Choose the set coefficients that correctly balances the following equation :
`xCr_(2)O_(7)^(2-)+yH^(+)+ze^(-)rarraCr^(+3)++bH_(2)O`

Options

A. `{:(x,y,z,a,b),(2,14,6,2,4):}`
B. `{:(x,y,z,a,b),(1,14,6,2,7):}`
C. `{:(x,y,z,a,b),(2,7,6,2,7):}`
D. `{:(x,y,z,a,b),(2,7,6,1,7):}`

Answer

To balance the redox reaction given in the question, we will follow these steps systematically:

### Step 1: Write the unbalanced equation
The unbalanced equation is:
\[ \text{Cr}_2\text{O}_7^{2-} + \text{H}^+ + e^- \rightarrow \text{Cr}^{3+} + \text{H}_2\text{O} \]

### Step 2: Identify the oxidation states
- In \(\text{Cr}_2\text{O}_7^{2-}\), the oxidation state of chromium (Cr) is +6.
- In \(\text{Cr}^{3+}\), the oxidation state of chromium is +3.

### Step 3: Balance the chromium atoms
There are 2 chromium atoms in \(\text{Cr}_2\text{O}_7^{2-}\), so we need to have 2 \(\text{Cr}^{3+}\) on the product side:
\[ \text{Cr}_2\text{O}_7^{2-} + \text{H}^+ + e^- \rightarrow 2 \text{Cr}^{3+} + \text{H}_2\text{O} \]

### Step 4: Balance the oxygen atoms
There are 7 oxygen atoms in \(\text{Cr}_2\text{O}_7^{2-}\). To balance the oxygen, we need to add 7 water molecules (\(\text{H}_2\text{O}\)) on the product side:
\[ \text{Cr}_2\text{O}_7^{2-} + \text{H}^+ + e^- \rightarrow 2 \text{Cr}^{3+} + 7 \text{H}_2\text{O} \]

### Step 5: Balance the hydrogen atoms
Now, we have 14 hydrogen atoms from 7 water molecules. To balance the hydrogen, we need to add 14 \(\text{H}^+\) ions on the reactant side:
\[ \text{Cr}_2\text{O}_7^{2-} + 14 \text{H}^+ + e^- \rightarrow 2 \text{Cr}^{3+} + 7 \text{H}_2\text{O} \]

### Step 6: Balance the charge
Now, let's balance the charge. The left side has a charge of:
- \(-2\) from \(\text{Cr}_2\text{O}_7^{2-}\)
- \(+14\) from \(14 \text{H}^+\)
- \(-1\) from \(e^-\)

Total charge on the left side = \(-2 + 14 - 1 = +11\).

The right side has a charge of:
- \(+6\) from \(2 \text{Cr}^{3+}\) (2 x +3)

Total charge on the right side = \(+6\).

To balance the charges, we need to add 6 electrons to the left side:
\[ \text{Cr}_2\text{O}_7^{2-} + 14 \text{H}^+ + 6 e^- \rightarrow 2 \text{Cr}^{3+} + 7 \text{H}_2\text{O} \]

### Step 7: Write the final balanced equation
The final balanced equation is:
\[ 1 \text{Cr}_2\text{O}_7^{2-} + 14 \text{H}^+ + 6 e^- \rightarrow 2 \text{Cr}^{3+} + 7 \text{H}_2\text{O} \]

### Step 8: Identify the coefficients
From the balanced equation, we can identify the coefficients:
- \(x = 1\)
- \(y = 14\)
- \(z = 6\)
- \(a = 2\)
- \(b = 7\)

### Final Answer
The coefficients are:
- \(x = 1\)
- \(y = 14\)
- \(z = 6\)
- \(a = 2\)
- \(b = 7\)