To find the value of [tex]\(\Delta G\)[/tex] at [tex]\(5000 \, \text{K}\)[/tex] given that [tex]\(\Delta H = -220 \, \text{kJ/mol}\)[/tex] and [tex]\(\Delta S = -0.05 \, \text{kJ/(mol} \cdot \text{K)}\)[/tex], we use the Gibbs free energy formula:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
Substitute the given values into the formula:
[tex]\[
\Delta G = -220 \, \text{kJ/mol} - (5000 \, \text{K} \times -0.05 \, \text{kJ/(mol} \cdot \text{K)})
\][/tex]
Next, we perform the multiplication inside the formula:
[tex]\[
5000 \, \text{K} \times -0.05 \, \text{kJ/(mol} \cdot \text{K)} = -250 \, \text{kJ/mol}
\][/tex]
So the equation now becomes:
[tex]\[
\Delta G = -220 \, \text{kJ/mol} - (-250 \, \text{kJ/mol})
\][/tex]
Subtracting a negative is the same as adding a positive:
[tex]\[
\Delta G = -220 \, \text{kJ/mol} + 250 \, \text{kJ/mol}
\][/tex]
Finally, combine the terms:
[tex]\[
\Delta G = 30 \, \text{kJ/mol}
\][/tex]
So, the value of [tex]\(\Delta G\)[/tex] at [tex]\(5000 \, \text{K}\)[/tex] is:
[tex]\[
\Delta G = 30 \, \text{kJ/mol}
\][/tex]
Therefore, the correct answer is:
D. [tex]\(30 \, \text{kJ}\)[/tex]
