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) \(1008 \mathrm{~K}\)

Short answer

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

Step-by-step solution

Understanding Spontaneity

For a reaction to be spontaneous, the Gibbs free energy change (\( \Delta G^\circ \)) must be negative. We can use the equation \( \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \) to analyze spontaneity based on temperature.

Rearrange the Equation

To find the temperature above which the reaction is spontaneous, set \( \Delta G^\circ \) to zero and solve for \( T \): \[0 = \Delta H^\circ - T \Delta S^\circ\] \[T = \frac{\Delta H^\circ}{\Delta S^\circ}\].

Plug in Values

Substitute \( \Delta H^\circ = 179.1 \times 10^3 \) J/mol (converted from kJ) and \( \Delta S^\circ = 160.2 \) J/K in the equation: \[T = \frac{179100}{160.2}\].

Calculate the Minimum Temperature

Calculate the value: \(T = \frac{179100}{160.2} \approx 1118 \) K. This is the temperature above which the reaction becomes spontaneous.

Key concepts

Spontaneity of Reactions

To determine if a chemical reaction will occur without being driven by external energy, we consider the spontaneity of reactions. A spontaneous reaction has a negative Gibbs Free Energy (\(\Delta G^\circ \)). In simple terms, it occurs naturally once begun. Gibbs Free Energy can predict spontaneity using the formula:

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

Here, \(\Delta H^\circ\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S^\circ\) is the change in entropy. When \(\Delta G^\circ\) is less than zero, the reaction is spontaneous. Therefore, to find when a reaction becomes spontaneous, you observe the temperature at which \(\Delta G^\circ\) switches from positive to negative, setting it to zero to solve for \(T\). This calculation shows at which temperature a reaction naturally progresses.

Enthalpy and Entropy

Enthalpy (\(\Delta H^\circ\)) and entropy (\(\Delta S^\circ\)) are vital concepts in understanding reactions. Let's break them down:

• Enthalpy (\(\Delta H^\circ\)): This represents the total heat content of a system. It helps determine whether a reaction absorbs or releases heat. A positive \(\Delta H^\circ\) indicates an endothermic reaction, absorbing heat, as seen in limestone to lime conversion.


• Entropy (\(\Delta S^\circ\)): Entropy measures the disorder or randomness in a system. A higher \(\Delta S^\circ\) signifies more disorder. In chemical reactions, gases generally contribute to higher entropy due to their dispersal characteristic.

These values help predict the direction of transformation, considering that both concepts are intermixed in the Gibbs Free Energy equation.

Thermodynamic Calculations

Thermodynamic calculations employ the Gibbs Free Energy equation to find key insights into chemical reactions under specific conditions. Calculating the temperature where a reaction becomes spontaneous, like the conversion of limestone to lime, involves several steps:

• Identify the Given Values: \(\Delta H^\circ = 179.1 \text{ kJ/mol} = 179100 \text{ J/mol} \) and \(\Delta S^\circ = 160.2 \text{ J/K} \).


• Use Gibbs Free Energy Formula: Set \(\Delta G^\circ\) to zero because we want to find the point of spontaneity:

\[0 = \Delta H^\circ - T \Delta S^\circ\]By solving \(T = \frac{\Delta H^\circ}{\Delta S^\circ}\), we can determine the spontaneous transition temperature. Plug in the values:\[T = \frac{179100}{160.2} \approx 1118\] K. This calculation shows that only above 1118 K, the reaction naturally proceeds, as spontaneity sets in.