Problem 41 - A Complete Resource Book in Chemistry for JEE Main

Question

A reaction occurs spontaneously if (a) \(\mathrm{T} \Delta \mathrm{S}<\Delta \mathrm{H}\) and both \(\Delta \mathrm{H}, \Delta \mathrm{S}\) are \(+\mathrm{ve}\) (b) \(\mathrm{T} \Delta \mathrm{S}>\Delta \mathrm{H}\) and \(\Delta \mathrm{H}=+\mathrm{ve}, \Delta \mathrm{S}=-\mathrm{ve}\) (c) \(\mathrm{T} \Delta \mathrm{S}>\Delta \mathrm{H}\) and both \(\Delta \mathrm{H}, \Delta \mathrm{S}\) are \(+\mathrm{ve}\) (d) \(\mathrm{T} \Delta \mathrm{S}=\Delta \mathrm{H}\) and both \(\Delta \mathrm{H}, \Delta \mathrm{S}\) are \(+\mathrm{ve}\)

Step-by-Step Solution

Answer

The correct option is (c): \( T \Delta S > \Delta H \) with \( \Delta H \) and \( \Delta S \) both positive.

Step 1: Understand Spontaneity in Terms of Gibbs Free Energy

For a reaction to occur spontaneously, the change in Gibbs Free Energy, \( \Delta G \), must be negative. The equation for Gibbs Free Energy is given by \( \Delta G = \Delta H - T\Delta S \), where \( \Delta H \) is the change in enthalpy, \( \Delta S \) is the change in entropy, and \( T \) is the temperature in Kelvin.

Step 2: Analyze Option A

Option (a) states that the reaction is spontaneous if \( T \Delta S < \Delta H \) with both \( \Delta H \) and \( \Delta S \) being positive. Substituting these into the Gibbs Free Energy equation yields \( \Delta G = \Delta H - T \Delta S > 0 \), indicating \( \Delta G \) is positive, making the reaction non-spontaneous. Hence, option (a) is incorrect.

Step 3: Analyze Option B

Option (b) specifies that a reaction is spontaneous if \( T \Delta S > \Delta H \) with \( \Delta H = +\) and \( \Delta S = -\). In this case, \( \Delta G = \Delta H - T \Delta S > 0 \) becomes positive since \( T \Delta S \) is negative which makes the reaction non-spontaneous. Option (b) is incorrect.

Step 4: Analyze Option C

Option (c) proposes that a reaction is spontaneous if \( T \Delta S > \Delta H \) with both \( \Delta H \) and \( \Delta S \) being positive. Here, \( \Delta G = \Delta H - T \Delta S < 0 \), yielding a negative free energy change, which means the reaction is spontaneous. Thus, option (c) is correct.

Step 5: Analyze Option D

Option (d) says the reaction is spontaneous if \( T \Delta S = \Delta H \) with both \( \Delta H \) and \( \Delta S \) positive. This gives \( \Delta G = \Delta H - T \Delta S = 0 \), indicating the reaction is at equilibrium and not spontaneous in one direction. Therefore, option (d) is incorrect.

Key Concepts

Spontaneity Criteria

To determine if a chemical reaction happens on its own, we look at the concept of spontaneity. Essentially, spontaneity refers to whether a reaction can occur without needing any external push or additional energy input. While some reactions happen spontaneously, others need a little help or don't occur naturally at all. The heart of understanding spontaneity lies in the Gibbs Free Energy change (\( \Delta G \)), a key factor that helps us predict the spontaneity of a reaction.

• If \( \Delta G \) is less than zero (negative), the reaction is spontaneous.
• If \( \Delta G \) is greater than zero (positive), the reaction is non-spontaneous.
• If \( \Delta G \) is zero, the reaction is at equilibrium.

The Gibbs Free Energy equation is given by \( \Delta G = \Delta H - T \Delta S \), where each component plays a vital part in determining spontaneity. Understanding these terms and how they interact is crucial for predicting if a reaction will take place naturally.

Enthalpy Change

Enthalpy (\( \Delta H \)) is one of the critical components of the Gibbs Free Energy equation, representing the total heat content in a system at constant pressure. When it comes to spontaneity, understanding \( \Delta H \) helps us see how energy flows in a reaction. Here's what you need to know:

• A positive \( \Delta H \) indicates an endothermic reaction, where the system absorbs heat from the surroundings.
• A negative \( \Delta H \) signals an exothermic reaction, where the system releases heat to the surroundings.

In the context of spontaneity, \( \Delta H \) can be seen as the driving force of the energy exchange. A negative enthalpy change often favors spontaneity, especially when paired with a significant entropy increase. However, it's not just \( \Delta H \) alone that determines if a reaction will be spontaneous. The other factors, \( T \) and \( \Delta S \), play equally essential roles in this thermodynamic story.

Entropy Change

Entropy (\( \Delta S \)) is another pivotal component of the Gibbs Free Energy equation that helps decide spontaneity. Entropy reveals the level of disorder or randomness in a system. Here's a simple way to think about it:

• A positive \( \Delta S \) means there is an increase in disorder.
• A negative \( \Delta S \) indicates there's a decrease in disorder.

In thermodynamics, nature tends to prefer randomness or chaos, so reactions with a positive entropy change are often more likely to be spontaneous. Let's think of \( \Delta S \) as the chaos component. This preference for increased disorder is one reason why reactions with a positive \( \Delta S \) can help make \( \Delta G \) negative, promoting spontaneity. It's important to remember that \( \Delta S \) and \( \Delta H \) together shape how the energy in a system is handled, influencing whether a process occurs on its own.

Temperature in Kelvin

The temperature in Kelvin (\( T \)) is integral to the Gibbs Free Energy equation as it links enthalpy and entropy to decide spontaneity. Kelvin is used because it avoids negative temperature values, simplifying thermodynamic calculations.How does temperature factor into spontaneity?

• Higher temperatures can amplify the effect of \( \Delta S \), making entropy changes more significant in determining \( \Delta G \).
• At lower temperatures, the enthalpy component might dominate, affecting spontaneity differently.

Consider \( \Delta G = \Delta H - T \Delta S \): an increase in \( T \) means that even a small \( \Delta S \) could lead to a negative \( \Delta G \) and thus a spontaneous reaction. That's why high temperatures often favor spontaneity when \( \Delta S \) is positive. Meanwhile, if \( \Delta S \) is negative, low temperatures might be needed for the reaction to proceed naturally. Temperature, enthalpy, and entropy form a dynamic interplay that governs the spontaneity of reactions.