`a MnO_(4)^(-)+bH_(2)O_(2)+cH^(+)todMn^(+2)+eO_(2)+fH_(2)O`
The value so `a,b,c,d,e` and `f` are

Options

A. `2,1,6,2,3` and `4` respectively
B. `2,3,6,2,4` and `6` respectively
C. `2,7,6,2,6` and `10` respectively
D. none

Explanation

To solve the redox reaction given in the question, we need to balance the equation involving the reactants and products. The reaction can be summarized as follows:

**Given Reaction:**
\[ a \text{MnO}_4^{-} + b \text{H}_2\text{O}_2 + c \text{H}^{+} \rightarrow d \text{Mn}^{2+} + e \text{O}_2 + f \text{H}_2\text{O} \]

### Step 1: Identify Oxidation States
- In \(\text{MnO}_4^{-}\), manganese (Mn) has an oxidation state of +7.
- In \(\text{Mn}^{2+}\), manganese has an oxidation state of +2.
- In \(\text{H}_2\text{O}_2\), oxygen has an oxidation state of -1 (in peroxide).
- In \(\text{O}_2\), oxygen has an oxidation state of 0.

### Step 2: Write Half-Reactions
1. **Reduction half-reaction** (MnO4^- to Mn^2+):
\[ \text{MnO}_4^{-} + 8 \text{H}^{+} + 5 \text{e}^{-} \rightarrow \text{Mn}^{2+} + 4 \text{H}_2\text{O} \]

2. **Oxidation half-reaction** (H2O2 to O2):
\[ 2 \text{H}_2\text{O}_2 \rightarrow 2 \text{O}_2 + 2 \text{H}^{+} + 2 \text{e}^{-} \]

### Step 3: Balance Electrons
To balance the electrons transferred in the half-reactions, we need to multiply the oxidation half-reaction by 5:
\[ 5(2 \text{H}_2\text{O}_2 \rightarrow 2 \text{O}_2 + 2 \text{H}^{+} + 2 \text{e}^{-}) \]
This gives:
\[ 10 \text{H}_2\text{O}_2 \rightarrow 10 \text{O}_2 + 10 \text{H}^{+} + 10 \text{e}^{-} \]

### Step 4: Combine Half-Reactions
Now, we can combine the two half-reactions:
\[ \text{MnO}_4^{-} + 8 \text{H}^{+} + 5 \text{e}^{-} + 10 \text{H}_2\text{O}_2 \rightarrow \text{Mn}^{2+} + 4 \text{H}_2\text{O} + 10 \text{O}_2 + 10 \text{H}^{+} + 10 \text{e}^{-} \]

### Step 5: Simplify the Equation
Cancel out the electrons and simplify:
\[ \text{MnO}_4^{-} + 10 \text{H}_2\text{O}_2 + 6 \text{H}^{+} \rightarrow \text{Mn}^{2+} + 10 \text{O}_2 + 4 \text{H}_2\text{O} \]

### Step 6: Identify Coefficients
From the balanced equation, we can identify the coefficients:
- \( a = 2 \) (for MnO4^-)
- \( b = 5 \) (for H2O2)
- \( c = 6 \) (for H+)
- \( d = 2 \) (for Mn2+)
- \( e = 5 \) (for O2)
- \( f = 8 \) (for H2O)

### Final Answer:
The values are:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 6 \)
- \( d = 2 \)
- \( e = 5 \)
- \( f = 8 \)