Answer :
To solve for the Gibbs free energy change ([tex]$\Delta G$[/tex]) and determine the spontaneity of the process, we start with the formula:
[tex]$$
\Delta G = \Delta H - T \Delta S
$$[/tex]
Step 1. Convert Units (if necessary):
- The enthalpy change is given as [tex]$\Delta H = 147\,\text{kJ}$[/tex]. We need to convert this to joules since the entropy change ([tex]$\Delta S$[/tex]) is in joules per kelvin. Thus,
[tex]$$
\Delta H = 147\,\text{kJ} = 147\,000\,\text{J}.
$$[/tex]
- Temperature is given as [tex]$149^\circ\text{C}$[/tex]. Converting to Kelvin:
[tex]$$
T = 149 + 273.15 = 422.15\,\text{K}.
$$[/tex]
Step 2. Calculate [tex]$T\Delta S$[/tex]:
- The entropy change is given as [tex]$\Delta S = -67.0\,\frac{\text{J}}{\text{K}}$[/tex], so
[tex]$$
T\Delta S = 422.15\,\text{K} \times (-67.0\,\frac{\text{J}}{\text{K}}) \approx -28\,284.05\,\text{J}.
$$[/tex]
Step 3. Calculate [tex]$\Delta G$[/tex]:
Substitute the values into the formula:
[tex]$$
\Delta G = 147\,000\,\text{J} - (-28\,284.05\,\text{J}) = 147\,000\,\text{J} + 28\,284.05\,\text{J} \approx 175\,284.05\,\text{J}.
$$[/tex]
Convert [tex]$\Delta G$[/tex] back to kilojoules:
[tex]$$
\Delta G \approx \frac{175\,284.05\,\text{J}}{1000} \approx 175\,\text{kJ}.
$$[/tex]
Step 4. Determine Spontaneity:
A process is spontaneous if [tex]$\Delta G$[/tex] is negative. Here, [tex]$\Delta G$[/tex] is positive ([tex]$175\,\text{kJ}$[/tex]), which means the process is not spontaneous.
Final Answer:
[tex]$$
\Delta G \approx 175\,\text{kJ}; \quad \text{the system is not spontaneous}.
$$[/tex]
[tex]$$
\Delta G = \Delta H - T \Delta S
$$[/tex]
Step 1. Convert Units (if necessary):
- The enthalpy change is given as [tex]$\Delta H = 147\,\text{kJ}$[/tex]. We need to convert this to joules since the entropy change ([tex]$\Delta S$[/tex]) is in joules per kelvin. Thus,
[tex]$$
\Delta H = 147\,\text{kJ} = 147\,000\,\text{J}.
$$[/tex]
- Temperature is given as [tex]$149^\circ\text{C}$[/tex]. Converting to Kelvin:
[tex]$$
T = 149 + 273.15 = 422.15\,\text{K}.
$$[/tex]
Step 2. Calculate [tex]$T\Delta S$[/tex]:
- The entropy change is given as [tex]$\Delta S = -67.0\,\frac{\text{J}}{\text{K}}$[/tex], so
[tex]$$
T\Delta S = 422.15\,\text{K} \times (-67.0\,\frac{\text{J}}{\text{K}}) \approx -28\,284.05\,\text{J}.
$$[/tex]
Step 3. Calculate [tex]$\Delta G$[/tex]:
Substitute the values into the formula:
[tex]$$
\Delta G = 147\,000\,\text{J} - (-28\,284.05\,\text{J}) = 147\,000\,\text{J} + 28\,284.05\,\text{J} \approx 175\,284.05\,\text{J}.
$$[/tex]
Convert [tex]$\Delta G$[/tex] back to kilojoules:
[tex]$$
\Delta G \approx \frac{175\,284.05\,\text{J}}{1000} \approx 175\,\text{kJ}.
$$[/tex]
Step 4. Determine Spontaneity:
A process is spontaneous if [tex]$\Delta G$[/tex] is negative. Here, [tex]$\Delta G$[/tex] is positive ([tex]$175\,\text{kJ}$[/tex]), which means the process is not spontaneous.
Final Answer:
[tex]$$
\Delta G \approx 175\,\text{kJ}; \quad \text{the system is not spontaneous}.
$$[/tex]
