For the reaction `Ni^(2+) +4NH_(3) iff [Ni(NH_(3))_(4)]^(2+)`, at equilibrium, if the solution contain `1.65xx10^(-4)%` of nickel in the free state, and the concentration of `NH_(3)` at equilibrium is 0.5 M. Then the instability constant of the complex will be approximately equal to :

Options

A. `1.0xx10^(-5)`
B. `1.5xx10^(-16)`
C. `1.0xx10^(-7)`
D. `1.5xx10^(-17)`

Explanation

To find the instability constant (β) for the reaction \( \text{Ni}^{2+} + 4 \text{NH}_3 \rightleftharpoons [\text{Ni}(\text{NH}_3)_4]^{2+} \), we will follow these steps:

### Step 1: Understand the given data
We are given:
- The percentage of nickel in the free state: \( 1.65 \times 10^{-4} \% \)
- The concentration of \( \text{NH}_3 \) at equilibrium: \( 0.5 \, \text{M} \)

### Step 2: Convert the percentage of free nickel to concentration
The percentage of nickel in the free state can be converted to a decimal fraction:
\[ \text{Fraction of free Ni}^{2+} = \frac{1.65 \times 10^{-4}}{100} = 1.65 \times 10^{-6} \]

### Step 3: Define the total concentration of nickel
Let \( [\text{Ni}^{2+}] \) be the concentration of free nickel ions and \( [\text{Ni}(\text{NH}_3)_4]^{2+} \) be the concentration of the complex. Since the concentration of free nickel is very small compared to the concentration of the complex, we can approximate:
\[ [\text{Ni}^{2+}] \approx 1.65 \times 10^{-6} \, \text{M} \]

### Step 4: Calculate the concentration of the complex
Let \( [\text{Ni}(\text{NH}_3)_4]^{2+} = x \). The total concentration of nickel species in the solution is:
\[ [\text{Ni}^{2+}] + [\text{Ni}(\text{NH}_3)_4]^{2+} \approx x \]
Since \( [\text{Ni}^{2+}] \) is very small, we can say:
\[ x \approx [\text{Ni}(\text{NH}_3)_4]^{2+} \approx [\text{NH}_3]^4 \]

### Step 5: Write the equilibrium expression
The equilibrium constant \( K \) for the reaction is given by:
\[ K = \frac{[\text{Ni}(\text{NH}_3)_4]^{2+}}{[\text{Ni}^{2+}][\text{NH}_3]^4} \]

### Step 6: Substitute the known values
Substituting the values we have:
\[ K = \frac{x}{(1.65 \times 10^{-6})(0.5)^4} \]
Calculating \( (0.5)^4 = 0.0625 \):
\[ K = \frac{x}{(1.65 \times 10^{-6})(0.0625)} \]

### Step 7: Calculate the equilibrium constant
Since \( x \) is approximately equal to \( [\text{Ni}(\text{NH}_3)_4]^{2+} \), we can assume \( x \) is significantly larger than \( [\text{Ni}^{2+}] \):
\[ K \approx \frac{x}{(1.65 \times 10^{-6})(0.0625)} \]

### Step 8: Calculate the instability constant
The instability constant \( \beta \) is the inverse of the equilibrium constant \( K \):
\[ \beta = \frac{1}{K} \]

### Final Calculation
Assuming \( K \) is a large number, we can estimate:
\[ \beta \approx \frac{(1.65 \times 10^{-6})(0.0625)}{x} \]
This gives us an approximate value for \( \beta \).

### Conclusion
The instability constant \( \beta \) for the complex \( [\text{Ni}(\text{NH}_3)_4]^{2+} \) is approximately \( 10^{-7} \).

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