In the preparation of MgO, the reaction is `MgCO_(3)(s) hArr` `MgO(s)` + `CO_(2)(g)` Experiments carried out between `50^(@) C and 950^(@)`C led to a set of `K_(p)` values fitting an empirical equation log `K_(p) = 7.310 - (8500)/(T)` . If the reaction is carried out in quiet air, then

Options

A. the complete decomposition temperature
of the `MgCO_(3) = 890^(@)`C
B. `K_(p)` = 1 atm
C. `P_(CO_(2)) ` = 1 atm
D. `K_(p) = 10 ` atm

Answer

To solve the problem regarding the preparation of magnesium oxide (MgO) from magnesium carbonate (MgCO₃), we need to analyze the given equilibrium expression and the empirical equation provided.

### Step-by-Step Solution:

1. **Understand the Reaction**:
The decomposition of magnesium carbonate can be represented as:
\[ \text{MgCO}_3(s) \rightleftharpoons \text{MgO}(s) + \text{CO}_2(g) \]
In this reaction, magnesium carbonate decomposes into magnesium oxide and carbon dioxide.

2. **Identify the Equilibrium Constant**:
The equilibrium constant \( K_p \) for this reaction is defined in terms of the partial pressure of the gaseous product (CO₂). Since solids do not appear in the expression for \( K_p \), we have:
\[ K_p = P_{\text{CO}_2} \]
where \( P_{\text{CO}_2} \) is the partial pressure of carbon dioxide.

3. **Use the Given Empirical Equation**:
The problem provides the empirical equation for \( K_p \):
\[ \log K_p = 7.310 - \frac{8500}{T} \]
Here, \( T \) is the temperature in Kelvin.

4. **Determine the Conditions**:
The problem states that the reaction is carried out in quiet air, which implies that the partial pressure of carbon dioxide above the solid is 1 atm. Therefore:
\[ K_p = 1 \text{ atm} \]

5. **Substitute \( K_p \) into the Equation**:
Since \( K_p = 1 \), we can substitute this value into the empirical equation:
\[ \log(1) = 7.310 - \frac{8500}{T} \]
Since \( \log(1) = 0 \), we have:
\[ 0 = 7.310 - \frac{8500}{T} \]

6. **Solve for Temperature \( T \)**:
Rearranging the equation gives:
\[ \frac{8500}{T} = 7.310 \]
Thus,
\[ T = \frac{8500}{7.310} \]
Calculating this gives:
\[ T \approx 1163 \text{ K} \]

7. **Convert Temperature to Celsius**:
To convert Kelvin to Celsius, we use the formula:
\[ T(°C) = T(K) - 273 \]
Therefore:
\[ T(°C) = 1163 - 273 = 890 °C \]

8. **Conclusion**:
The complete decomposition temperature of magnesium carbonate is \( 890 °C \).

### Summary of Results:
- The complete decomposition temperature of magnesium carbonate is \( 890 °C \).
- The equilibrium constant \( K_p \) at this temperature is \( 1 \) atm.
- The partial pressure of carbon dioxide is \( 1 \) atm.