To determine the Gibbs free energy change [tex]\(\Delta G\)[/tex] at a temperature of 5000 K, given the change in enthalpy [tex]\(\Delta H = -220 \text{ kJ/mol}\)[/tex] and the change in entropy [tex]\(\Delta S = -0.05 \text{ kJ/(mol⋅K)}\)[/tex], we use the fundamental thermodynamic relation:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
Here's the step-by-step process to solve this:
1. Identify the given values:
- Temperature [tex]\(T = 5000 \text{ K}\)[/tex]
- Change in enthalpy [tex]\(\Delta H = -220 \text{ kJ/mol}\)[/tex]
- Change in entropy [tex]\(\Delta S = -0.05 \text{ kJ/(mol⋅K)}\)[/tex]
2. Substitute the given values into the equation:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
Substituting the given values:
[tex]\[
\Delta G = -220 \text{ kJ/mol} - (5000 \text{ K} \times -0.05 \text{ kJ/(mol⋅K)})
\][/tex]
3. Perform the multiplication inside the parenthesis:
[tex]\[
5000 \text{ K} \times -0.05 \text{ kJ/(mol⋅K)} = -250 \text{ kJ/mol}
\][/tex]
4. Subtract this value from [tex]\(\Delta H\)[/tex]:
[tex]\[
\Delta G = -220 \text{ kJ/mol} - (-250 \text{ kJ/mol})
\][/tex]
Simplifying the subtraction:
[tex]\[
\Delta G = -220 \text{ kJ/mol} + 250 \text{ kJ/mol}
\][/tex]
5. Add the values:
[tex]\[
\Delta G = 30 \text{ kJ/mol}
\][/tex]
So, the value for [tex]\(\Delta G\)[/tex] at 5000 K is:
[tex]\(\boxed{30 \text{ kJ}}\)[/tex]
Hence, the correct answer is:
B. 30 kJ
