Step 1: Understanding the extraction of iron in a blast furnace.
In a blast furnace, calcium carbonate (CaCO\(_3\)) is heated to produce calcium oxide (CaO) and carbon dioxide (CO\(_2\)). This reaction is essential for the reduction of iron ore in the furnace. The reaction typically occurs at high temperatures (around 2000 K).
Step 2: Analyzing the options.
(A) CaO + SiO\(_2\) \rightarrow CaSiO\(_3\): This reaction involves the formation of calcium silicate, but it does not occur at the high temperature needed for iron extraction.
(B) CaCO\(_3\) \rightarrow CaO + CO\(_2\): This is correct. Calcium carbonate decomposes into calcium oxide and carbon dioxide in the blast furnace, providing the necessary components for the reduction of iron ore.
(C) Fe\(_2\)O\(_3\) + 3CO \rightarrow 2Fe + 3CO\(_2\): This reaction represents the reduction of iron ore, but it is not the reaction that primarily occurs at the high temperatures in the furnace.
(D) 2C + O\(_2\) \rightarrow 2CO: This reaction represents the formation of carbon monoxide, which occurs in the blast furnace, but it is not the primary reaction for extraction.
Step 3: Conclusion.
The correct answer is (B) CaCO\(_3\) \rightarrow CaO + CO\(_2\).