Problem 34 - Chemistry: An Atoms-Focused Approach

Question

The value of \(\Delta S_{\mathrm{rxn}}\) of the spontaneous reaction \(\mathrm{D}+\mathrm{E} \rightarrow \mathrm{F}\) is \(72.0 \mathrm{J} / \mathrm{K} .\) What is the minimum value of the entropy change in the reaction's surroundings?

Step-by-Step Solution

Answer

Answer: The minimum value of the entropy change in the surroundings for this reaction is -72.0 J/K.

Step 1: Write down the given values

We are given the change in entropy of the reaction, ΔS_rxn = 72.0 J/K.

Step 2: Write down the second law of thermodynamics equation

The second law of thermodynamics states that the total entropy change (ΔS_total) in the system and surroundings must be greater than 0, so we can write the equation as: ΔS_total = ΔS_rxn + ΔS_surroundings

Step 3: Set the total entropy change to 0

In order to find the minimum value of ΔS_surroundings, we will set ΔS_total equal to 0: 0 = ΔS_rxn + ΔS_surroundings

Step 4: Solve for the minimum value of the entropy change in the surroundings

Now we can solve for ΔS_surroundings: ΔS_surroundings = -ΔS_rxn ΔS_surroundings = -72.0 J/K The minimum value of the entropy change in the reaction's surroundings is -72.0 J/K.

Key Concepts

Entropy Change

Entropy change is a core concept in thermodynamics, representing the measure of disorder within a system. When a reaction occurs, it either increases or decreases the entropy, depending on how the arrangement of particles is affected. In a chemical context, if a system's disorder decreases during the reaction, the entropy change is negative, and vice versa.
Entropically speaking, reactions can be:

• Endothermic: When a system absorbs heat, it often leads to an increase in disorder, resulting in a positive entropy change.
• Exothermic: When heat is released, the surroundings may become more disordered, increasing the system's entropy change.

For any reaction, knowing the entropy change provides insight into the reaction's direction and feasibility, especially under the umbrella of the second law of thermodynamics.

Spontaneous Reaction

A spontaneous reaction is one that occurs naturally without needing additional energy input once underway. The second law of thermodynamics plays a critical role here, declaring that for a process to be spontaneous, the total entropy change ( ΔS_{total} ) must be greater than zero.
For our discussed reaction ( ΔS_{rxn} = 72.0 J/K ), it is spontaneous because this entropy change contributes positively towards the total entropy.
The driving factors for spontaneity include:

• Gibbs Free Energy ( ΔG eq 0): A negative Gibbs free energy change confirms spontaneity.
• Entropy Increase: Typically, an increase in entropy signifies natural progress, fulfilling the spontaneity criterion.

Understanding spontaneity helps predict which reactions will naturally occur without external pressure or influence.

System and Surroundings

In thermodynamics, understanding the system and its surroundings is vital. The system is the part of the universe we focus on, like the reaction in question. Conversely, the surroundings include everything else that can interact with the system.
During reactions like D + E ightarrow F , both the system and the surroundings undergo entropy changes. While our focus is on the system's entropy change ( ΔS_{rxn} ), the surroundings have an equally important role. The second law of thermodynamics frames these interactions with the stipulation that the sum of the entropy change of the system and surroundings:

• Must yield a positive total: Ensuring the natural direction of energy dispersal across the universe.

Balancing these two changes is crucial for predicting the behavior of reactions and energies.

Thermodynamics Equation

The thermodynamics equation central to assessing entropy is derived from the second law and formulates as: ΔS_{total} = ΔS_{rxn} + ΔS_{surroundings} This equation shows that the total entropy change must always be zero or positive for processes to exist naturally. Understanding each component of this equation helps in comprehensively evaluating reactions:

• ΔS_{rxn}: Captures how the reaction itself affects the system's entropy.
• ΔS_{surroundings}: Reflects the reaction's influence on the environment.

By solving this equation, one can discern the minimum entropy change required for a process to remain feasible—in this particular scenario, it is -72.0 J/K for the surroundings. Mastering this fundamental equation provides a powerful tool for predicting chemical processes across various conditions.