To determine the Gibbs free energy change ([tex]\(\Delta G_{\text{system}}\)[/tex]) for the given system, we can use the formula:
[tex]\[ \Delta G_{\text{system}} = \Delta H_{\text{system}} - T \Delta S_{\text{system}} \][/tex]
Here are the given values:
- [tex]\(\Delta H_{\text{system}} = -232 \, \text{kJ}\)[/tex]
- [tex]\(T = 293 \, \text{K}\)[/tex]
- [tex]\(\Delta S_{\text{system}} = 195 \, \text{J/K}\)[/tex]
First, we need to ensure that the units are consistent. Notably, [tex]\(\Delta H_{\text{system}}\)[/tex] is in kilojoules (kJ) and [tex]\(\Delta S_{\text{system}}\)[/tex] is in joules per Kelvin (J/K). Hence, we need to convert [tex]\(\Delta S_{\text{system}}\)[/tex] from joules to kilojoules:
[tex]\[ \Delta S_{\text{system}} = 195 \, \text{J/K} = 0.195 \, \text{kJ/K} \][/tex]
(This is done by dividing 195 by 1000 because there are 1000 joules in a kilojoule.)
Now we can plug these values into the [tex]\(\Delta G_{\text{system}}\)[/tex] formula:
[tex]\[ \Delta G_{\text{system}} = \Delta H_{\text{system}} - T \Delta S_{\text{system}} \][/tex]
[tex]\[ \Delta G_{\text{system}} = -232 \, \text{kJ} - 293 \times 0.195 \, \text{kJ/K} \][/tex]
Next, let's calculate the term [tex]\(T \Delta S_{\text{system}}\)[/tex]:
[tex]\[ 293 \times 0.195 = 57.135 \, \text{kJ} \][/tex]
Finally, we subtract this value from [tex]\(\Delta H_{\text{system}}\)[/tex]:
[tex]\[ \Delta G_{\text{system}} = -232 \, \text{kJ} - 57.135 \, \text{kJ} \][/tex]
[tex]\[ \Delta G_{\text{system}} = -232 - 57.135 \][/tex]
[tex]\[ \Delta G_{\text{system}} = -289.135 \, \text{kJ} \][/tex]
Hence, the correct value of [tex]\(\Delta G_{\text{system}}\)[/tex] is [tex]\(-289 \, \text{kJ}\)[/tex], which matches the first option. The closest numerical answer provided in the options and the true Gibbs free energy value is:
[tex]\[ -289 \, \text{kJ} \][/tex]
So, the answer is:
[tex]\[ -289 \, \text{kJ} \][/tex]
