Problem 67 - Chemistry The Central Science In Si Units

Question

For a particular reaction, \(\Delta H=30.0 \mathrm{~kJ}\) and \(\Delta S=90.0 \mathrm{~J} / \mathrm{K}\). Assume that \(\Delta H\) and \(\Delta S\) do not vary with temperature. (a) At what temperature will the reaction have \(\Delta G=0 ?\) (b) If T is increased from that in part (a), will the reaction be spontaneous or nonspontaneous?

Step-by-Step Solution

Answer

(a) 333.33 K, (b) The reaction is spontaneous at temperatures above 333.33 K.

Step 1: Understand the Relationship

We know the Gibbs free energy equation: \( \Delta G = \Delta H - T \Delta S \). To find when \( \Delta G = 0 \), we set the equation to zero: \( 0 = \Delta H - T \Delta S \) and solve for \( T \).

Step 2: Solve for Temperature

Rearrange the equation to solve for \( T \): \( T = \frac{\Delta H}{\Delta S} \). Given \( \Delta H = 30.0 \text{ kJ} = 30000 \text{ J} \) and \( \Delta S = 90.0 \text{ J/K} \), find \( T \): \( T = \frac{30000}{90} = 333.33 \text{ K} \).

Step 3: Determine Reaction Spontaneity Above Calculated Temperature

Once \( \Delta G = 0 \) at \( T = 333.33 \text{ K} \), if the temperature increases, \( \Delta G = \Delta H - T \Delta S \) becomes negative because \( T \Delta S > \Delta H \). Therefore, the reaction becomes spontaneous at temperatures above 333.33 K.

Key Concepts

Enthalpy

Enthalpy is a key concept in thermodynamics, representing the total heat content of a system. When we talk about reactions, enthalpy changes (Delta H) tell us whether a reaction absorbs or releases energy in the form of heat.
Enthalpy is denoted as H and is measured in kilojoules per mole (kJ/mol). When Delta H is positive, the reaction absorbs energy, known as an endothermic reaction. Conversely, a negative Delta H indicates an exothermic reaction, where energy is released.

• Endothermic Reaction: Energy is absorbed; Delta H > 0
• Exothermic Reaction: Energy is released; Delta H < 0

In the context of Gibbs free energy, enthalpy difference becomes crucial in determining reaction feasibility and spontaneity. When evaluating the given exercise, we see Delta H = 30.0  kJ or 30000 J. This means, initially, the reaction requires energy to proceed but does not determine alone if the reaction will be spontaneous at a certain temperature.

Entropy

Entropy (S) is another fundamental concept in thermodynamics. It describes the level of disorder or randomness within a system. Generally, systems tend to progress toward states with maximum entropy, or maximum disorder, according to the second law of thermodynamics.
Entropy changes (Delta S) can have significant effects on a reaction’s spontaneity. An increase in entropy (Delta S > 0) suggests a transition to a more disordered state, while a decrease (Delta S < 0) denotes a move to a more ordered state.
It's important to note that entropy is heavily temperature-dependent; meaning any change in temperature can greatly affect Delta S's influence on Gibbs free energy. In our exercise, Delta S = 90.0  J/K indicates an increase in disorder with the progress of the reaction. This increase in entropy is favorable for the reaction's spontaneity when considered at higher temperatures. When entropy contributes significantly in the Gibbs equation, we find crucial insights into how temperature shifts affect reaction paths.

Reaction Spontaneity

Reaction spontaneity is determined using the Gibbs free energy change (Delta G). A reaction is spontaneous if Delta G < 0, meaning the process occurs without needing added energy. In simple terms, spontaneous reactions occur naturally.
The Gibbs free energy equation, Delta G = Delta H - T Delta S, helps predict reaction spontaneity by balancing the effects of enthalpy and entropy across temperatures (T):

• If Delta G > 0: Reaction is non-spontaneous.
• If Delta G = 0: Reaction is at equilibrium.
• If Delta G < 0: Reaction is spontaneous.

In the exercise, Delta G = 0 when T = 333.33  K, suggesting equilibrium. Raising temperature beyond this point lets T Delta S overshadow Delta H, making Delta G negative, thus spontaneous for the reaction. Understanding these dynamics allows deeper recognition of how conditions affect whether a reaction will naturally occur.