Question

Sodium carbonate, also called "washing soda," can be made by heating sodium hydrogen carbonate: $$ \begin{array}{c} 2 \mathrm{NaHCO}_{3}(s) \longrightarrow \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \\ \Delta H^{\circ}=+135.6 \mathrm{k} J ; \Delta G^{\circ}=+34.6 \mathrm{~kJ} \text { at } 25^{\circ} \mathrm{C} \end{array} $$ (a) Calculate \(\Delta S^{\circ}\) for this reaction. Is the sign reasonable? (b) Calculate \(\Delta G^{\circ}\) at \(0 \mathrm{~K} ;\) at \(1000 \mathrm{~K}\).

Short answer

Question: Calculate the entropy change (∆S°) for the decomposition of sodium hydrogen carbonate and check if its sign is reasonable. Calculate the Gibbs free energy change (∆G°) at 0K and 1000K. Answer: The entropy change (∆S°) for the decomposition of sodium hydrogen carbonate is 0.34 kJ/mol·K, and its sign is positive, which is reasonable. The Gibbs free energy change (∆G°) at 0K is 135.6 kJ/mol, and at 1000K, it is -204 kJ/mol.

Step-by-step solution

(a) Calculating \(\Delta S^°\)

To calculate \(\Delta S^°\), we will use the relation \(\Delta G^° = \Delta H^° - T\Delta S^°\): 1. Convert the temperature from Celsius to Kelvin: \(T_1 = 25+273.15 = 298.15K\) 2. Rearrange the formula to isolate \(\Delta S^°\): \(\Delta S^° = (\Delta H^° - \Delta G^°)/T\) 3. Plug in the given values: \(\Delta S^°= (\Delta H^° - \Delta G^°)/T = (135.6 - 34.6)/298.15 = 0.34 \,\mathrm{kJ/mol\cdot K}\) Now let's analyze the sign of \(\Delta S^°\). In the given reaction, 2 moles of solid reactant transform into 1 mole of solid, 1 mole of gas, and 1 mole of liquid products. Since the solid is transforming into a gas and liquid, the randomness of the system is increasing, which means that the entropy change should be positive. Since the calculated value of \(\Delta S^°\) is positive, the sign is reasonable.

(b) Calculating \(\Delta G^°\) at 0K and 1000K

To calculate \(\Delta G^°\) at 0K and 1000K, we will use the relation \(\Delta G^° = \Delta H^° - T\Delta S^°\) with the obtained value of \(\Delta S^°\): 1. Calculate \(\Delta G^°\) at 0K: \(\Delta G^° = \Delta H^° - T\Delta S^° = 135.6 - 0\cdot 0.34 = 135.6 \,\mathrm{kJ/mol}\) 2. Calculate \(\Delta G^°\) at 1000K: \(\Delta G^° = \Delta H^° - T\Delta S^° = 135.6 - 1000\cdot 0.34 = -204 \,\mathrm{kJ/mol}\) So, at 0K, the Gibbs free energy change \((\Delta G^°)\) is 135.6 kJ/mol, and at 1000K, it is -204 kJ/mol.

Key concepts

Gibbs Free Energy

Gibbs free energy, denoted as \( G \), is a thermodynamic quantity that corresponds to the energy of a system that can be used to do work at a constant temperature and pressure. It combines enthalpy, entropy, and temperature into a single value, defined by the equation \( \Delta G^° = \Delta H^° - T\Delta S^° \). In chemistry, Gibbs free energy is pivotal for predicting the direction of chemical reactions. A negative value indicates that the reaction is spontaneous, while a positive value implies that it is nonspontaneous.

Considering the exercise on the decomposition of sodium hydrogen carbonate, we see that the Gibbs free energy change (\( \Delta G^° \) at 298.15 K is positive, indicating that under standard conditions, the reaction is nonspontaneous. This insight is crucial in identifying that additional energy is required for the reaction to proceed.

Entropy

Entropy, symbolized as \( S \), is a measure of disorder or randomness in a system. In thermodynamics, it is a state function, implying that its change between two states is independent of the process taken. A positive entropy change \( (\Delta S) \) means that the disorder of the system has increased during the reaction, which is common when solids turn into liquids or gases. Conversely, a negative entropy change indicates a reduction in disorder.

The exercise shows a positive \( \Delta S^° \) for the decomposition of sodium hydrogen carbonate, reflecting the increase in disorder when a solid decomposes into a gas and a liquid. This increase in disorder is expected, as gases have higher entropy than liquids or solids, due to more molecular randomness.

Enthalpy

Enthalpy, represented by \( H \), is essentially the heat content of a system at constant pressure. It is a state property meaning it depends only on the current state of the system, not on how the system reached that state. The change in enthalpy (\( \Delta H \)) during a chemical reaction indicates whether the reaction is exothermic (releases heat) or endothermic (absorbs heat).

In the exercise we see that the reaction has a positive \( \Delta H^° \) value, meaning it is endothermic and requires heat to proceed. This correlates with the scenario, where heating the sodium hydrogen carbonate provides the energy needed for the endothermic decomposition.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products and include changes in enthalpy, entropy, and Gibbs free energy. The direction and spontaneity of a reaction are not solely determined by enthalpy changes but also by entropy changes, which together affect the Gibbs free energy. Reactions proceed towards a state of lower free energy.

This is exemplified by the decomposition reaction in the exercise where sodium hydrogen carbonate reacts to form sodium carbonate, carbon dioxide, and water. The various states of matter involved lead to changes in enthalpy and entropy, influencing the Gibbs free energy and thus the reaction's spontaneity.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is vital in thermodynamics as many equations, such as those for Gibbs free energy, use the absolute temperature scale (Kelvin). To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. Conversely, to convert from Kelvin to Celsius, subtract 273.15 from the Kelvin temperature.

In the given exercise, the reaction's temperature is initially provided in Celsius and must be converted to Kelvin in order to calculate the entropy change. This conversion highlights the importance of using the appropriate temperature scale for accurate thermodynamic calculations.