Problem 53 - Chemistry The Molecular Science

Question

Use a mathematical equation to show how the statement leads to the conclusion cited: If a reaction is exothermic (negative \(\Delta_{1} H\) ) and if the entropy of the system increases (positive \(\Delta_{\\{} S\) ), then \(\Delta_{r} G\) must be negative, and the reaction is product-favored.

Step-by-Step Solution

Answer

When \\( \\Delta_{1} H \\) is negative and \\( \\Delta_{\{} S \\) is positive, \\( \\Delta_{r} G \\) is negative, favoring products.

Step 1: Understanding the Problem

We need to show how an exothermic reaction and an increase in entropy lead to a negative Gibbs free energy change, \( \Delta_{r} G \), indicating that the reaction is product-favored. We will use the Gibbs free energy equation to do this.

Step 2: Writing the Gibbs Free Energy Equation

The Gibbs free energy change for a reaction is given by the equation: \[ \Delta_{r} G = \Delta_{1} H - T\Delta_{\{} S \] where \( \Delta_{1} H \) is the enthalpy change, \( \Delta_{\{} S \) is the entropy change, and \( T \) is the temperature in Kelvin.

Step 3: Substituting Given Values

For an exothermic reaction, \( \Delta_{1} H \) is negative, and if the entropy increases, \( \Delta_{\{} S \) is positive. Substituting these conditions into the equation gives: \[ \Delta_{r} G = (\text{negative value}) - T(\text{positive value}) \]

Step 4: Analyzing the Equation

Since \( T \) and \( \Delta_{\{} S \) are both positive, their product is positive. The equation thus becomes: \[ \Delta_{r} G = \text{negative} - \text{positive} = \text{more negative} \] Therefore, \( \Delta_{r} G \) must be negative, meaning the reaction is product-favored.

Key Concepts

Gibbs Free Energy

Gibbs Free Energy is a key concept in thermodynamics. It tells us if a chemical reaction can occur on its own or not. The symbol for Gibbs Free Energy is \( \Delta G \).
It combines enthalpy, entropy, and temperature to tell us about the energy changes in a system.
The equation for Gibbs Free Energy is:

• \( \Delta G = \Delta H - T \Delta S \)

Where:- \( \Delta H \) is the change in enthalpy- \( T \) is the temperature in Kelvin- \( \Delta S \) is the change in entropy
When \( \Delta G \) is negative, the reaction happens spontaneously. It means that the products are more stable than the reactants. In the context of an exothermic reaction with increasing entropy, \( \Delta G \) often ends up being negative. That indicates the reaction is product-favored.

Entropy

Entropy is all about disorder and randomness.
In a simple way, you can think of it as the amount of chaos or unpredictability in a system.
The symbol for entropy is \( S \).
In chemical reactions, if entropy increases, it means the products are more disordered than the reactants.

• For example, when ice melts to form water, the entropy increases. The water molecules move around more freely compared to ice.

If a reaction has a positive change in entropy \( (\Delta S > 0) \), the system has become more disordered. Adding positive entropy into the Gibbs Free Energy equation helps make \( \Delta G \) more negative, which favors the formation of products.
This is important for reactions like combustion, where gases are produced, leading to higher entropy.

Enthalpy

Enthalpy deals with heat changes at constant pressure. The symbol for enthalpy is \( H \).
In chemistry, enthalpy is about understanding heat absorbed or released during a reaction.
If a reaction is exothermic, it releases heat, and \( \Delta H \) becomes negative.

• This is why exothermic reactions feel warm to the touch, like combustion or rusting of iron.

Inserting a negative \( \Delta H \) into the Gibbs Free Energy equation can cause \( \Delta G \) to be negative.
A negative \( \Delta G \) signifies that the reaction may spontaneously form products. So, when a reaction is exothermic (gives off heat) and even slightly increases in entropy, the chances of it being product-favored go up significantly.
In a nutshell, enthalpy helps determine if the reaction releases energy or absorbs it.