Question

Under what conditions will the reaction be spontaneous? (a) \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) $$ \begin{aligned} \Delta S<0 \text { and } \Delta H<0 \end{aligned} $$ (b) \(\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}(s) \longrightarrow \mathrm{Ba}^{2+}(a q)+2 \mathrm{NO}_{3}^{-}(a q)\) \(\Delta S>0\) and \(\Delta H>0\)

Short answer

The reaction \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) is spontaneous at all temperatures. The dissociation of \(\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}(s)\) is spontaneous only at high temperatures.

Step-by-step solution

Understanding Spontaneity

Determine the conditions for spontaneity using the Gibbs free energy equation, which is \(\Delta G = \Delta H - T\Delta S\). A spontaneous reaction occurs when \(\Delta G < 0\).

Analyzing the First Reaction

For the reaction \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\), given that \(\Delta S<0\) and \(\Delta H<0\), it will be spontaneous at all temperatures because a negative \(\Delta H\) coupled with a negative \(\Delta S\) can only make the \(\Delta G\) more negative as temperature increases, ensuring \(\Delta G<0\).

Analyzing the Second Reaction

For the dissociation of \(\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}(s)\), given that \(\Delta S>0\) and \(\Delta H>0\), the reaction will be spontaneous only at high temperatures. Spontaneity is favored by an increase in temperature since a positive \(\Delta S\) contributes to a more negative \(\Delta G\) when temperature is increased according to the equation \(\Delta G = \Delta H - T\Delta S\).

Key concepts

Gibbs Free Energy

Gibbs free energy (\textbf{G}) is the amount of usable energy that can be harnessed from a chemical reaction to perform work. The concept of Gibbs free energy is crucial in predicting the spontaneity of a reaction without needing to perform the reaction. The formula for calculating Gibbs free energy is \begin{align*}\Delta G = \Delta H - T\Delta S\end{align*}where \(\Delta G\) represents the change in free energy, \(\Delta H\) is the change in enthalpy, or heat content, \(T\) is the absolute temperature measured in Kelvin (K), and \(\Delta S\) is the change in entropy, or disorder.

Entropy and Enthalpy

Entropy (\textbf{S}) is a measure of the disorder or randomness of a system. An increase in entropy (\textbf{\(\Delta S > 0\)}) generally means the system is becoming more disordered. For example, when a solid melts into a liquid, it's entropy increases because the molecules move from a more ordered to a less ordered state.

Enthalpy (\textbf{H}) reflects the total heat content of a system, equivalent to its internal energy plus the product of volume and pressure. A negative change in enthalpy (\textbf{\(\Delta H < 0\)}) implies that a reaction is exothermic, releasing heat to its surroundings, whereas a positive change (\textbf{\(\Delta H > 0\)}) indicates an endothermic reaction that absorbs heat.

Understanding both concepts is vital because they help predict whether a reaction will occur spontaneously based on the signs of \(\Delta S\) and \(\Delta H\), which directly affect \(\Delta G\).

Chemical Thermodynamics

Chemical thermodynamics is the area of science that deals with the interrelation of heat with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It provides a framework for relating the energy changes involved in reactions (\textbf{enthalpy}, \(\Delta H\)), the randomness (\textbf{entropy}, \(\Delta S\)), and the energy that can be used to do work (\textbf{Gibbs free energy}, \(\Delta G\)).

The first and second laws of thermodynamics are particularly important. The first law, which is the law of conservation of energy, states that energy cannot be created or destroyed. This principle helps us understand that the energy change in a system must equal the energy transferred. The second law introduces the concept of entropy and posits that for any spontaneous process, the total entropy of the system and its surroundings always increases.

Temperature Dependence of Spontaneity

The spontaneity of a chemical reaction often depends on temperature because temperature directly affects the value of the Gibbs free energy (\textbf{\(\Delta G\)}). From the equation\begin{align*}\Delta G = \Delta H - T\Delta S\end{align*}we see that the sign of \(\Delta G\) can change with varying temperature due to the \(T\Delta S\) term. For a reaction with a positive \(\Delta S\) (increasing entropy) and a positive \(\Delta H\) (absorbing heat), increasing the temperature can lead to a spontaneous reaction because the positive entropy term becomes more significant as it gets multiplied by a higher temperature, potentially making \(\Delta G\) negative.

Conversely, a reaction with negative \(\Delta S\) and \(\Delta H\) will become less favourable at high temperatures because the negative \(\Delta S\) term will contribute to an increase in \(\Delta G\). This temperature dependence is essential for understanding how external conditions affect the spontaneity of reactions.