`aK_(2)Cr_(2)O_(7)+bKCl+cH_(2)SO_(4)rarrxCrO_(2)Cl_(2)+yKHSO_(4)+zH_(2)O`
The above equation balances when

Options

A. `a=2, b=4,c=6 and x=2, y=6, z=3`
B. `a=4, b=2,c=6 and x=6,y=2,z=3`
C. `a=6,b=4,c=2 and x=6,y=3,z=2`
D. `a=1,b=4,c=6 and x=2,y=6,z=3`

Explanation

To balance the given chemical equation:

\[ aK_2Cr_2O_7 + bKCl + cH_2SO_4 \rightarrow xCrO_2Cl_2 + yKHSO_4 + zH_2O \]

we need to determine the coefficients \( a, b, c, x, y, z \) such that the number of atoms of each element is equal on both sides of the equation.

### Step 1: Identify the number of atoms for each element in the reactants and products.

**Reactants:**
- Potassium (K): \( 2a + b \)
- Chromium (Cr): \( 2a \)
- Oxygen (O): \( 7a \)
- Chlorine (Cl): \( b \)
- Hydrogen (H): \( 2c \)
- Sulfur (S): \( c \)

**Products:**
- Potassium (K): \( y \)
- Chromium (Cr): \( x \)
- Oxygen (O): \( 2x + 4y + z \)
- Chlorine (Cl): \( 2x \)
- Hydrogen (H): \( y + 2z \)
- Sulfur (S): \( y \)

### Step 2: Set up equations based on the conservation of atoms.

1. **For Potassium (K):**
\[ 2a + b = y \quad \text{(1)} \]

2. **For Chromium (Cr):**
\[ 2a = x \quad \text{(2)} \]

3. **For Oxygen (O):**
\[ 7a + 4c = 2x + 4y + z \quad \text{(3)} \]

4. **For Chlorine (Cl):**
\[ b = 2x \quad \text{(4)} \]

5. **For Hydrogen (H):**
\[ 2c = y + 2z \quad \text{(5)} \]

6. **For Sulfur (S):**
\[ c = y \quad \text{(6)} \]

### Step 3: Solve the equations.

From equation (2):
\[ x = 2a \]

From equation (4):
\[ b = 2(2a) = 4a \]

From equation (6):
\[ c = y \]

Substituting \( c \) in equation (5):
\[ 2y = y + 2z \implies y = 2z \quad \text{(7)} \]

Now substitute \( b \) and \( c \) into equation (1):
\[ 2a + 4a = y \implies 6a = y \]

Substituting \( y \) from equation (7):
\[ 6a = 2z \implies z = 3a \]

Now we have:
- \( x = 2a \)
- \( b = 4a \)
- \( c = 6a \)
- \( y = 6a \)
- \( z = 3a \)

### Step 4: Choose a value for \( a \).

Let’s set \( a = 1 \):
- \( x = 2(1) = 2 \)
- \( b = 4(1) = 4 \)
- \( c = 6(1) = 6 \)
- \( y = 6(1) = 6 \)
- \( z = 3(1) = 3 \)

### Final Balanced Equation:

The balanced equation is:
\[ 1K_2Cr_2O_7 + 4KCl + 6H_2SO_4 \rightarrow 2CrO_2Cl_2 + 6KHSO_4 + 3H_2O \]

### Summary of Coefficients:
- \( a = 1 \)
- \( b = 4 \)
- \( c = 6 \)
- \( x = 2 \)
- \( y = 6 \)
- \( z = 3 \)