The stepwise and overall stability constants of coordination compounds are related as:

Question

Accepted Answer

The relationship between stepwise stability constants and overall stability constants in coordination chemistry is a fundamental concept. Let's explore why Option 2 is the correct answer and why the other options are incorrect.

### Explanation for Option 2:

**Option 2: $\beta_{4}=K_{1}\times K_{2}\times K_{3}\times K_{4}$**

This option correctly represents the relationship between the stepwise stability constants and the overall stability constant for a coordination compound. In coordination chemistry, the overall stability constant ($\beta_n$) is the product of the stepwise stability constants ($K_1, K_2, K_3, \ldots, K_n$). This is because each stepwise constant represents the equilibrium constant for the addition of one ligand to the metal complex, and the overall stability constant is the equilibrium constant for the formation of the entire complex from the free metal ion and free ligands. Therefore, the overall stability constant is the product of all the individual stepwise constants:

$$
\beta_4 = K_1 \times K_2 \times K_3 \times K_4
$$

### Explanation for why other options are incorrect:

**Option 1: $\beta_{4}=K_{1}+K_{2}+K_{3}+K_{4}$**

This option suggests that the overall stability constant is the sum of the stepwise stability constants. However, this is incorrect because the stability of a complex is not additive in nature. The formation of each successive complex is an independent equilibrium process, and their constants multiply rather than add.

**Option 3: $\beta_{4}=K_{1}-K_{2}-K_{3}-K_{4}$**

This option is incorrect because it implies that the overall stability constant is derived by subtracting the stepwise constants from each other. This does not make sense in the context of equilibrium constants, as they represent multiplicative relationships in chemical equilibria, not subtractive ones.

**Option 4: $\beta_{4}=\frac{K_{3}\times K_{4}}{K_{2}\times K_{1}}$**

This option incorrectly suggests that the overall stability constant is a ratio of products of some stepwise constants to others. This is not how stability constants are related. The overall stability constant is a straightforward product of all stepwise constants, not a fraction or ratio.

In conclusion, Option 2 is the correct representation of the relationship between stepwise and overall stability constants in coordination chemistry, as it accurately reflects the multiplicative nature of these constants.