For the redox reaction
`Cr_(2)O_(7)^(-2)+H^(+)+Ni rarr Cr^(3)+Ni^(2+)+H_(2)O`
The correct coefficients of the reactions for the balanced reaction are

Options

A. (`Cr_(2)O_(7)^(2-)=1`, `Ni=3`, `H^(+)=14`)
B. (`Cr_(2)O_(7)^(2-)=2`, `Ni=3`, `H^(+)=14`)
C. (`Cr_(2)O_(7)^(2-)=1`, `Ni=1`, `H^(+)=16`)
D. (`Cr_(2)O_(7)^(2-)=3`, `Ni=3`, `H^(+)=12`)

Explanation

To balance the given redox reaction `Cr_(2)O_(7)^(-2) + H^(+) + Ni → Cr^(3+) + Ni^(2+) + H_(2)O`, we will use the ion-electron method. Here’s a step-by-step breakdown of the balancing process:

### Step 1: Identify Oxidation and Reduction Half-Reactions
- **Oxidation Half-Reaction**: Nickel (Ni) is oxidized to Nickel ion (Ni²⁺).
\[ \text{Ni} \rightarrow \text{Ni}^{2+} + 2e^- \]

- **Reduction Half-Reaction**: Dichromate ion (Cr₂O₇²⁻) is reduced to Chromium ion (Cr³⁺).
\[ \text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ + 6e^- \rightarrow 2\text{Cr}^{3+} + 7\text{H}_2\text{O} \]

### Step 2: Balance the Electrons
- The oxidation half-reaction produces 2 electrons per Ni atom. For the reduction half-reaction, we need to ensure that the number of electrons lost in oxidation matches the number gained in reduction.
- To balance the electrons, we need 3 Ni atoms to match the 6 electrons from the reduction half-reaction. Therefore, we multiply the oxidation half-reaction by 3:
\[ 3\text{Ni} \rightarrow 3\text{Ni}^{2+} + 6e^- \]

### Step 3: Combine the Half-Reactions
Now we can combine the balanced half-reactions:
\[ 3\text{Ni} + \text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ \rightarrow 3\text{Ni}^{2+} + 2\text{Cr}^{3+} + 7\text{H}_2\text{O} \]

### Step 4: Write the Final Balanced Equation
The final balanced equation is:
\[ 3\text{Ni} + \text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ \rightarrow 3\text{Ni}^{2+} + 2\text{Cr}^{3+} + 7\text{H}_2\text{O} \]

### Step 5: Identify Coefficients
The coefficients in the balanced reaction are:
- Cr₂O₇²⁻: 1
- Ni: 3
- H⁺: 14
- Ni²⁺: 3
- Cr³⁺: 2
- H₂O: 7

### Final Answer
The correct coefficients for the balanced reaction are:
- Cr₂O₇²⁻: 1
- Ni: 3
- H⁺: 14
- Ni²⁺: 3
- Cr³⁺: 2
- H₂O: 7