The stepwise formation of `[Cu(NH_3)_4]^(2+)` is given below `Cu^(2+)+NH_3overset(K_1)hArr[Cu(NH_3)]^(2+]`
`[Cu(NH_3)]^(2+)+NH_3overset(K_2)hArr[Cu(NH_3)_2]^(2+)`
`[Cu(NH_3)_2]^(2+)+NH_3overset(K_3)hArr[Cu(NH_3)_3]^(2+)`
`[Cu(NH_3)_3]^(2+)+NH_3overset(K_4)hArr[Cu(NH_3)_4]^(2+)`
The value of stability constants `K_1,K_2,K_3 and K_4` are `10^4,1.58xx10^3,5xx10^2 and 0^2` respectively. The overall equilibrium constant for dissociation of `[Cu(NH_3)_4]^(2+)" is ""x"xx10^(-12)`. The value of x is `"______"`. (Rounded off to the nearest integer)

Answer

To find the value of \( x \) in the overall equilibrium constant for the dissociation of \([Cu(NH_3)_4]^{2+}\), we will follow these steps:

### Step 1: Write the formation reactions and their equilibrium constants
The formation of the complex \([Cu(NH_3)_4]^{2+}\) occurs in four steps:

1. \( Cu^{2+} + NH_3 \overset{K_1}{\rightleftharpoons} [Cu(NH_3)]^{2+} \) with \( K_1 = 10^4 \)
2. \( [Cu(NH_3)]^{2+} + NH_3 \overset{K_2}{\rightleftharpoons} [Cu(NH_3)_2]^{2+} \) with \( K_2 = 1.58 \times 10^3 \)
3. \( [Cu(NH_3)_2]^{2+} + NH_3 \overset{K_3}{\rightleftharpoons} [Cu(NH_3)_3]^{2+} \) with \( K_3 = 5 \times 10^2 \)
4. \( [Cu(NH_3)_3]^{2+} + NH_3 \overset{K_4}{\rightleftharpoons} [Cu(NH_3)_4]^{2+} \) with \( K_4 = 10^2 \)

### Step 2: Calculate the overall formation constant
The overall formation constant \( K_f \) for the complex \([Cu(NH_3)_4]^{2+}\) is the product of the individual formation constants:

\[ K_f = K_1 \times K_2 \times K_3 \times K_4 \]

Substituting the values:

\[ K_f = (10^4) \times (1.58 \times 10^3) \times (5 \times 10^2) \times (10^2) \]

### Step 3: Perform the multiplication
Calculating step by step:

1. \( K_1 \times K_2 = 10^4 \times 1.58 \times 10^3 = 1.58 \times 10^7 \)
2. \( K_3 \times K_4 = 5 \times 10^2 \times 10^2 = 5 \times 10^4 \)
3. Now multiply these two results:
\[ K_f = (1.58 \times 10^7) \times (5 \times 10^4) = 7.9 \times 10^{11} \]

### Step 4: Calculate the dissociation constant
The dissociation constant \( K_d \) is the reciprocal of the formation constant:

\[ K_d = \frac{1}{K_f} = \frac{1}{7.9 \times 10^{11}} \]

### Step 5: Calculate \( K_d \)
Calculating \( K_d \):

\[ K_d \approx 1.27 \times 10^{-12} \]

### Step 6: Express \( K_d \) in the required form
The question states that \( K_d \) can be expressed as \( x \times 10^{-12} \). From our calculation:

\[ K_d \approx 1.27 \times 10^{-12} \]

Thus, \( x \approx 1.27 \).

### Step 7: Round off to the nearest integer
Rounding \( 1.27 \) gives us \( 1 \).

### Final Answer
The value of \( x \) is \( \boxed{1} \).

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