Balance equations by oxidation number method or ion electron method.
S+HNO3→H2SO4+NO2+H2O Fe2++MnO−4+H+→Fe3++Mn2++H2O

To balance the given chemical equations using the oxidation number method, we need to first assign oxidation numbers to each element in the compounds involved. Then, we'll determine the change in oxidation numbers for the elements that undergo oxidation or reduction. Finally, we'll balance the equations by adding coefficients to the compounds to ensure that both the mass and charge are conserved.

Let's start with the first equation:

S + HNO₃ → H₂SO₄ + NO₂ + H₂O
We need to assign oxidation numbers:

For Sulfur (S) in H₂SO₄, it's +6.
For Nitrogen (N) in HNO₃, it's +5 in NO₃⁻.
For Oxygen (O) in H₂SO₄, it's -2.
For Hydrogen (H), it's +1 in HNO₃ and H₂SO₄.
For Nitrogen (N) in NO₂, it's +4.
For Oxygen (O) in NO₂, it's -2.
For Oxygen (O) in H₂O, it's -2.
For Hydrogen (H) in H₂O, it's +1.
Next, let's determine the changes in oxidation numbers:

Sulfur in S undergoes oxidation from 0 to +6.
Nitrogen in HNO₃ undergoes reduction from +5 to +2.
Nitrogen in NO₂ undergoes oxidation from +5 to +4.
Now, we can balance the equation:

Step 1: Balance the atoms other than hydrogen and oxygen.
S + HNO₃ → H₂SO₄ + NO₂ + H₂O
1S + 2HNO₃ → H₂SO₄ + NO₂ + H₂O

Step 2: Balance the oxygen atoms by adding water on the right side:
1S + 2HNO₃ → H₂SO₄ + NO₂ + 3H₂O

Step 3: Balance the hydrogen atoms by adding H⁺ ions on the left side:
S + 2HNO₃ + 2H⁺ → H₂SO₄ + NO₂ + 3H₂O

Step 4: Balance the charges by adding electrons:
S + 2HNO₃ + 2H⁺ + 6e⁻ → H₂SO₄ + NO₂ + 3H₂O

The equation is now balanced by the oxidation number method.

Now, let's balance the second equation:

Fe²⁺ + MnO₄⁻ + H⁺ → Fe³⁺ + Mn²⁺ + H₂O
Assign oxidation numbers:

For Iron (Fe) in Fe²⁺, it's +2.
For Manganese (Mn) in MnO₄⁻, it's +7.
For Oxygen (O) in MnO₄⁻, it's -2.
For Hydrogen (H), it's +1.
For Iron (Fe) in Fe³⁺, it's +3.
For Manganese (Mn) in Mn²⁺, it's +2.
For Hydrogen (H) in H₂O, it's +1.
For Oxygen (O) in H₂O, it's -2.
Now, let's determine the changes in oxidation numbers:

Iron in Fe undergoes oxidation from +2 to +3.
Manganese in MnO₄⁻ undergoes reduction from +7 to +2.
Now, we can balance the equation:

Step 1: Balance the atoms other than hydrogen and oxygen.
Fe²⁺ + MnO₄⁻ + H⁺ → Fe³⁺ + Mn²⁺ + H₂O
Fe²⁺ + MnO₄⁻ + 8H⁺ → Fe³⁺ + Mn²⁺ + H₂O

Step 2: Balance the oxygen atoms by adding water on the right side:
Fe²⁺ + MnO₄⁻ + 8H⁺ → Fe³⁺ + Mn²⁺ + 4H₂O

Step 3: Balance the hydrogen atoms by adding H⁺ ions on the left side:
Fe²⁺ + MnO₄⁻ + 8H⁺ → Fe³⁺ + Mn²⁺ + 4H₂O

Step 4: Balance the charges by adding electrons:
Fe²⁺ + MnO₄⁻ + 8H⁺ + 5e⁻ → Fe³⁺ + Mn²⁺ + 4H₂O

The equation is now balanced by the oxidation number method.