Which of the following equations is a balanced one?

Options

A. `5BiO_(3)^(-)+22H^(+)+Mn^(2+)rarr5Bi^(3+)+7H_(2)O+MnO_(4)^(-)`
B. `5BiO_(3)^(-)+14H^(+)+2Mn^(2+)rarr5Bi^(3+)+7H_(2)O+2MnO_(4)^(-)`
C. `2BiO_(3)^(-)+4H^(+)+Mn^(2+)+Mn^(2+)rarr2Bi^(3+)+2H_(2)O+MnO_(4)^(-)`
D. `6BiO_(3)^(-)+12H^(+)+3Mn^(2+)rarr6Bi^(3+)+6H_(2)O+3MnO_(4)^(-)`

Answer

To determine which of the given equations is balanced, we will analyze each option by counting the number of atoms of each element on both the reactant (left-hand side) and product (right-hand side) sides of the equation. A balanced equation will have the same number of each type of atom on both sides.

### Step-by-Step Solution:

**Step 1: Analyze Option A**
- **Left-Hand Side (LHS)**:
- Bismuth (Bi): 5
- Oxygen (O): 15
- Hydrogen (H): 22
- Manganese (Mn): 1
- **Right-Hand Side (RHS)**:
- Bismuth (Bi): 5
- Oxygen (O): 11
- Hydrogen (H): 14
- Manganese (Mn): 1
- **Conclusion**: The number of hydrogen atoms is not equal (22 on LHS vs. 14 on RHS). Thus, **Option A is unbalanced**.

**Step 2: Analyze Option B**
- **Left-Hand Side (LHS)**:
- Bismuth (Bi): 5
- Oxygen (O): 15
- Hydrogen (H): 14
- Manganese (Mn): 2
- **Right-Hand Side (RHS)**:
- Bismuth (Bi): 5
- Oxygen (O): 15
- Hydrogen (H): 14
- Manganese (Mn): 2
- **Conclusion**: All elements are balanced (5 Bi, 15 O, 14 H, and 2 Mn on both sides). Thus, **Option B is balanced**.

**Step 3: Analyze Option C**
- **Left-Hand Side (LHS)**:
- Bismuth (Bi): 2
- Oxygen (O): 6
- Hydrogen (H): 4
- Manganese (Mn): 1
- **Right-Hand Side (RHS)**:
- Bismuth (Bi): 2
- Oxygen (O): 6
- Hydrogen (H): 4
- Manganese (Mn): 1
- **Conclusion**: All elements are balanced (2 Bi, 6 O, 4 H, and 1 Mn on both sides). Thus, **Option C is balanced**.

**Step 4: Analyze Option D**
- **Left-Hand Side (LHS)**:
- Bismuth (Bi): 6
- Oxygen (O): 18
- Hydrogen (H): 12
- Manganese (Mn): 3
- **Right-Hand Side (RHS)**:
- Bismuth (Bi): 6
- Oxygen (O): 18
- Hydrogen (H): 12
- Manganese (Mn): 3
- **Conclusion**: All elements are balanced (6 Bi, 18 O, 12 H, and 3 Mn on both sides). Thus, **Option D is balanced**.

### Final Conclusion:
The balanced equations are **Option B** and **Option D**.