a `Cr_(2)O_(7)^(2-) + b H^(+) + cS^(2-) rarr x Cr^(3+) + yS + zH_(2)O`
The value of a, b, c, x, y and z in the balanced chemical reaction are :

Options

A. `{:("(a)","(b)","(c)","(x)","(y)","(z)"),(1,14,3,1,14,3):}`
B. `{:("(a)","(b)","(c)","(x)","(y)","(z)"),(1,14,3,2,3,7):}`
C. `{:("(a)","(b)","(c)","(x)","(y)","(z)"),(1,7,3,2,7,6):}`
D. `{:("(a)","(b)","(c)","(x)","(y)","(z)"),(1,7,3,2,14,3):}`

Explanation

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

### Step 1: Identify the oxidation states
- For \( \text{Cr}_2\text{O}_7^{2-} \):
- Let the oxidation state of Cr be \( x \).
- The equation for the oxidation state is:
\[ 2x + 7(-2) = -2 \implies 2x - 14 = -2 \implies 2x = 12 \implies x = +6 \]
- Therefore, each Cr is in the +6 oxidation state.

- For \( \text{S}^{2-} \):
- The oxidation state of S is -2.

### Step 2: Determine the changes in oxidation states
- Chromium changes from +6 to +3 (a decrease of 3).
- Sulfur changes from -2 to 0 (an increase of 2).

### Step 3: Calculate the n-factor
- For \( \text{Cr}_2\text{O}_7^{2-} \):
- There are 2 Cr atoms, each undergoing a change of 3:
\[ \text{Total change for Cr} = 2 \times 3 = 6 \]
- For \( \text{S}^{2-} \):
- There are 1 S atom undergoing a change of 2:
\[ \text{Total change for S} = 1 \times 2 = 2 \]

### Step 4: Write the half-reactions
- Reduction half-reaction for Cr:
\[ \text{Cr}_2\text{O}_7^{2-} + 6\text{e}^- \rightarrow 2\text{Cr}^{3+} \]
- Oxidation half-reaction for S:
\[ 3\text{S}^{2-} \rightarrow 3\text{S} + 6\text{e}^- \]

### Step 5: Combine the half-reactions
- The balanced equation combining both half-reactions is:
\[ \text{Cr}_2\text{O}_7^{2-} + 3\text{S}^{2-} \rightarrow 2\text{Cr}^{3+} + 3\text{S} \]

### Step 6: Balance the hydrogen and oxygen
- There are 7 oxygen atoms in \( \text{Cr}_2\text{O}_7^{2-} \), so we need to add \( 14 \text{H}^+ \) to balance the hydrogen:
\[ \text{Cr}_2\text{O}_7^{2-} + 3\text{S}^{2-} + 14\text{H}^+ \rightarrow 2\text{Cr}^{3+} + 3\text{S} + 7\text{H}_2\text{O} \]

### Final balanced equation
\[ \text{Cr}_2\text{O}_7^{2-} + 3\text{S}^{2-} + 14\text{H}^+ \rightarrow 2\text{Cr}^{3+} + 3\text{S} + 7\text{H}_2\text{O} \]

### Step 7: Identify the coefficients
From the balanced equation:
- \( a = 1 \) (for \( \text{Cr}_2\text{O}_7^{2-} \))
- \( b = 14 \) (for \( \text{H}^+ \))
- \( c = 3 \) (for \( \text{S}^{2-} \))
- \( x = 2 \) (for \( \text{Cr}^{3+} \))
- \( y = 3 \) (for \( \text{S} \))
- \( z = 7 \) (for \( \text{H}_2\text{O} \))

### Summary of values
- \( a = 1 \)
- \( b = 14 \)
- \( c = 3 \)
- \( x = 2 \)
- \( y = 3 \)
- \( z = 7 \)