To determine the standard Gibbs free energy change ([tex]\(\Delta G^\circ\)[/tex]) for the given reaction at equilibrium, follow these steps:
### Step 1: Write down the balanced chemical equation.
For the reaction:
[tex]\[ P_2(g) + 3 Cl_2(g) \rightarrow 2 PCl_3(g) \][/tex]
### Step 2: Note the partial pressures of each gas at equilibrium.
Given:
- [tex]\( P_{P_2} = 0.0268 \)[/tex] atm
- [tex]\( P_{Cl_2} = 0.0689 \)[/tex] atm
- [tex]\( P_{PCl_3} = 0.399 \)[/tex] atm
### Step 3: Determine the reaction quotient, [tex]\( Q \)[/tex].
For the reaction quotient, [tex]\( Q \)[/tex], we need to use the partial pressures provided. The reaction quotient for gases is given by:
[tex]\[ Q = \frac{(P_{PCl_3})^2}{P_{P_2} \times (P_{Cl_2})^3} \][/tex]
Substitute the given pressures into the equation:
[tex]\[ Q = \frac{(0.399)^2}{(0.0268) \times (0.0689)^3} \][/tex]
### Step 4: Calculate [tex]\( Q \)[/tex].
Using the provided result:
[tex]\[ Q \approx 18161.5676 \][/tex]
### Step 5: Use the relationship between [tex]\( \Delta G^\circ \)[/tex] and [tex]\( Q \)[/tex].
The standard Gibbs free energy change is calculated using the equation:
[tex]\[ \Delta G^\circ = -RT \ln(Q) \][/tex]
Where:
- [tex]\( R \)[/tex] is the universal gas constant [tex]\( 8.314 \)[/tex] J/(mol·K)
- [tex]\( T \)[/tex] is the temperature in Kelvin, which is [tex]\( 880 \)[/tex] K
- [tex]\( Q \)[/tex] is the reaction quotient
### Step 6: Calculate [tex]\( \Delta G^\circ \)[/tex].
Plug in the values:
[tex]\[ \Delta G^\circ = - (8.314 \, \text{J/(mol·K)}) \times 880 \, \text{K} \times \ln(18161.5676) \][/tex]
Referring to the provided answer:
[tex]\[ \Delta G^\circ \approx -71751.61 \, \text{J/mol} \][/tex]
OR
[tex]\[ \Delta G^\circ \approx -71.75 \, \text{kJ/mol} \][/tex] (since [tex]\( 1 \, \text{kJ} = 1000 \, \text{J} \)[/tex]).
### Final Answer:
The standard Gibbs free energy change ([tex]\(\Delta G^\circ\)[/tex]) for the reaction at 880 K is approximately [tex]\(-71751.61 \, \text{J/mol}\)[/tex] or [tex]\(-71.75 \, \text{kJ/mol}\)[/tex].
