In which of the following is there an element with the same oxidation state as that of chromium in `K_(2)Cr_(2)O_(7) ?`

Options

A. `Cl_(2)O_(7)`
B. `[Fe(CN)_(6)]^(3-)`
C. `VO^(2+)`
D. `K_(2)MnO_(4)`

Answer

To determine which of the given compounds has an element with the same oxidation state as chromium in \( K_2Cr_2O_7 \), we first need to find the oxidation state of chromium in \( K_2Cr_2O_7 \).

### Step 1: Calculate the oxidation state of chromium in \( K_2Cr_2O_7 \)

1. **Identify the charges of other elements**:
- Potassium (K) has an oxidation state of +1.
- Oxygen (O) has an oxidation state of -2.

2. **Set up the equation**:
- The formula for potassium dichromate is \( K_2Cr_2O_7 \).
- There are 2 potassium atoms, contributing \( 2 \times (+1) = +2 \).
- There are 7 oxygen atoms, contributing \( 7 \times (-2) = -14 \).
- Let the oxidation state of chromium be \( x \). Since there are 2 chromium atoms, they contribute \( 2x \).

3. **Overall charge balance**:
- The overall charge of the compound is neutral (0), so we can set up the equation:
\[ 2 + 2x - 14 = 0 \]

4. **Solve for \( x \)**:
\[ 2x - 12 = 0 \\ 2x = 12 \\ x = 6 \]
Thus, the oxidation state of chromium in \( K_2Cr_2O_7 \) is +6.

### Step 2: Check the oxidation states of the elements in the other compounds

Now we will check the oxidation states of the elements in the other given compounds to find one with +6 oxidation state.

#### Option 1: \( Fe(CN)_6^{3-} \)

1. **Identify the charges**:
- The cyanide ion (CN) has a charge of -1.
- There are 6 cyanide ions, contributing \( 6 \times (-1) = -6 \).

2. **Set up the equation**:
- Let the oxidation state of iron be \( x \):
\[ x - 6 = -3 \]

3. **Solve for \( x \)**:
\[ x = -3 + 6 = +3 \]
The oxidation state of iron is +3.

#### Option 2: \( VO^{2+} \)

1. **Identify the charges**:
- Oxygen has an oxidation state of -2.

2. **Set up the equation**:
- Let the oxidation state of vanadium be \( x \):
\[ x + (-2) = +2 \]

3. **Solve for \( x \)**:
\[ x - 2 = 2 \\ x = 4 \]
The oxidation state of vanadium is +4.

#### Option 3: \( K_2MnO_4 \)

1. **Identify the charges**:
- Potassium has an oxidation state of +1.
- Oxygen has an oxidation state of -2.

2. **Set up the equation**:
- There are 2 potassium atoms and 4 oxygen atoms:
\[ 2 \times (+1) + x + 4 \times (-2) = 0 \]

3. **Solve for \( x \)**:
\[ 2 + x - 8 = 0 \\ x - 6 = 0 \\ x = +6 \]
The oxidation state of manganese is +6.

### Conclusion

The element with the same oxidation state as chromium in \( K_2Cr_2O_7 \) is manganese in \( K_2MnO_4 \), which also has an oxidation state of +6.

**Answer: Option 4: \( K_2MnO_4 \)**