The equation Fe + Cl$_2$ $\rightarrow$ FeCl$_3$ is balanced because it has equal numbers of iron and chlorine atoms on both sides of the equation. On the other hand, the other equations are not balanced, as they do not have the same number of atoms for each element on both sides. Thus, the correct answer is \( \mathbf{(C) Fe + Cl_2 \rightarrow FeCl_3} \).