Animated Solution for Chemistry - Coordination Compounds: The overall stability constant of the complex ion $[Cu(N{H}_3{)}_4{]}^{2+}$ is $2.1\times 10^{13}$. The overall dissociations constant is $y\times 10^{-14}$. Then, $y$ is ......... . (Nearest integer)

Imagine a bustling dance floor where a central metal ion, like $C{u}^{2+}$, is surrounded by eager partners, the ligands, such as $N{H}_3$. When they come together, they form a beautiful, stable entity known as a coordination complex. But this dance isn't permanent; it's a dynamic equilibrium. The partners can join hands, and they can also let go.

In chemistry, we quantify this dynamic relationship using two very important constants: the Stability Constant ($K_f$) and the Dissociation Constant ($K_d$).

The stability constant, $K_f$, tells us how much the equilibrium favors the formation of the complex. A massive $K_f$ value, like the $2.1\times 10^{13}$ given in our problem, means the complex is incredibly stable. The metal and ligands really want to stay together!

On the flip side, the dissociation constant, $K_d$, measures the tendency of the complex to break apart into its constituent ions. Because formation and dissociation are exact reverse processes of the same chemical equilibrium, their constants are mathematically reciprocal to each other.

This gives us our master equation:

Let's substitute the given stability constant into our equation to find the dissociation constant:

To solve this without breaking a sweat, we can separate the numbers from the powers of ten:

Dividing $1$ by $2.1$ gives us approximately $0.476$. So, we have:

The question is a bit sneaky; it asks for the answer in a specific scientific notation format: $y\times 10^{-14}$. To match this, we need to shift our decimal point one place to the right, which decreases the exponent by one:

By comparing this to $y\times 10^{-14}$, we can clearly see that $y=4.76$.

But wait, there's a final catch! The question demands the nearest integer. Since the first decimal digit is $7$ (which is $5$ or greater), we round up the number.

Therefore, $4.76$ rounds off to $5$.

And there we have it! A seemingly complex equilibrium problem dismantled into a simple reciprocal calculation.