The addition of four amine groups to a metal ion `(X^(2+))` shows a stability constants of `2 xx 10^4`, `1.5 xx10^3`, `1.2 xx10^2` and `1.4 xx10^1` respectively. Then, the overall complex dissociation equilibrium constant for `[X(NH_3)_4]^(2+)` ion is

Options

A. `5.04xx 10^(-10)`
B. `19.8 xx10^(-10)`
C. `1.98 xx 10^(-11)`
D. `50.4 xx10^(-9)`

Answer

To solve the problem, we need to find the overall complex dissociation equilibrium constant for the complex ion \([X(NH_3)_4]^{2+}\) based on the given stability constants for the formation of the complex.

### Step-by-Step Solution:

1. **Identify the Stability Constants**:
We are given the stability constants for the formation of the complex in four steps:
- \(K_1 = 2 \times 10^4\)
- \(K_2 = 1.5 \times 10^3\)
- \(K_3 = 1.2 \times 10^2\)
- \(K_4 = 1.4 \times 10^1\)

2. **Write the Formation Reaction**:
The formation of the complex can be represented as:
\[ X^{2+} + 4 NH_3 \rightleftharpoons [X(NH_3)_4]^{2+} \]

3. **Calculate the Overall Formation Constant**:
The overall formation constant \(K_f\) for the reaction can be calculated by multiplying the individual stability constants:
\[ K_f = K_1 \times K_2 \times K_3 \times K_4 \]
Substituting the values:
\[ K_f = (2 \times 10^4) \times (1.5 \times 10^3) \times (1.2 \times 10^2) \times (1.4 \times 10^1) \]

4. **Perform the Multiplication**:
Let's calculate it step by step:
- First, calculate \(K_1 \times K_2\):
\[ K_1 \times K_2 = (2 \times 10^4) \times (1.5 \times 10^3) = 3 \times 10^7 \]
- Next, multiply by \(K_3\):
\[ (3 \times 10^7) \times (1.2 \times 10^2) = 3.6 \times 10^9 \]
- Finally, multiply by \(K_4\):
\[ (3.6 \times 10^9) \times (1.4 \times 10^1) = 5.04 \times 10^{10} \]

5. **Determine the Overall Dissociation Constant**:
The overall dissociation constant \(K_d\) is the inverse of the formation constant:
\[ K_d = \frac{1}{K_f} \]
Therefore:
\[ K_d = \frac{1}{5.04 \times 10^{10}} = 1.98 \times 10^{-11} \]

6. **Final Answer**:
The overall complex dissociation equilibrium constant for \([X(NH_3)_4]^{2+}\) is:
\[ K_d = 1.98 \times 10^{-11} \]