To find the value of [tex]\(\Delta G\)[/tex] at [tex]\(500 \, K\)[/tex] given [tex]\(\Delta H = 27 \, kJ / mol\)[/tex] and [tex]\(\Delta S = 0.09 \, kJ / ( mol \cdot K )\)[/tex], we use the Gibbs free energy equation:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
1. Identify the given data:
- [tex]\(\Delta H = 27 \, kJ / mol\)[/tex]
- [tex]\(\Delta S = 0.09 \, kJ / ( mol \cdot K )\)[/tex]
- [tex]\(T = 500 \, K\)[/tex]
2. Substitute the values into the equation:
[tex]\[
\Delta G = 27 - 500 \times 0.09
\][/tex]
3. Calculate the product of [tex]\(T\)[/tex] and [tex]\(\Delta S\)[/tex]:
[tex]\[
500 \times 0.09 = 45
\][/tex]
4. Subtract this product from [tex]\(\Delta H\)[/tex]:
[tex]\[
\Delta G = 27 - 45
\][/tex]
5. Perform the subtraction:
[tex]\[
\Delta G = -18 \, kJ / mol
\][/tex]
So, the value of [tex]\(\Delta G\)[/tex] is [tex]\(-18 \, kJ / mol\)[/tex], which corresponds to option C:
C. [tex]\(\Delta G = -18 \, kJ / mol\)[/tex]
