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 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 half-reaction involves the reduction of permanganate ion (\(\text{MnO}_4^{-}\)) to manganese ion (\(\text{Mn}^{2+}\)):
\[ \text{MnO}_4^{-} \rightarrow \text{Mn}^{2+} \]

The second half-reaction involves the oxidation of oxalate ion (\(\text{C}_2\text{O}_4^{2-}\)) to carbon dioxide (\(\text{CO}_2\)):
\[ \text{C}_2\text{O}_4^{2-} \rightarrow \text{CO}_2 \]

### Step 2: Balance the first half-reaction
1. **Balance oxygen by adding water**:
\[ \text{MnO}_4^{-} + 4 \text{H}_2\text{O} \rightarrow \text{Mn}^{2+} \]
2. **Balance hydrogen by adding protons**:
\[ \text{MnO}_4^{-} + 4 \text{H}_2\text{O} + 8 \text{H}^{+} \rightarrow \text{Mn}^{2+} \]
3. **Balance charge by adding electrons**:
The oxidation state of Mn changes from +7 to +2, which means it gains 5 electrons:
\[ \text{MnO}_4^{-} + 8 \text{H}^{+} + 5 e^{-} \rightarrow \text{Mn}^{2+} + 4 \text{H}_2\text{O} \]

### Step 3: Balance the second half-reaction
1. **Balance carbon**:
\[ \text{C}_2\text{O}_4^{2-} \rightarrow 2 \text{CO}_2 \]
2. **Balance charge by adding electrons**:
The oxidation state of carbon changes from -2 to +4, which means it loses 2 electrons for each carbon:
\[ \text{C}_2\text{O}_4^{2-} \rightarrow 2 \text{CO}_2 + 2 e^{-} \]

### Step 4: Equalize the number of electrons
To balance the electrons, we need to multiply the second half-reaction by 5:
\[ 5 \text{C}_2\text{O}_4^{2-} \rightarrow 10 \text{CO}_2 + 10 e^{-} \]

### Step 5: Combine the half-reactions
Now we can combine the balanced half-reactions:
\[ 2 \text{MnO}_4^{-} + 8 \text{H}^{+} + 10 \text{C}_2\text{O}_4^{2-} \rightarrow 2 \text{Mn}^{2+} + 10 \text{CO}_2 + 8 \text{H}_2\text{O} \]

### Step 6: Identify the coefficients
From the balanced equation, we can identify the coefficients:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 8 \)
- \( d = 2 \)
- \( e = 10 \)
- \( f = 8 \)

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