To balance the chemical equation \( CH_4 + 2 O_2 \rightarrow CO_2 + H_2O \), we need to ensure that the number of atoms for each element is the same on both sides of the equation.
1. Balance Carbon (C) atoms:
- On the left side (reactants), we have 1 carbon atom in \( CH_4 \).
- On the right side (products), we need 1 carbon atom in \( CO_2 \).
- The carbon atom is already balanced (1 carbon atom on both sides).
2. Balance Hydrogen (H) atoms:
- On the left side, we have 4 hydrogen atoms in \( CH_4 \).
- On the right side, each \( H_2O \) molecule has 2 hydrogen atoms.
- To balance the 4 hydrogen atoms on the left side, we need 2 \( H_2O \) molecules on the right side. This gives us a total of \( 2 \times 2 = 4 \) hydrogen atoms.
3. Balance Oxygen (O) atoms:
- On the left side, we have 2 \( O_2 \) molecules, each containing 2 oxygen atoms, giving a total of \( 2 \times 2 = 4 \) oxygen atoms.
- On the right side, we already have:
- \( CO_2 \) with 2 oxygen atoms.
- Each \( H_2O \) molecule has 1 oxygen atom, and we need 2 \( H_2O \) molecules to balance the hydrogen atoms. Thus, we have \( 2 \times 1 = 2 \) oxygen atoms from water.
- In total, \( 2 \) (from \( CO_2 \)) + \( 2 \) (from \( 2 H_2O \)) = 4 oxygen atoms.
- The oxygen atoms are now balanced with 4 oxygen atoms on both sides.
Therefore, the coefficient that must be placed in front of water to balance the equation is [tex]\( \boxed{2} \)[/tex].
