Question

Given below are two statements labeled Assertion A and the labeled as Reason R : 
Assertion A: In the equation \(\Delta_rG=-nFE_{cell}\) value of \(\Delta_rG\) depends on n. 
Reasons R: \(E_{cell}\) is an intensive property and \(\Delta_rG\) is an extensive property. 
In the light of the above statements, choose the correct answer from the options below :

• A. Both A and R are true and R is the correct explanation of A.
• B. A is true but R is false.
• C. A is false but R is true.
• D. Both A and R are true and R is the correct explanation of A.

Answer 1

The Correct Option is D

To solve this assertion-reason type question, we need to understand the relationship between the Gibbs free energy change (\(\Delta_rG\)), the cell potential (\(E_{cell}\)), and the number of moles of electrons transferred (n) in an electrochemical reaction. The equation in question is:
\(\Delta_rG=-nFE_{cell}\)  
Here, \(\Delta_rG\) represents the Gibbs free energy change which is an extensive property. Extensive properties depend on the amount of substance present. The number of moles of electrons, n, is directly proportional to the change in Gibbs free energy, meaning as n increases, \(\Delta_rG\) becomes more negative.
On the other hand, \(E_{cell}\) is the electromotive force or cell potential, an intensive property, which is independent of the amount of substance.
 

Properties	Relation in \(\Delta_rG\) equation
Intensive Property	\(E_{cell}\)
Extensive Property	\(\Delta_rG\)

In the given context:
Assertion A: States that \(\Delta_rG\) depends on n, which is true because \(\Delta_rG\) increases in magnitude with an increase in n, keeping all other factors constant.
Reason R: Points out that \(E_{cell}\) is an intensive property and \(\Delta_rG\) is an extensive property. This distinction accurately supports the mechanism of how \(\Delta_rG\) is calculated in relation to n and \(E_{cell}\).
Thus, both Assertion A and Reason R are true, and Reason R correctly explains why Assertion A is accurate.
Therefore, the correct conclusion is:
Both A and R are true and R is the correct explanation of A.

Answer 2

The correct option is (D): Both A and R are true and R is the correct explanation of A. 

The formula for Gibbs Free Energy change in a reaction is given by:

\(\Delta_rG = -nFE_{cell}\)

Step 1: Understanding the Formula

- \(\Delta_rG\) represents the change in Gibbs free energy for the reaction. - \(n\) is the number of moles of electrons transferred in the reaction. - \(F\) is the Faraday constant, and \(E_{cell}\) is the cell potential. This equation shows the relationship between the change in Gibbs free energy (\(\Delta_rG\)) and the cell potential (\(E_{cell}\)).

Step 2: Intensive and Extensive Properties

The key to understanding the problem is the distinction between intensive and extensive properties: - **Intensive Property:** A property that does not depend on the amount of substance, such as \(E_{cell}\) (cell potential). The value of \(E_{cell}\) remains the same regardless of the amount of material used in the electrochemical cell. - **Extensive Property:** A property that depends on the amount of substance, such as \(\Delta_rG\) (Gibbs free energy change). The change in Gibbs free energy is directly proportional to the amount of material involved in the reaction, which is why it is considered an extensive property.

Step 3: Explanation of \(e^{\theta}\)

The term \(e^{\theta}\) refers to the number of electrons transferred during the cell reaction. The value of \(\Delta_rG\) depends on the number of electrons transferred in the reaction, hence it is an extensive property.

Conclusion:

Based on the above analysis: - \(E_{cell}\) is an intensive property. - \(\Delta_rG\) is an extensive property, as it depends on the number of electrons transferred during the cell reaction. Therefore, the correct option is:

Final Answer:

The correct option is (D): Both A and R are true and R is the correct explanation of A.