To determine the base dissociation constant ($K_b$) of ammonia ($NH_3$) from the given information, we can follow these steps:

Step 1: Use the pOH to find the hydroxide ion concentration $[\text{OH}^-]$

We are given that the pOH is 2.72. The concentration of hydroxide ions $[\text{OH}^-]$ can be found using the formula:

$[\text{OH}^-] = 10^{-\text{pOH}}$

Substitute the given pOH value:

$[\text{OH}^-] = 10^{-2.72}$

Now, let's calculate that value.

$[\text{OH}^-] = 1.905 \times 10^{-3} \, \text{M}$

Step 2: Set up the base dissociation equation for ammonia

The base dissociation of ammonia in water is represented as:

$NH_3 + H_2O \rightleftharpoons NH_4^+ + OH^-$

Let $x$ represent the concentration of $[\text{OH}^-]$, which is also the concentration of $[NH_4^+]$ due to the stoichiometry of the reaction. This means that:

$x = [\text{OH}^-] = 1.905 \times 10^{-3} \, \text{M}$

Step 3: Use the ICE table to set up the expression for $K_b$

Species	Initial (M)	Change (M)	Equilibrium (M)
$NH_3$	0.21	$-x$	$0.21 - x$
$NH_4^+$	0	$+x$	$x$
$OH^-$	0	$+x$	$x$

Since $x = [\text{OH}^-]$, we have:

$x = 1.905 \times 10^{-3} \, \text{M}$

The equilibrium concentration of $NH_3$ is:

$[ NH_3 ] = 0.21 - 1.905 \times 10^{-3} \approx 0.2081 \, \text{M}$

Step 4: Write the expression for $K_b$

The expression for the base dissociation constant $K_b$ is:

[ K_b = \frac{[NH_4^+][OH^-]}{[NH_3]} ]

Substituting the values:

$K_b = \frac{(1.905 \times 10^{-3})(1.905 \times 10^{-3})}{0.2081}$

Now, calculate the value of $K_b$:

$K_b = \frac{3.630 \times 10^{-6}}{0.2081} = 1.744 \times 10^{-5}$

Final Answer:

The base dissociation constant $K_b$ for ammonia is:

$K_b = 1.74 \times 10^{-5}$

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the relationship between pOH and pH used to find the dissociation constants?
2. What role does ammonia's structure play in its basicity in water?
3. How can the $K_b$ of a base be related to the $K_a$ of its conjugate acid?
4. How does changing the concentration of ammonia affect the pOH of the solution?
5. What are the units of the base dissociation constant $K_b$?

Tip: The relationship between pH and pOH can always be used to find unknowns, as they are connected by $\text{pH} + \text{pOH} = 14$.