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: Review the Clausius-Clapeyron Equation

The equation in option (a), \( \Delta \mathrm{G} = \Delta \mathrm{H} + \Delta \mathrm{nRT} \), resembles a form related to the ideal gas law where \( \Delta \mathrm{n} \) represents the change in moles of gas. This is not a standard representation for Gibbs free energy change and is generally incorrect without additional context.

Step 2: Analyze the Correct Gibbs Free Energy Equation

The Gibbs free energy change is given by the equation: \( \Delta \mathrm{G} = \Delta \mathrm{H} - \mathrm{T} \Delta \mathrm{S} \). Therefore, option (d) accurately represents the Gibbs free energy equation.

Step 3: Evaluate Each Option for Correctness

- Option (a) is not the standard form for Gibbs free energy and lacks context, making it incorrect.- Option (b) \( \Delta \mathrm{G} = \Delta \mathrm{H} + \mathrm{T} \Delta \mathrm{S} \) is not correct because the sign before \( T \Delta S \) should be negative.- Option (c) equates \( \Delta G \) with a derivative term which is not a typical representation in thermodynamics.- Option (d) is correct as it matches the correct equation for Gibbs free energy change.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and the different forms of energy. It helps us understand how energy transformations take place in various physical and chemical processes. In the realm of thermodynamics, several fundamental laws govern the principles of energy exchange:

• The First Law, also known as the Law of Energy Conservation, states that energy cannot be created or destroyed; it can only change forms.
• The Second Law states that in any natural process, the total entropy, or disorder, of a system and its surroundings always increases.
• The Third Law introduces the concept that as the temperature approaches absolute zero, the entropy of a perfect crystal approaches zero.

These principles are crucial for understanding how energy balance is maintained in chemical reactions and physical changes, making thermodynamics essential in explaining aspects like heat engines, refrigerators, and phase transitions.

Clausius-Clapeyron Equation

The Clausius-Clapeyron Equation is an essential tool in thermodynamics, particularly useful for understanding phase transitions. It describes how the pressure and temperature of a substance are interrelated during a phase change, like from liquid to vapor. The equation provides a way to calculate the latent heat of transformation through the relation:

\[\frac{dP}{dT} = \frac{L}{T(V_2 - V_1)}\]

Here, \(dP/dT\) is the slope of the phase boundary line on a pressure-temperature diagram, \(L\) is the latent heat of the phase change, \(T\) is temperature, and \((V_2 - V_1)\) represents the change in specific volume.

This equation is vitally used in meteorology to predict weather patterns and in engineering to design equipment that involves heating and cooling. Its application spans into fields like chemistry and physics, playing a crucial role in understanding the dynamics of evaporation and condensation.

Ideal Gas Law

The Ideal Gas Law provides a vital equation connecting various state properties of an ideal gas: pressure, volume, temperature, and the number of moles. The equation states:

\[ PV = nRT \]

Where \(P\) represents pressure, \(V\) is volume, \(n\) is the number of moles of the gas, \(R\) is the universal gas constant, and \(T\) is temperature. This equation assumes that gas molecules do not interact in any significant way with each other and occupy no volume themselves.

Although no real gaseous system perfectly conforms to these assumptions, the Ideal Gas Law approximates the behavior of most gases under a variety of conditions fairly well. It forms the basis for more complex models like the Van der Waals equation, which adjusts for real-world deviations. The Ideal Gas Law is crucial in fields such as chemistry, where understanding the behavior of gases under different conditions is necessary for experiments and processes.

Gibbs Free Energy Equation

The Gibbs Free Energy Equation is a central concept in thermodynamics, crucial for understanding the direction of chemical processes and reaction feasibility. It is expressed as:

\[ \Delta G = \Delta H - T \Delta S \]

Where \(\Delta G\) is the change in Gibbs free energy, \(\Delta H\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the change in entropy. This equation helps predict whether a reaction will occur spontaneously. A negative \(\Delta G\) indicates a spontaneous process at constant temperature and pressure, while a positive \(\Delta G\) suggests non-spontaneity.

The Gibbs Free Energy also links multiple thermodynamic properties, making it a crucial equation for systems where chemical reactions, phase changes, or other processes occur. Understanding this concept allows scientists and engineers to design processes that maximize energy efficiency and predict the behavior of materials under various conditions.