Consider the following transformation:
`Cr(CO)_(x) to Cr(CO)_(y)(NO)_(x)`
If both reactant and product follow EAN rule, then calculate value of `x+z-y` (where x,y and z are natural numbers).

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the EAN Rule
The Effective Atomic Number (EAN) rule states that the stability of a coordination compound is determined by the total number of electrons surrounding the central metal atom. The EAN is calculated as:
\[ \text{EAN} = \text{Number of electrons in the metal atom} + \text{Number of electrons donated by ligands} \]

### Step 2: Determine the EAN for the Reactant
For the reactant \( \text{Cr(CO)}_x \):
- Chromium (Cr) has an atomic number of 24, so it has 24 electrons.
- Each CO ligand donates 2 electrons. Therefore, if there are \( x \) CO ligands, the total contribution from CO is \( 2x \).

Thus, the EAN for the reactant can be expressed as:
\[ \text{EAN}_{\text{reactant}} = 24 + 2x \]

### Step 3: Set the EAN for the Product
For the product \( \text{Cr(CO)}_y(\text{NO})_z \):
- The contribution from \( y \) CO ligands is \( 2y \).
- Each neutral NO ligand donates 3 electrons, so the contribution from \( z \) NO ligands is \( 3z \).

Thus, the EAN for the product can be expressed as:
\[ \text{EAN}_{\text{product}} = 24 + 2y + 3z \]

### Step 4: Apply the EAN Rule
According to the problem, both the reactant and product follow the EAN rule, meaning:
\[ \text{EAN}_{\text{reactant}} = \text{EAN}_{\text{product}} \]
This gives us the equation:
\[ 24 + 2x = 24 + 2y + 3z \]

### Step 5: Simplify the Equation
By simplifying the equation, we can eliminate 24 from both sides:
\[ 2x = 2y + 3z \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 2x - 2y = 3z \]
or
\[ x - y = \frac{3z}{2} \]

### Step 7: Finding Natural Number Values
Since \( x, y, z \) are natural numbers, \( 3z \) must be even, which means \( z \) must be even. Let’s assume \( z = 2 \) (the smallest even natural number):
- Then \( 3z = 6 \), and we have:
\[ x - y = 3 \]
This implies:
\[ x = y + 3 \]

### Step 8: Substitute Values
Now we can substitute \( z = 2 \) into our earlier equations. Let’s say \( y = 3 \) (a natural number):
- Then \( x = 3 + 3 = 6 \).

### Step 9: Calculate \( x + z - y \)
Now we can calculate:
\[ x + z - y = 6 + 2 - 3 = 5 \]

### Final Answer
Thus, the value of \( x + z - y \) is:
\[ \boxed{5} \]