To determine the values for $a$, $b$, $c$, and $d$ in the expression for the base dissociation constant ($K_b$) of the nitrite ion ($\text{NO}_2^-$), we can use the relationship between the base dissociation constant $K_b$ and the acid dissociation constant $K_a$ of the conjugate acid.

The conjugate acid of $\text{NO}_2^-$ is nitrous acid ($\text{HNO}_2$). The relationship between $K_a$ and $K_b$ is given by:

$K_a \times K_b = K_w$

Where $K_w$ is the ionization constant of water, which is $1.0 \times 10^{-14}$ at 25°C.

Step 1: Find the $K_a$ for $\text{HNO}_2$.

The $K_a$ value for nitrous acid ($\text{HNO}_2$) is commonly known and can be found in tables of acid dissociation constants. It is approximately:

$K_a = 4.5 \times 10^{-4}$

Step 2: Calculate $K_b$ for $\text{NO}_2^-$.

Using the relationship $K_a \times K_b = K_w$:

$K_b = \frac{K_w}{K_a} = \frac{1.0 \times 10^{-14}}{4.5 \times 10^{-4}}$

Now, let's calculate $K_b$:

$K_b = \frac{1.0 \times 10^{-14}}{4.5 \times 10^{-4}} = 2.22 \times 10^{-11}$

Step 3: Identify the values of $a$, $b$, $c$, and $d$.

In the expression $a.b \times 10^{-cd}$, we have:

$K_b = 2.22 \times 10^{-11}$

Thus:

• $a = 2$,
• $b = 2$,
• $c = 1$, and
• $d = 1$.

So, the values are $a = 2$, $b = 2$, $c = 1$, $d = 1$.

Would you like further details on this calculation or have any other questions?

Here are some related questions:

1. How do the values of $K_a$ and $K_b$ relate to the strength of acids and bases?
2. Why is $K_w$ temperature-dependent?
3. What is the significance of the $pK_a$ value in acid-base chemistry?
4. How does the conjugate base's $K_b$ affect the pH of a solution?
5. Can you explain how to determine $K_b$ for polyprotic acids?

Tip: Always remember that strong acids have weak conjugate bases with very small $K_b$ values and vice versa.