To find the standard free energy change (ΔG°) for the decomposition of solid calcium bicarbonate (Ca(HCO₃)₂), we can follow these steps:
### Step 1: Write the balanced chemical equation
The decomposition reaction is given as:
\[ \text{Ca(HCO}_3\text{)}_2(s) \rightleftharpoons \text{CaCO}_3(s) + \text{CO}_2(g) + \text{H}_2O(g) \]
### Step 2: Identify the total pressure and the equilibrium condition
It is given that the total pressure at equilibrium is \( P_{total} = 0.2 \, \text{bar} \) at \( T = 420 \, \text{K} \).
### Step 3: Determine the partial pressures of the gaseous products
Since the reaction produces equal moles of CO₂ and H₂O, we can denote their partial pressures as:
- \( P_{\text{CO}_2} = x \)
- \( P_{\text{H}_2O} = x \)
The total pressure is the sum of the partial pressures of the gaseous products:
\[ P_{total} = P_{\text{CO}_2} + P_{\text{H}_2O} = x + x = 2x \]
Setting this equal to the total pressure:
\[ 2x = 0.2 \, \text{bar} \]
Thus,
\[ x = \frac{0.2}{2} = 0.1 \, \text{bar} \]
So,
\[ P_{\text{CO}_2} = 0.1 \, \text{bar} \]
\[ P_{\text{H}_2O} = 0.1 \, \text{bar} \]
### Step 4: Calculate the equilibrium constant (Kp)
The equilibrium constant \( K_p \) for the reaction is given by:
\[ K_p = P_{\text{CO}_2} \times P_{\text{H}_2O} \]
Substituting the values we found:
\[ K_p = (0.1 \, \text{bar}) \times (0.1 \, \text{bar}) = 0.01 \, \text{bar}^2 \]
### Step 5: Use the relationship between ΔG° and Kp
The standard free energy change is related to the equilibrium constant by the equation:
\[ \Delta G° = -RT \ln K_p \]
Where:
- \( R = 8.314 \, \text{J/mol·K} \)
- \( T = 420 \, \text{K} \)
- \( K_p = 0.01 \)
### Step 6: Calculate ΔG°
Substituting the values into the equation:
\[ \Delta G° = - (8.314 \, \text{J/mol·K}) \times (420 \, \text{K}) \times \ln(0.01) \]
Calculating \( \ln(0.01) \):
\[ \ln(0.01) = -4.605 \]
Now substituting this back:
\[ \Delta G° = - (8.314 \times 420 \times -4.605) \]
Calculating the product:
\[ \Delta G° = 8.314 \times 420 \times 4.605 \]
\[ \Delta G° \approx 16081.66 \, \text{J/mol} \]
### Step 7: Convert ΔG° to kilojoules
To convert from joules to kilojoules:
\[ \Delta G° = \frac{16081.66}{1000} \approx 16.082 \, \text{kJ/mol} \]
### Final Answer
Thus, the standard free energy change for the given reaction is:
\[ \Delta G° \approx 16.082 \, \text{kJ/mol} \]
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