`aK_(2)Cr_(2)O_(7)+bKCl+cH_(2)SO_(4) to xCrO_(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 ad x=6, y=3,z=2
D. a=1,bb=4,c=6 and x=2, y=6, z=3

Answer

To balance the chemical equation given in the question, we need to follow a systematic approach. Let's break it down step by step.

### Step 1: Write the unbalanced equation
The unbalanced equation is:
\[ aK_{2}Cr_{2}O_{7} + bKCl + cH_{2}SO_{4} \rightarrow xCrO_{2}Cl_{2} + yKHSO_{4} + zH_{2}O \]

### Step 2: Identify the number of atoms for each element
We need to identify how many atoms of each element are present on both sides of the equation.

- **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 3: Set up equations based on the conservation of mass
From the above counts, we can set up the following equations:

1. **Potassium:**
\[ 2a + b = y \]

2. **Chromium:**
\[ 2a = x \]

3. **Oxygen:**
\[ 7a + 4c = 2x + 4y + z \]

4. **Chlorine:**
\[ b = 2x \]

5. **Hydrogen:**
\[ 2c = y + 2z \]

6. **Sulfur:**
\[ c = y \]

### Step 4: Substitute and solve the equations
From equation (2), we can express \(x\) in terms of \(a\):
\[ x = 2a \]

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

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

Now substituting \(b\) into equation (1):
\[ 2a + 4a = y \]
\[ 6a = y \]

Now substituting \(y\) into equation (5):
\[ 2c = 6a + 2z \]
Since \(c = y = 6a\):
\[ 2(6a) = 6a + 2z \]
\[ 12a = 6a + 2z \]
\[ 6a = 2z \]
\[ z = 3a \]

### Step 5: Substitute back to find coefficients
Now we can summarize the coefficients:
- \(a = 1\)
- \(b = 4\)
- \(c = 6\)
- \(x = 2\)
- \(y = 6\)
- \(z = 3\)

### Final Balanced Equation
The balanced equation is:
\[ K_{2}Cr_{2}O_{7} + 4KCl + 6H_{2}SO_{4} \rightarrow 2CrO_{2}Cl_{2} + 6KHSO_{4} + 3H_{2}O \]

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