 \text{ kJ/mol} \) (which is equivalent to \( 179100 \text{ J/mol} \)), and \( \Delta S^\circ = 160.2 \text{ J/K} \).

Step 3: Solve the inequality for T

Rearrange the inequality \( 179100 - T(160.2) < 0 \) to solve for \( T \):\[ T > \frac{179100}{160.2} \]Calculate \( T \) to find that \( T > 1118 \text{ K} \).

Step 4: Determine the correct answer choice

Comparing the calculated temperature of \( 1118 \text{ K} \) with the provided options, the temperature corresponding to the reaction becoming spontaneous is option (c) 1118 K.

Key Concepts

Spontaneity Conditions

When we talk about whether a chemical reaction will proceed without any external input of energy, we are referring to the reaction's spontaneity. The Gibbs free energy change (\( \Delta G \)) is the main indicator of spontaneity. The reaction will happen on its own if \( \Delta G \) is less than zero.
\( \Delta G \) is determined by the equation:

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

where:

• \( \Delta H \) is the enthalpy change.
• \( T \) is the temperature in Kelvin.
• \( \Delta S \) is the entropy change.

This equation helps us understand that spontaneity is influenced by both the energy absorbed or released (enthalpy) and the disorder (entropy) in the system. A reaction is often spontaneous at high temperatures if the entropy change \( \Delta S \) is positive because \( T \Delta S \) then dominates the equation, making \( \Delta G \) negative.

Thermodynamic Calculations

Performing thermodynamic calculations involves using known values to predict the behavior of a system. In the provided exercise, we calculated the temperature above which the limestone to lime conversion becomes spontaneous using the Gibbs free energy equation.
First, the provided values for enthalpy \( (\Delta H) \) and entropy \( (\Delta S) \) must be substituted into the equation \( \Delta G = \Delta H - T \Delta S \). Since we want to find the critical temperature \( T \) that will make \( \Delta G < 0 \), we rearrange the formula to solve for \( T \):

• \( \Delta G < 0 \)
• \( \Rightarrow \Delta H < T \Delta S \)
• \( \Rightarrow T > \frac{\Delta H}{\Delta S} \)

In the example of limestone to lime conversion:

• \( \Delta H = 179100 \text{ J/mol} \)
• \( \Delta S = 160.2 \text{ J/K} \)

Plugging these values into the inequality gives \( T > \frac{179100}{160.2} \), resulting in \( T > 1118 \text{ K} \). This means that above 1118 K, the reaction becomes spontaneous.

Limestone to Lime Conversion

The conversion of limestone (\( \text{CaCO}_3 \)) to lime (\( \text{CaO} \)) is a classical chemical reaction with practical importance in industries such as construction and materials manufacturing. This reaction is represented by:

• \( \text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g) \)

In this process, limestone decomposes into lime and carbon dioxide gas when subjected to high temperatures. This decomposition is crucial for producing lime, which is used in cement, glassmaking, and more.
Understanding the conditions under which this reaction is spontaneous is vital for industry. It determines the temperature at which the reaction must be carried out to be efficient. The energetic analysis involving Gibbs free energy helps find the threshold temperature, ensuring the process is economical and sustainable. In our example, above 1118 K, the conversion happens naturally, which is critical for optimizing production.