Problem 6 - Chemistry The Molecular Science

Question

In terms of values of \(\Delta_{r} H^{\circ}\) and \(\Delta_{r} S^{\circ},\) under what conditions can you be sure that a reaction is product-favored? When can you be sure that it is not product-favored?

Step-by-Step Solution

Answer

A reaction is product-favored if \\(\Delta_{r} H^{\circ} < T \Delta_{r} S^{\circ}\\); not product-favored if \\(\Delta_{r} H^{\circ} > T \Delta_{r} S^{\circ}\\).

Step 1: Understand Gibbs Free Energy

The spontaneity of a chemical reaction can be determined by the Gibbs free energy change, \(\Delta G^{\circ}\), which is related to the enthalpy change \(\Delta_{r} H^{\circ}\) and the entropy change \(\Delta_{r} S^{\circ}\) by the equation: \[ \Delta G^{\circ} = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ} \] where \(T\) is the temperature in Kelvin.

Step 2: Condition for Product-Favored Reaction

For a reaction to be product-favored (spontaneous), \(\Delta G^{\circ}\) must be negative. This occurs when \[ \Delta_{r} H^{\circ} < T \Delta_{r} S^{\circ} \] or equivalently when \(\Delta_{r} H^{\circ}\) is negative and \(\Delta_{r} S^{\circ}\) is positive.

Step 3: Condition Against Product-Favored Reaction

A reaction is not product-favored (non-spontaneous) when \(\Delta G^{\circ}\) is positive, which occurs when \[ \Delta_{r} H^{\circ} > T \Delta_{r} S^{\circ} \] or equivalently when \(\Delta_{r} H^{\circ}\) is positive and \(\Delta_{r} S^{\circ}\) is negative.

Key Concepts

Spontaneity of Chemical Reactions

Understanding the spontaneity of chemical reactions is crucial to predict whether a reaction will proceed without additional input of energy. We determine spontaneity using the Gibbs free energy change, denoted as \( \Delta G^{\circ} \). If \( \Delta G^{\circ} \) is negative, the reaction is considered spontaneous or product-favored. This means that the products have lower energy than the reactants and the process releases energy. Conversely, a positive \( \Delta G^{\circ} \) indicates a non-spontaneous reaction, requiring additional energy to proceed.Three components influence spontaneity:

• Enthalpy Change (\( \Delta_{r} H^{\circ}\))
• Entropy Change (\( \Delta_{r} S^{\circ}\))
• Temperature (\( T \))

The relationship between these factors is summarized by the equation: \[ \Delta G^{\circ} = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ} \]This formula shows how the enthalpy and entropy changes, alongside temperature, influence whether \( \Delta G^{\circ} \) is negative or positive, thus affecting the reaction's spontaneity.

Enthalpy Change

Enthalpy change, denoted as \( \Delta_{r} H^{\circ} \), is a measure of the total heat content change during a reaction. It reflects the energy absorbed or released to break and form chemical bonds. If \( \Delta_{r} H^{\circ} \) is negative, the reaction is exothermic, releasing heat. Such reactions often tend to be product-favored because they help make \( \Delta G^{\circ} \) negative.When \( \Delta_{r} H^{\circ} \) is positive, the reaction is endothermic and requires external energy. This is often the case with non-spontaneous reactions.It's crucial to remember that while a negative enthalpy change favors spontaneity, it's not the sole factor. Temperature and entropy changes also significantly contribute to determining whether \( \Delta G^{\circ} \) becomes negative.

Entropy Change

Entropy change, represented by \( \Delta_{r} S^{\circ} \), indicates the change in disorder or randomness in a system during a reaction. Higher entropy means greater disorder. An increase in entropy (positive \( \Delta_{r} S^{\circ} \)) can help drive a reaction to spontaneity by making \( \Delta G^{\circ} \) more negative.For example, when a solid goes into solution or a complex molecule breaks into simpler ones, entropy usually increases, favoring the spontaneity of the reaction.Even if a reaction is endothermic, an increase in entropy can still make it spontaneous, provided that the temperature \( T \) is sufficiently high. Thus, entropy is a critical factor in determining reaction spontaneity alongside enthalpy change and temperature.