Question

Consider the following equation for a cell reaction:
$\begin{align}
& \text{ A + B }\to \text{ C + D };\text{ }{{\text{E}}^{\text{o}}}\text{ = x volts },\text{ }\Delta \text{G = }\Delta {{\text{G}}_{\text{1}}}\text{ } \\
& \text{ 2A+2B}\to \text{ 2C + 2D };\text{ }{{\text{E}}^{\text{o}}}\text{ = y volts , }\Delta \text{G = }\Delta {{\text{G}}_{2}}\text{ } \\
\end{align}$
A)$\text{ x = y }$, $\text{ }\Delta {{\text{G}}_{2}}\text{ = }\Delta {{\text{G}}_{1}}\text{ }$
B)$\text{ x }>\text{ y }$, $\text{ }\Delta {{\text{G}}_{2}}>\Delta {{\text{G}}_{1}}\text{ }$
C)$\text{ x = y }$, $\text{ }\Delta {{\text{G}}_{2}}\text{ = 2}\Delta {{\text{G}}_{1}}\text{ }$
D)$\text{ x }<\text{ y }$, $\text{ }\Delta {{\text{G}}_{2}}\text{ = 2}\Delta {{\text{G}}_{1}}\text{ }$

Answer 1

Hint: The electrode standard potential values do not depend on the stoichiometric coefficients for a half-reaction .as it is an intensive property. However, Gibbs free energy is an extensive property and it depends on the stoichiometric coefficients of the cell reaction.

Complete step by step answer:
We are given the following cell reaction:
1)\[\text{ A + B }\to \text{ C + D }\]
The standard electrode potential for the reaction is $\text{ }{{\text{E}}^{\text{o}}}\text{ = x volts }$ and the change in the Gibbs free energy for the reaction is equal to $\text{ }\Delta \text{G = }\Delta {{\text{G}}_{\text{1}}}\text{ }$
2)$\text{ 2A+2B}\to \text{ 2C + 2D }$
The standard electrode potential for the reaction is $\text{ }{{\text{E}}^{\text{o}}}\text{ = y volts }$ and the change in the Gibbs free energy for the reaction is equal to, $\text{ }\Delta \text{G = }\Delta {{\text{G}}_{2}}\text{ }$
The EMF and its temperature of a cell used to determine the various thermodynamic quantities in a standard state such as a standard free energy change. It is denoted as$\text{ }\Delta {{\text{G}}^{\text{o}}}\text{ }$, standard entropy change, etc.
The standard free energy change for a cell reaction is given by the relation stated below,
$\text{ }\!\!\Delta\!\!\text{ }{{\text{G}}^{\text{o}}}\text{=}-\text{nF}{{\text{E}}^{\text{o}}}\text{ }$
Where, $\text{ }\Delta {{\text{G}}^{\text{o}}}\text{ }$is standard free energy change, n is associated with the number of electron involved in reaction F is faraday's constant and $\text{ }{{\text{E}}^{\text{o}}}\text{ }$ is a standard electrode potential of a cell reaction.
The standard electrode potential $\text{ }{{\text{E}}^{\text{o}}}\text{ }$is an intensive property.it does not depend on the amount of the reactant and the product in a cell reaction. Thus, there is no change in the electrode potential .thus the cell reaction for the reaction 1) and 2) we have,
$\text{ }{{\text{E}}^{\text{o}}}\text{ = x = y }$
The $\text{ }\Delta \text{G }$is an extensive property and electrode potential is an intensive property. It means the Gibbs free energy depends on the number of moles of reactant taking part in the reaction and forming a product. It depends on the coefficient of the reactant and the product of the cell reaction. We are interested to determine the relationship between the change in the free energy difference between the reaction 1) and 2).
Let's multiply the cell reaction 1) by the two. we have,
$\text{ 2 }\left( \text{ A + B }\to \text{ C + D } \right)\text{ = 2A + 2B }\to \text{ 2C + 2D }$
The gigs free energy of the reaction also gets doubled. The Gibbs free energy for the reaction 1) would be,
$\text{ }\Delta {{\text{G}}_{\text{(new)}}}\text{ = 2}\times \Delta {{\text{G}}_{1}}\text{ }$
From the given data, it is clear that new gigs free energy $\text{ }\Delta {{\text{G}}_{\text{(new)}}}\text{ }$ is equal to the Gibbs free energy of the reaction 2). That is,
$\text{ }\Delta {{\text{G}}_{2}}\text{ = 2}\Delta {{\text{G}}_{1}}\text{ }$
Thus, for the given cell reaction, $\text{ x = y }$ and$\text{ }\Delta {{\text{G}}_{2}}\text{ = 2}\Delta {{\text{G}}_{1}}\text{ }$.

Hence, (C) is the correct option.

Note: Extensive properties are those which depend on the amount of matter and intensive properties. Some of the examples are, entropy, internal energy, enthalpy, heat capacity, etc. the properties which do not depend on the amount of the substance present in the system. Some of the examples are temperature, density, pressure, chemical potential, etc. the ratio of two extensive properties of the system is an intensive property.