Consider the equilibrium
`Ni(s)+4CO(g)hArrNi(CO)_(4)(g), K_(p)=0.125atm^(-3)`
If equal number of moles of `CO "and" Ni(CO)_(4)` (ideal gases) are mixed in a small container fitted with a piston, find the maximum total pressure (in atm) to which this mixture must be brough in order to just precipitate out metallic `Ni`?

Explanation

To solve the problem, we need to analyze the given equilibrium reaction and apply the concepts of chemical equilibrium, particularly the equilibrium constant \( K_p \).

### Step-by-Step Solution:

1. **Write the Equilibrium Reaction:**
The equilibrium reaction is given as:
\[ \text{Ni(s)} + 4 \text{CO(g)} \rightleftharpoons \text{Ni(CO)}_4(g) \]

2. **Identify the Equilibrium Constant:**
The equilibrium constant \( K_p \) for the reaction is provided as:
\[ K_p = 0.125 \, \text{atm}^{-3} \]

3. **Define the Initial Conditions:**
Let's assume we mix equal moles of CO and Ni(CO)₄. Let the number of moles of CO and Ni(CO)₄ be \( n \). Thus:
- Moles of CO = \( n \)
- Moles of Ni(CO)₄ = \( n \)

4. **Calculate the Partial Pressures:**
The total number of moles in the system is:
\[ n + n = 2n \]
The mole fraction of CO and Ni(CO)₄ will both be:
\[ \text{Mole fraction of CO} = \frac{n}{2n} = \frac{1}{2} \]
\[ \text{Mole fraction of Ni(CO)₄} = \frac{n}{2n} = \frac{1}{2} \]

5. **Express Partial Pressures in Terms of Total Pressure:**
The partial pressures can be expressed as:
\[ P_{\text{CO}} = \text{Mole fraction of CO} \times P_T = \frac{1}{2} P_T \]
\[ P_{\text{Ni(CO)}_4} = \text{Mole fraction of Ni(CO)₄} \times P_T = \frac{1}{2} P_T \]

6. **Substitute into the Expression for \( K_p \):**
The expression for \( K_p \) is:
\[ K_p = \frac{P_{\text{Ni(CO)}_4}}{(P_{\text{CO}})^4} \]
Substituting the partial pressures:
\[ K_p = \frac{\frac{1}{2} P_T}{\left(\frac{1}{2} P_T\right)^4} \]
Simplifying this gives:
\[ K_p = \frac{\frac{1}{2} P_T}{\frac{1}{16} P_T^4} = \frac{16}{2} \cdot \frac{P_T}{P_T^4} = 8 \cdot \frac{1}{P_T^3} \]

7. **Set Up the Equation:**
Now, we can set up the equation using the value of \( K_p \):
\[ 0.125 = \frac{8}{P_T^3} \]

8. **Solve for \( P_T^3 \):**
Rearranging gives:
\[ P_T^3 = \frac{8}{0.125} = 64 \]

9. **Calculate \( P_T \):**
Taking the cube root:
\[ P_T = \sqrt[3]{64} = 4 \, \text{atm} \]

### Final Answer:
The maximum total pressure to which this mixture must be brought in order to just precipitate out metallic Ni is:
\[ \boxed{4 \, \text{atm}} \]