To balance the chemical equation \( \text{MnO}_2 + \text{HCl} \rightarrow \text{MnCl}_2 + \text{H}_2\text{O} + \text{Cl}_2 \) while preserving the law of conservation of matter, we need to make sure that there are equal numbers of each type of atom on both sides of the equation.
- Mn (Manganese): 1 on both sides.
- O (Oxygen): 2 on the left (from \( \text{MnO}_2 \)) and 1 on the right (from \( \text{H}_2\text{O} \)).
- H (Hydrogen): 1 on the left (from \( \text{HCl} \)) and 2 on the right (from \( \text{H}_2\text{O} \)).
- Cl (Chlorine): 1 on the left (from \( \text{HCl} \)) and 2 on the right (1 from \( \text{MnCl}_2 \) and 1 from \( \text{Cl}_2 \)).
To balance this reaction, we can try the coefficients \( 2, 4, 2, 4, 1 \) in the following way:
\[
2 \text{MnO}_2 + 4 \text{HCl} \rightarrow 2 \text{MnCl}_2 + 2 \text{H}_2\text{O} + \text{Cl}_2
\]
Now let's check the atom counts:
- Mn: \( 2 \) (left) vs \( 2 \) (right)
- O: \( 4 \) (left) vs \( 2 \) (right)
- H: \( 4 \) (left) vs \( 4 \) (right, \( 2 \) from \( H_2O \))
- Cl: \( 4 \) (left) vs \( 4 \) (right, \( 2 \) in \( MnCl_2 \) and \( 2 \) in \( Cl_2 \))
Thus, after balancing, the correct coefficients to represent the complete and balanced equation are:
2, 4, 2, 4, 1
So the answer is \( 2, 4, 2, 4, 1 \).