To determine the Gibbs free energy ([tex]\( \Delta G \)[/tex]) of the reaction, we can use the Gibbs free energy equation:
[tex]\[ \Delta G = \Delta H - T \Delta S \][/tex]
where:
- [tex]\( \Delta G \)[/tex] is the Gibbs free energy change,
- [tex]\( \Delta H \)[/tex] is the enthalpy change,
- [tex]\( T \)[/tex] is the temperature in Kelvin,
- [tex]\( \Delta S \)[/tex] is the entropy change.
Given the values:
- Temperature, [tex]\( T = 298 \)[/tex] K,
- Enthalpy change, [tex]\( \Delta H = -1652 \)[/tex] kJ/mol,
- Entropy change, [tex]\( \Delta S = 0.097 \)[/tex] kJ/(K·mol),
So, we need to plug these values into the equation:
[tex]\[ \Delta G = (-1652) - (298 \times 0.097) \][/tex]
First, calculate [tex]\( 298 \times 0.097 \)[/tex]:
[tex]\[ 298 \times 0.097 = 28.906 \][/tex]
Next, substitute this value back into the equation for [tex]\( \Delta G \)[/tex]:
[tex]\[ \Delta G = -1652 - 28.906 \][/tex]
Subtracting 28.906 from -1652:
[tex]\[ \Delta G = -1652 - 28.906 = -1680.906 \][/tex]
The Gibbs free energy change ([tex]\( \Delta G \)[/tex]) for the reaction is thus:
[tex]\[ \Delta G = -1680.906 \, \text{kJ/mol} \][/tex]
From the provided choices, the correct one closely matching this calculated value is:
A. [tex]$-745 kJ$[/tex]
B. [tex]$225 kJ$[/tex]
C. [tex]$-907 kJ$[/tex]
D. [tex]$67,000 kJ$[/tex]
As none of these directly match our calculated value, it appears there has been a mistake in the provided multiple-choice options. The correct Gibbs free energy change, based on the calculation, is indeed:
[tex]\[ \Delta G = -1680.906 \, \text{kJ/mol} \][/tex]
