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: Determine $\Delta {{\text{G}}^ \circ }$ for the following reaction: ${\text{CO}}\left( {\text{g}} \right) + \dfrac{{\text{1}}}{{\text{2}}}{{\text{O}}_{\text{2}}}\left( {\text{g}} \right) \to {\text{C}}{{\text{O}}_{\text{2}}}\left( {\text{g}} \right);{\text{ }}\Delta {{\text{H}}^ \circ } = - 282.84{\text{ kJ}}$[Given: ${\text{S}}_{{\text{C}}{{\text{O}}_2}}^ \circ = 213.8{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$, ${\text{S}}_{{\text{CO}}}^ \circ = 197.9{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$, ${\text{S}}_{{{\text{O}}_2}}^ \circ = 205.8{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$](1)$- 157.33{\text{ kJ}}$(2)$+ 201.033{\text{ kJ}}$(3)$- 256.91{\text{ kJ}}$(4)$+ 257.033{\text{ kJ}}$

Last updated date: 26th Feb 2024
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Hint: The measure of randomness or disordered distribution is known as entropy. The randomness is always higher in a gaseous state. More the number of gaseous molecules higher is the entropy. To solve this we must know the expression that gives the relation between free energy, entropy and enthalpy.
Formula Used:$\Delta {\text{S}}_{{\text{reaction}}}^ \circ = \Delta {{\text{S}}_{{\text{products}}}} - \Delta {{\text{S}}_{{\text{reactants}}}}$
$\Delta {{\text{G}}^ \circ } = \Delta {{\text{H}}^ \circ } - {\text{T}}\Delta {{\text{S}}^ \circ }$

Complete step-by-step answer:VWe are given the reaction as follows:
${\text{CO}}\left( {\text{g}} \right) + \dfrac{{\text{1}}}{{\text{2}}}{{\text{O}}_{\text{2}}}\left( {\text{g}} \right) \to {\text{C}}{{\text{O}}_{\text{2}}}\left( {\text{g}} \right)$
We know that the measure of randomness or disordered distribution is known as entropy.
We will first calculate the change in entropy of the reaction using the equation as follows:
$\Delta {\text{S}}_{{\text{reaction}}}^ \circ = \Delta {{\text{S}}_{{\text{products}}}} - \Delta {{\text{S}}_{{\text{reactants}}}}$
Where, $\Delta {\text{S}}_{{\text{reaction}}}^ \circ$ is the standard change in entropy of the reaction.
We are given the values of standard entropies as ${\text{S}}_{{\text{C}}{{\text{O}}_2}}^ \circ = 213.8{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$, ${\text{S}}_{{\text{CO}}}^ \circ = 197.9{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$, ${\text{S}}_{{{\text{O}}_2}}^ \circ = 205.8{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$. Thus,
$\Delta {\text{S}}_{{\text{reaction}}}^ \circ = {\text{213}}{\text{.8}} - \left( {197.9 + \dfrac{1}{2} \times 205.8} \right)$
$\Delta {\text{S}}_{{\text{reaction}}}^ \circ = - 87{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$
Thus, the standard change in entropy of the reaction is $- 87{\text{ J }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}} = - 87 \times {10^{ - 3}}{\text{ kJ }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}$.
We know the expression that gives the relation between free energy, entropy and enthalpy is as follows:
$\Delta {{\text{G}}^ \circ } = \Delta {{\text{H}}^ \circ } - {\text{T}}\Delta {{\text{S}}^ \circ }$
Where, $\Delta {{\text{G}}^ \circ }$ is the standard change in Gibb’s free energy,
$\Delta {{\text{H}}^ \circ }$ is the standard change in enthalpy,
${\text{T}}$ is the temperature,
$\Delta {{\text{S}}^ \circ }$ is the standard change in entropy.
We are given the standard values. At standard condition, ${\text{T}} = {25^ \circ }{\text{C}} = 298{\text{ K}}$. We are given that $\Delta {{\text{H}}^ \circ } = - 282.84{\text{ kJ}}$. Thus,
$\Delta {{\text{G}}^ \circ } = - 282.84{\text{ kJ}} - \left[ {{\text{298 K}}\left( { - 87 \times {{10}^{ - 3}}{\text{ kJ }}{{\text{K}}^{ - 1}}{\text{ mo}}{{\text{l}}^{ - 1}}} \right)} \right]$
$\Delta {{\text{G}}^ \circ } = - 256.914{\text{ kJ}}$
Thus, $\Delta {{\text{G}}^ \circ }$ for the given reaction is $- 256.91{\text{ kJ}}$.
Thus, the correct option is (3) $- 256.91{\text{ kJ}}$.

Note:: We know that if the free energy change is negative then the reaction is spontaneous. If the free energy change is positive then the reaction is non-spontaneous. Here, $\Delta {{\text{G}}^ \circ } = - 256.914{\text{ kJ}}$ which is a negative value. Thus, the given reaction is a spontaneous reaction.