What is the pH of a 0.50 M solution of NaNO2 if the Ka of H20x = 4.5 x 10^-4. I know that you use the Ka to find Kb and solve from there, but why?

Why not?

NO2^- is a base; it attracts a H^+ from water and Kb is the strength as a base.

So you're saying that if you have the Ka of any acid and you can use the ionization constant of water to find Kb of its conjugate base?

I didn't say that in so many words but, yes, you can do that. Why? Because

KaKb = Kw.
If you know EITHER Ka or Kb you can calculate the other one from this relationship and Kw.

To find the pH of a solution of NaNO2, you can use the concept of hydrolysis. In this case, NaNO2 is a salt that comes from a weak base (NO2-) and a strong acid (Na+). When this salt dissolves in water, it undergoes hydrolysis, leading to the formation of hydroxide ions (OH-) and hydronium ions (H3O+).

To determine the pH, we first need to calculate the concentration of hydroxide ions (OH-) in the solution. Since NaNO2 is the salt of a weak base (NO2- is the conjugate base of HNO2) and a strong acid (Na+ derived from NaOH), the NO2- anions will react with water to form OH- ions via hydrolysis.

The hydrolysis reaction can be represented as follows:
NO2- + H2O ⇌ HNO2 + OH-

Since the hydrolysis reaction involves the formation of OH- ions, we can think of this process as the salt acting as a weak base and producing OH- ions. This allows us to consider the equilibrium constant, Kb, for the reaction of the NO2- ion with water.

However, in this case, the Ka of HNO2 (the acid from which NO2- originates) is given (Ka = 4.5 x 10^-4), which describes the dissociation of HNO2 into H+ and NO2-. To find the Kb, we can use the relationship between Ka and Kb:

Kw = Ka * Kb

Since Kw is the equilibrium constant for water and is always equal to 1.0 x 10^-14 at 25°C, we can rearrange the equation to solve for Kb:

Kb = Kw / Ka

Now that we have Kb, we can use it to determine the concentration of OH- ions in the solution. Once we have the concentration of OH-, we can convert it to the concentration of H3O+ ions and then calculate the pH using the equation:

pH = -log[H3O+]

Therefore, in this case, we use the given Ka of HNO2 to calculate the Kb for the hydrolysis of NO2-, which allows us to determine the concentration of OH- ions and ultimately find the pH of the NaNO2 solution.