Balance the following equations by ion electron (half reaction) method for each of the following equations:
`Cr_(2)O_(7)^(2-) + Fe^(2+) rarr cr^(3+) + Fe^(3+) + H_(2)O`

Explanation

To balance the equation \( \text{Cr}_2\text{O}_7^{2-} + \text{Fe}^{2+} \rightarrow \text{Cr}^{3+} + \text{Fe}^{3+} + \text{H}_2\text{O} \) using the ion-electron (half-reaction) method, we will follow these steps:

### Step 1: Identify the oxidation and reduction half-reactions
- **Oxidation half-reaction**: The oxidation involves the conversion of \( \text{Fe}^{2+} \) to \( \text{Fe}^{3+} \).
- **Reduction half-reaction**: The reduction involves the conversion of \( \text{Cr}_2\text{O}_7^{2-} \) to \( \text{Cr}^{3+} \).

### Step 2: Write the oxidation half-reaction
The oxidation half-reaction can be written as:
\[ \text{Fe}^{2+} \rightarrow \text{Fe}^{3+} + e^- \]
Here, one electron is lost during the oxidation of \( \text{Fe}^{2+} \) to \( \text{Fe}^{3+} \).

### Step 3: Write the reduction half-reaction
The reduction half-reaction involves \( \text{Cr}_2\text{O}_7^{2-} \):
\[ \text{Cr}_2\text{O}_7^{2-} + 14 \text{H}^+ + 6 e^- \rightarrow 2 \text{Cr}^{3+} + 7 \text{H}_2\text{O} \]
In this half-reaction, \( \text{Cr}_2\text{O}_7^{2-} \) is reduced to \( \text{Cr}^{3+} \) by gaining 6 electrons, and we add \( 14 \text{H}^+ \) to balance the hydrogen atoms and \( 7 \text{H}_2\text{O} \) to balance the oxygen atoms.

### Step 4: Balance the electrons
To balance the overall reaction, we need to ensure that the number of electrons lost in the oxidation half-reaction equals the number of electrons gained in the reduction half-reaction. Since the oxidation half-reaction involves 1 electron and the reduction half-reaction involves 6 electrons, we multiply the oxidation half-reaction by 6:
\[ 6 \text{Fe}^{2+} \rightarrow 6 \text{Fe}^{3+} + 6 e^- \]

### Step 5: Combine the half-reactions
Now, we can add the two half-reactions together:
\[ 6 \text{Fe}^{2+} + \text{Cr}_2\text{O}_7^{2-} + 14 \text{H}^+ \rightarrow 6 \text{Fe}^{3+} + 2 \text{Cr}^{3+} + 7 \text{H}_2\text{O} \]

### Step 6: Write the balanced equation
The balanced equation is:
\[ \text{Cr}_2\text{O}_7^{2-} + 6 \text{Fe}^{2+} + 14 \text{H}^+ \rightarrow 2 \text{Cr}^{3+} + 6 \text{Fe}^{3+} + 7 \text{H}_2\text{O} \]