Problem 123 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following relation is/are incorrect? (a) \(\Delta \mathrm{G}=\Delta \mathrm{H}+\Delta \mathrm{nRT}\) (b) \(\Delta \mathrm{G}=\Delta \mathrm{H}+\mathrm{T} \Delta \mathrm{S}\) (c) \(\Delta \mathrm{G}=\Delta \mathrm{H}+\mathrm{T}[\delta\\{\Delta \mathrm{G}\\} / \delta \mathrm{T}]_{\mathrm{P}}\) (d) \(\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \Delta \mathrm{S}\)

Step-by-Step Solution

Answer

Options (a), (b), and (c) are incorrect; option (d) is correct.

Step 1: Understand the Primary Equation

The standard relationship between Gibbs free energy change \( \Delta G \), enthalpy change \( \Delta H \), and entropy change \( \Delta S \) is given by:\[ \Delta G = \Delta H - T \Delta S \] This will be used as the basis to evaluate the correctness of each provided relation.

Step 2: Analyze Option (a)

Option (a) is \( \Delta G = \Delta H + \Delta nRT \). In this form, it incorrectly uses \( + \Delta nRT \). There is no standard formula in thermodynamics that supports this equation as accurate in comparison with the standard relation.

Step 3: Analyze Option (b)

Option (b) is \( \Delta G = \Delta H + T \Delta S \). This equation has the wrong sign in front of the \( T \Delta S \) term. The accepted relationship is \( \Delta G = \Delta H - T \Delta S \), making this relation incorrect.

Step 4: Analyze Option (c)

Option (c) is \( \Delta G = \Delta H + T[\delta \Delta G / \delta T]_P \). This form derives from a variation on the standard equation and implies a dependence on temperature. However, this exact equation is non-standard and does not reflect the typical thermodynamic relationship for \( \Delta G \).

Step 5: Analyze Option (d)

Option (d) is \( \Delta G = \Delta H - T \Delta S \). This equation matches the standard Gibbs free energy relationship and is, therefore, correct.

Key Concepts

Enthalpy Change

Enthalpy change is a key concept in understanding energy transformations during chemical reactions. Enthalpy, often represented as \( \Delta H \), is a measure of the total heat content of a system. It indicates the amount of energy absorbed or released when a chemical reaction takes place at constant pressure.

Using enthalpy, we can predict whether a reaction is exothermic or endothermic. In an exothermic reaction, the system releases heat, and \( \Delta H \) is negative. For instance, combustion reactions are typically exothermic. In contrast, an endothermic reaction absorbs heat, leading to a positive \( \Delta H \), such as in the melting of ice. Enthalpy change provides valuable insight into the energy dynamics of reactions.

Understanding \( \Delta H \) is crucial when calculating Gibbs free energy, \( \Delta G \). It directly influences \( \Delta G \) in the standard equation \( \Delta G = \Delta H - T \Delta S \), where it contributes to determining whether a process is spontaneous.

Entropy Change

Entropy change, symbolized as \( \Delta S \), measures the degree of disorder or randomness within a system. In thermodynamics, entropy is a crucial factor that influences the feasibility of a process or reaction. Systems tend to move from a state of order to disorder, which can be quantified by changes in entropy.

When a system undergoes a change, such as melting or evaporation, entropy generally increases (\( \Delta S \) > 0). This indicates a shift towards greater disorder, aligning with the Second Law of Thermodynamics. Conversely, a decrease in entropy (\( \Delta S \) < 0) suggests a transition to a more ordered state, typical of processes like freezing.

Entropy change plays a critical role in the calculation of Gibbs free energy \( \Delta G \). The formula \( \Delta G = \Delta H - T \Delta S \) includes \( T \Delta S \), which affects whether a reaction is spontaneous. At higher temperatures, entropy change can even dominate the entropy term in this equation, influencing the direction of spontaneous reactions.

Thermodynamic Equations

Thermodynamic equations allow us to understand the energy and entropy changes within systems. One of the most essential equations in thermodynamics is the Gibbs free energy equation: \( \Delta G = \Delta H - T \Delta S \). Each component of this equation serves a distinct purpose:

• \( \Delta G \) measures the maximum reversible work obtainable from a process at constant temperature and pressure.
• \( \Delta H \) represents the enthalpy change, providing insight into the heat absorbed or released.
• \( T \Delta S \) accounts for the temperature's impact on entropy change, affecting system disorder.

Understanding this equation is crucial for chemistry and physics students as it predicts the spontaneity of reactions. A negative \( \Delta G \) signals a spontaneous process, while a positive \( \Delta G \) suggests non-spontaneity.

There are variations and extensions of the Gibbs free energy equation, like including terms for pressure or concentration changes in non-standard conditions. However, the primary formula remains fundamental when assessing reaction feasibility and calculating other thermodynamic properties.