Question

Determine whether each of the following reactions is spontaneous. a. \(\Delta H_{\text { system }}=-75.9 \mathrm{k} J, T=273 \mathrm{K}, \Delta S_{\text { system }}=138 \mathrm{J} / \mathrm{K}\) b. \(\Delta H_{\text { system }}=-27.6 \mathrm{k} \mathrm{J}, T=535 \mathrm{K}, \Delta S_{\mathrm{system}}=-55.2 \mathrm{J} / \mathrm{K}\) c. \(\Delta H_{\text { system }}=365 \mathrm{kJ}, T=388 \mathrm{K}, \Delta S_{\text { system }}=-55.2 \mathrm{J} / \mathrm{K}\) d. \(\Delta H_{\text { system }}=452 \mathrm{kJ}, T=165 \mathrm{K}, \Delta S_{\text { system }}=55.7 \mathrm{J} / \mathrm{K}\)

Short answer

a. Reaction is spontaneous (\(\Delta G < 0\)) b. Reaction is spontaneous (\(\Delta G < 0\)) c. Reaction is non-spontaneous (\(\Delta G > 0\)) d. Reaction is non-spontaneous (\(\Delta G > 0\))

Step-by-step solution

Calculate Gibbs free energy change for reaction a

For the first reaction, we have the following given values: - \(\Delta H_{\text { system }} = -75.9 \mathrm{k} J\) - \(T = 273 \mathrm{K}\) - \(\Delta S_{\text { system }} = 138 \mathrm{J} / \mathrm{K}\) Using the Gibbs free energy change formula, we can calculate \(\Delta G\) for reaction a. \[ \Delta G = \Delta H - T \Delta S = -75.9 \times 10^{3} \mathrm{J} - (273 \mathrm{K})(138 \mathrm{J} / \mathrm{K}) \]

Calculate Gibbs free energy change for reaction b

For the second reaction, we have the following given values: - \(\Delta H_{\text { system }} = -27.6 \mathrm{k} \mathrm{J}\) - \(T = 535 \mathrm{K}\) - \(\Delta S_{\text { system }} = -55.2 \mathrm{J} / \mathrm{K}\) Using the Gibbs free energy change formula, we can calculate \(\Delta G\) for reaction b. \[ \Delta G = \Delta H - T \Delta S = -27.6 \times 10^3 \mathrm{J} - (535 \mathrm{K})(-55.2 \mathrm{J} / \mathrm{K}) \]

Calculate Gibbs free energy change for reaction c

For the third reaction, we have the following given values: - \(\Delta H_{\text { system }} = 365 \mathrm{kJ}\) - \(T = 388 \mathrm{K}\) - \(\Delta S_{\text { system }} = -55.2 \mathrm{J} / \mathrm{K}\) Using the Gibbs free energy change formula, we can calculate \(\Delta G\) for reaction c. \[ \Delta G = \Delta H - T \Delta S = 365 \times 10^3 \mathrm{J} - (388 \mathrm{K})(-55.2 \mathrm{J} / \mathrm{K}) \]

Calculate Gibbs free energy change for reaction d

For the fourth reaction, we have the following given values: - \(\Delta H_{\text { system }} = 452 \mathrm{kJ}\) - \(T = 165 \mathrm{K}\) - \(\Delta S_{\text { system }} = 55.7 \mathrm{J} / \mathrm{K}\) Using the Gibbs free energy change formula, we can calculate \(\Delta G\) for reaction d. \[ \Delta G = \Delta H - T \Delta S = 452 \times 10^3 \mathrm{J} - (165 \mathrm{K})(55.7 \mathrm{J} / \mathrm{K}) \]

Determine spontaneity of each reaction

Calculate the values of \(\Delta G\) for each of the reactions in steps 1 through 4. Then, determine whether each reaction is spontaneous, non-spontaneous, or at equilibrium based on the sign of \(\Delta G\). If \(\Delta G<0\), the reaction is spontaneous; if \(\Delta G>0\), the reaction is non-spontaneous; if \(\Delta G=0\), the reaction is at equilibrium.

Key concepts

Enthalpy Change

In thermodynamics, enthalpy change is a measurement of heat absorbed or released in a system at constant pressure. Represented by the symbol \( \Delta H \), it helps us understand whether a chemical reaction is endothermic (absorbing heat) or exothermic (releasing heat).
Typically, when \( \Delta H \) is negative, the reaction is exothermic, meaning heat is released as bonds are formed. This often indicates a potential for spontaneity, depending on other factors.

• Exothermic reactions: \( \Delta H < 0 \)
• Endothermic reactions: \( \Delta H > 0 \)

Understanding enthalpy is important because it helps us predict how heat energy changes during chemical processes. Remember, enthalpy alone doesn't determine if a reaction will happen spontaneously; it must be considered with entropy and temperature in the Gibbs free energy equation.

Entropy Change

Entropy is a measure of disorder or randomness in a system. In a chemical reaction, the entropy change, denoted as \( \Delta S \), indicates how the disorder changes from reactants to products.
When \( \Delta S \) is positive, the system becomes more disordered, which often favors spontaneity, especially at higher temperatures. Conversely, a negative \( \Delta S \) implies the system becomes more ordered.

• Increasing disorder: \( \Delta S > 0 \)
• Decreasing disorder: \( \Delta S < 0 \)

Entropy and temperature influence the spontaneity of reactions through their contribution to Gibbs free energy. These two factors, when combined, can override the effects of enthalpy.

Spontaneity of Reactions

To predict whether a chemical reaction is spontaneous, we rely on the Gibbs free energy change (\( \Delta G \)). The formula \( \Delta G = \Delta H - T\Delta S \) helps determine spontaneity by combining enthalpy, temperature, and entropy.
If \( \Delta G \) is negative, the reaction occurs spontaneously. This means at the given temperature, the system's free energy decreases, favoring product formation.

• Spontaneous reaction: \( \Delta G < 0 \)
• Non-spontaneous reaction: \( \Delta G > 0 \)
• Equilibrium: \( \Delta G = 0 \)

It's crucial to know that spontaneity doesn't speak to the speed of a reaction. A spontaneous reaction might still occur slowly if under kinetic constraints.

Thermodynamics

Thermodynamics is the branch of physics dealing with energy and its transformations. It's essential in evaluating chemical reactions for energy changes, particularly focusing on enthalpy and entropy.

• The first law of thermodynamics, the principle of conservation of energy, states that energy cannot be created or destroyed, only transformed.
• The second law establishes that entropy in an isolated system always increases over time.

In the context of reactions, thermodynamic principles help us understand how energy and matter interact, informing us about the feasibility and direction of processes. Using thermodynamic analyses, we can predict how reactions exchange energy with their surroundings, providing insights into their potential spontaneity.