To find the Gibbs free energy change ([tex]\(\Delta G\)[/tex]) at a given temperature ([tex]\(T\)[/tex]), we can use the following formula:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
Where:
- [tex]\(\Delta H\)[/tex] is the enthalpy change.
- [tex]\(T\)[/tex] is the temperature in Kelvin.
- [tex]\(\Delta S\)[/tex] is the entropy change.
We are given the following values:
- [tex]\(\Delta H = -220 \text{ kJ/mol}\)[/tex]
- [tex]\(T = 1000 \text{ K}\)[/tex]
- [tex]\(\Delta S = -0.05 \text{ kJ/(mol K)}\)[/tex]
Let's substitute these values into the formula step-by-step:
Step 1: Write down the formula and substitute the values:
[tex]\[ \Delta G = -220 \text{ kJ/mol} - (1000 \text{ K} \times -0.05 \text{ kJ/(mol K)}) \][/tex]
Step 2: Calculate the term [tex]\(T \Delta S\)[/tex]:
[tex]\[ 1000 \text{ K} \times -0.05 \text{ kJ/(mol K)} = -50 \text{ kJ/mol} \][/tex]
Step 3: Substitute this result back into the formula:
[tex]\[ \Delta G = -220 \text{ kJ/mol} - (-50 \text{ kJ/mol}) \][/tex]
Step 4: Simplify the expression:
[tex]\[ \Delta G = -220 \text{ kJ/mol} + 50 \text{ kJ/mol} = -170 \text{ kJ/mol} \][/tex]
Therefore, the value of [tex]\(\Delta G\)[/tex] at [tex]\(1000\)[/tex] K is [tex]\(-170 \text{ kJ/mol}\)[/tex].
The correct answer is:
D. [tex]\(-170 \text{ kJ}\)[/tex]
