In conversion of limestone to lime , `CaCO_3(s) rarrCaO(s)+CO_2(g)` the values of `DeltaH^@ and DeltaS^@` are +179.1 KJ/mol , 160.2J/K respectively at 298 K and 1 bar. Assuming that `DeltaH^@ and DeltaS^@` do not change with temperature , temperature above which conversion of limestone to lime will be spontaneous is

Options

A. 1008 K
B. 1200 K
C. 845 K
D. 1118 K

Answer

To determine the temperature above which the conversion of limestone to lime becomes spontaneous, we will use the Gibbs free energy equation:

\[ \Delta G = \Delta H - T \Delta S \]

### Step 1: Understand the spontaneity condition
For a reaction to be spontaneous, the change in Gibbs free energy (\(\Delta G\)) must be less than zero:

\[ \Delta G < 0 \]

This implies:

\[ \Delta H - T \Delta S < 0 \]

### Step 2: Rearranging the equation
Rearranging the equation gives us:

\[ \Delta H < T \Delta S \]

or

\[ T > \frac{\Delta H}{\Delta S} \]

### Step 3: Substitute the values
We are given:

- \(\Delta H = +179.1 \, \text{kJ/mol}\)
- \(\Delta S = 160.2 \, \text{J/K}\)

Since \(\Delta H\) is in kilojoules, we need to convert it to joules:

\[ \Delta H = 179.1 \, \text{kJ/mol} \times 1000 \, \text{J/kJ} = 179100 \, \text{J/mol} \]

Now, substituting the values into the equation:

\[ T > \frac{179100 \, \text{J/mol}}{160.2 \, \text{J/K}} \]

### Step 4: Calculate the temperature
Now we perform the calculation:

\[ T > \frac{179100}{160.2} \approx 1118.4 \, \text{K} \]

### Step 5: Conclusion
Thus, the temperature above which the conversion of limestone to lime will be spontaneous is approximately:

\[ T > 1118 \, \text{K} \]

### Final Answer:
The temperature above which the conversion of limestone to lime will be spontaneous is approximately **1118 K**.

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