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.

Step 1: Understanding the Terms

We need to determine which of the given relations are incorrect. This involves understanding the terms: - \( \Delta G \) is the change in Gibbs free energy, - \( \Delta H \) is the change in enthalpy, - \( T \) is temperature, - \( \Delta S \) is the change in entropy, - \( \Delta n \) is the change in moles of gas, and - \( R \) is the universal gas constant.

Step 2: Analyzing Option (a)

The relation \( \Delta G = \Delta H + \Delta nRT \) suggests that Gibbs free energy changes are calculated by adding \( \Delta H \) and \( \Delta nRT \). This is not a standard thermodynamic equation, especially because \( \Delta nRT \) considers only gaseous systems in variable moles context. This relation is incorrect as it does not account for the entropy change \( \Delta S \).

Step 3: Analyzing Option (b)

The relation \( \Delta G = \Delta H + T \Delta S \) does not conform to the correct formula for Gibbs free energy, which is \( \Delta G = \Delta H - T \Delta S \). Therefore, this relation is incorrect since the sign before \( T \Delta S \) should be a negative, not positive.

Step 4: Analyzing Option (c)

The relation \( \Delta G = \Delta H + T[\delta{\Delta G} / \delta T]_{P} \) is an alternate form expression under constant pressure scenarios and is more advanced. It compares how changes in temperature impact Gibbs free energy, a deviation from conventional introductory equations.

Step 5: Analyzing Option (d)

The relation \( \Delta G = \Delta H - T \Delta S \) is correct. This is the fundamental expression for calculating Gibbs free energy changes at constant temperature and pressure. This equation accurately reflects the balance between enthalpy and entropy contributions.

Key Concepts

Thermodynamics

Thermodynamics is a scientific discipline focused on the study of energy transformations and how these relate to matter. It provides insight into how physical and chemical processes take place, particularly under certain conditions like pressure and temperature.

• Energy transformations: These involve heat transfers, mechanical work, and more within systems or between systems and their surroundings.
• Systems and surroundings: A system in thermodynamics might refer to a specific substance we're studying, while everything else is its surroundings.
• State functions: These are properties like internal energy, enthalpy, and entropy that define the state of a system.

In the context of Gibbs free energy, thermodynamics helps us understand why certain reactions or processes occur spontaneously under given conditions, while others do not.

Enthalpy

Enthalpy ( H) is a key concept in thermodynamics that denotes the heat content of a system at constant pressure. It is a state function, meaning it depends only on the initial and final states of the system, not on the pathway taken.

• Internal Energy and Pressure-Volume Work: Enthalpy combines internal energy (the energy contained within the system) with the product of pressure and volume, expressed as: \[ H = U + PV \]
• Enthalpy Changes: During reactions,  H measures absorbed or released heat, helping us determine whether reactions are endothermic (heat absorption, positive  H) or exothermic (heat release, negative  H).

Calculating changes in enthalpy is crucial for predicting reaction spontaneity when combined with entropy in the Gibbs free energy equation.

Entropy

Entropy ( S) measures the randomness or disorder of a system. It is one of the fundamental pillars of thermodynamics, vital for understanding how and why processes occur.

• Second Law of Thermodynamics: This states that the entropy of an isolated system always increases over time, leading systems to a state of maximum disorder.
• Entropy Changes: Changes in entropy determine whether processes are spontaneous. An increase in entropy ( S) typically implies that a process is favorable.
• Implications in Reactions: The interplay of entropy and enthalpy in the Gibbs free energy equation determines whether a reaction is spontaneous ( G is negative).

Entropy is a measure of molecular disorder, which, when factored with enthalpy and temperature, predicts chemical and physical reaction spontaneity through Gibbs free energy.

Universal Gas Constant

The Universal Gas Constant ( R) is a key factor in the ideal gas law and other equations in thermodynamics. It provides a linkage between universal properties of gases like pressure, volume, and temperature.

• Value and Units: The universal gas constant is approximately 8.314 J/mol·K.
• Ideal Gas Law: In the equation  PV = nRT,  R relates the quantities of pressure ( P), volume ( V), amount of gas ( n), and temperature ( T).
• Importance in Thermodynamics: It is widely used beyond ideal gas scenarios, like in calculations involving equilibrium and Gibbs free energy changes, especially when considering gaseous systems.

The Universal Gas Constant is a foundation in both thermodynamics and kinetics, enabling us to quantify dynamics of gaseous systems and link them to chemical reactions.