What will be the value of a, b, c, d, e, and f in the following equation -
`aMnO_(4)^(-)+bC_(2)O_(4)^(--)+CH^(+)rarrdMn^(++)+eCO_(2)+fH_(2)O`

Options

A. `2,2,10,8,5,16`
B. `2,5,16,2,10,8`
C. `2,5,10,2,8`
D. `2,8,16,2,5,10`

Explanation

To solve the given redox reaction and find the values of a, b, c, d, e, and f in the equation:

\[ a \text{MnO}_4^{-} + b \text{C}_2\text{O}_4^{2-} + c \text{H}^{+} \rightarrow d \text{Mn}^{2+} + e \text{CO}_2 + f \text{H}_2\text{O} \]

we will follow these steps:

### Step 1: Identify the half-reactions
The first step is to identify the oxidation and reduction half-reactions.

**Reduction Half-Reaction:**
\[ \text{MnO}_4^{-} + 8 \text{H}^{+} + 5 \text{e}^{-} \rightarrow \text{Mn}^{2+} + 4 \text{H}_2\text{O} \]

**Oxidation Half-Reaction:**
\[ \text{C}_2\text{O}_4^{2-} \rightarrow 2 \text{CO}_2 + 2 \text{e}^{-} \]

### Step 2: Balance the half-reactions
We need to balance the half-reactions for the number of electrons transferred.

- The reduction half-reaction involves 5 electrons.
- The oxidation half-reaction involves 2 electrons.

To equalize the number of electrons, we can multiply the oxidation half-reaction by 5 and the reduction half-reaction by 2.

### Step 3: Multiply the half-reactions
**Oxidation Half-Reaction (multiplied by 5):**
\[ 5 \text{C}_2\text{O}_4^{2-} \rightarrow 10 \text{CO}_2 + 10 \text{e}^{-} \]

**Reduction Half-Reaction (multiplied by 2):**
\[ 2 \text{MnO}_4^{-} + 16 \text{H}^{+} + 10 \text{e}^{-} \rightarrow 2 \text{Mn}^{2+} + 8 \text{H}_2\text{O} \]

### Step 4: Combine the half-reactions
Now, we can combine the balanced half-reactions:

\[ 2 \text{MnO}_4^{-} + 5 \text{C}_2\text{O}_4^{2-} + 16 \text{H}^{+} \rightarrow 2 \text{Mn}^{2+} + 10 \text{CO}_2 + 8 \text{H}_2\text{O} \]

### Step 5: Identify the coefficients
From the combined balanced equation, we can identify the coefficients:

- \( a = 2 \) (for MnO4^-)
- \( b = 5 \) (for C2O4^2-)
- \( c = 16 \) (for H+)
- \( d = 2 \) (for Mn2+)
- \( e = 10 \) (for CO2)
- \( f = 8 \) (for H2O)

### Final Values
Thus, the values are:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 16 \)
- \( d = 2 \)
- \( e = 10 \)
- \( f = 8 \)