Question

From the data given in the following table, determine \(\Delta S^{\circ} \quad\) for the reaction \(\quad \mathrm{NH}_{3}(\mathrm{g})+\mathrm{HCl}(\mathrm{g}) \longrightarrow\) \(\mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) .\) All data are at \(298 \mathrm{K}\) $$\begin{array}{lcc} \hline & \Delta H_{f}^{\circ}, \mathrm{kJ} \mathrm{mol}^{-1} & \Delta G_{f,}^{\circ} \mathrm{kJ} \mathrm{mol}^{-1} \\ \hline \mathrm{NH}_{3}(\mathrm{g}) & -46.11 & -16.48 \\ \mathrm{HCl}(\mathrm{g}) & -92.31 & -95.30 \\ \mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) & -314.4 & -202.9 \\ \hline \end{array}$$

Short answer

\(\Delta S^{\circ} = -0.284 \ K^{-1} mol^{-1}\)

Step-by-step solution

Identify the relevant thermodynamic relation

The relationship between Gibbs Free Energy (\( \Delta G \)), Enthalpy (\( \Delta H \)), Temperature (T) and Entropy (\(\Delta S\)) is given by the equation: \[ \Delta G = \Delta H -T \Delta S \]Considering we are dealing with standard conditions (298 K), this can be adapted to:\[ \Delta G^{\circ} = \Delta H^{\circ} -T \Delta S^{\circ} \]To find \(\Delta S^{\circ}\), we rearrange the equation like so:\[ \Delta S^{\circ} = (\Delta H^{\circ} - \Delta G^{\circ}) / T \]

Calculate the change in enthalpy and Gibbs free energy of the reaction

Next, calculate the enthalpy (\(\Delta H_f^{\circ}\)) and Gibbs free energy (\(\Delta G_f^{\circ}\)) for the reaction. This can be done using the provided tables and the formulas:\[\Delta H^{\circ} = \sum \Delta H_f^{\circ} (products) - \sum \Delta H_f^{\circ} (reactants)\]\[\Delta G^{\circ} = \sum \Delta G_f^{\circ} (products) - \sum \Delta G_f^{\circ} (reactants)\]For our reaction:\[\Delta H^{\circ}= \Delta H_f^{\circ}(NH_{4}Cl) -(\Delta H_f^{\circ}(NH_{3}) + \Delta H_f^{\circ}(HCl)) = -314.4-(-46.11-92.31) = -175.98 \,kJ/mol \]\[\Delta G^{\circ}= \Delta G_f^{\circ}(NH_{4}Cl)- (\Delta G_f^{\circ}(NH_{3}) + \Delta G_f^{\circ}(HCl)) = -202.9 -(-16.48-95.3) = -91.12 \, kJ/mol \]

Determine the standard entropy change

Finally, calculate the standard entropy change (\(\Delta S^{\circ}\)) using the formula from Step 1:\[\Delta S^{\circ} = (\Delta H^{\circ} - \Delta G^{\circ}) / T = (-175.98 - (-91.12)) / 298 = -0.284 \ K^{-1} mol^{-1} \]Notice that the units of \(\Delta S^{\circ}\) are K^{-1}mol^{-1}, which are the proper units for entropy

Key concepts

Gibbs Free Energy

Gibbs Free Energy, denoted as \( \Delta G \), is a vital concept in thermodynamics that helps us understand the spontaneity of a reaction. It combines enthalpy, entropy, and temperature into a single value.
This concept is key because it predicts whether a reaction can occur without any external input. If \( \Delta G \) is negative, the reaction proceeds spontaneously.

• Formula: \( \Delta G = \Delta H - T \Delta S \)
• \( \Delta H \): Change in enthalpy (heat content)
• \( T \): Absolute temperature in Kelvin
• \( \Delta S \): Change in entropy (disorder)

In our example, substituting the values gives \( \Delta G^{\circ} = -91.12 \text{ kJ/mol} \).
This negative value indicates the formation of \( \text{NH}_4\text{Cl} \) from \( \text{NH}_3 \) and \( \text{HCl} \) is spontaneous.
Understanding Gibbs Free Energy can help you predict the feasibility of various chemical reactions in both industrial processes and natural phenomena.

Enthalpy

Enthalpy, represented as \( \Delta H \), is a measurement of heat content in a chemical system. It reflects the total energy change during a reaction, including heat absorbed or released.
Enthalpy change can be either positive (endothermic) or negative (exothermic).

• Endothermic: Absorbs heat, \( \Delta H > 0 \)
• Exothermic: Releases heat, \( \Delta H < 0 \)

For the reaction \( \text{NH}_3(\text{g}) + \text{HCl}(\text{g}) \rightarrow \text{NH}_4\text{Cl}(\text{s}) \), we calculated \( \Delta H^{\circ} = -175.98 \text{ kJ/mol} \).
This exothermic reaction releases heat, indicating that forming ammonium chloride from ammonia and hydrochloric acid is heat-releasing.
Enthalpy is crucial for understanding energy dynamics in reactions and helps in designing chemical processes where temperature control is essential.

Entropy

Entropy, denoted as \( \Delta S \), is a measure of disorder or randomness in a system. In chemistry, it helps us understand how energy disperses among molecules.

• Increased entropy: Greater disorder, \( \Delta S > 0 \)
• Decreased entropy: Greater order, \( \Delta S < 0 \)

For the given chemical reaction, we found \( \Delta S^{\circ} = -0.284 \text{ K}^{-1} \text{mol}^{-1} \).
This negative change suggests the system becomes more ordered as gases combine to form a solid (\( \text{NH}_4\text{Cl} \)).
Entropy is a fundamental concept because it affects the direction and feasibility of reactions. In combination with enthalpy, it helps chemists predict the behavior of chemical systems under different conditions.