The pH of an aqueous 0.10 M nitrite ion is 8.17. What is the base dissociation constant of the base?

Here this question states, the pH of an aqueous 0.10 molar nitrite ion is 8.17. Here we're asked, what is the base dissociation constant of the base? We're asked to find Kb. Nitride ion is NO2 minus. A negatively charged species usually indicates that we have a base which is why they're asking us to find the Kb of the base. So this is n o two minus plus remember, here they're asking us to figure out k b. All our answer options are less than 1 so we know that this is a weak base. Weak bases, weak acids react with water in solution to establish equilibrium. Here, this NO2 minus will react with water. Here, it is the base. So water will act as the acid. Water will donate an H+ over. We're gonna get here HNO2 +OH minus. We're dealing with an ICE chart because it's a weak species, so we have initial change equilibrium. We're gonna ignore water because it's a liquid. Our initial concentration is 0.10 molar. These are 0 initially. We lose reactants to make product. 0.10 minuteus x plus x plus x. Here, we're looking to find k b. K b equals products over reactants just like k a. So it's HNO 2 concentration times o h minus concentration divided by NO2 minus concentration. Now here, because we're looking for our equilibrium constant k b, we cannot ignore any variables. So it's x squared divided by 0.10 minus x. We're gonna have to solve for x in order to find our Kb value. How do we do that? Well, they tell us what the pH is for our nitrite ion. But remember, pH is usually associated with an acid and hydronium ion concentration. In this question, from our equation, we have Oh minus concentration. That means that we're gonna have to find actually pOH first. With pOH, we'll be able to find o h minus. So remember, we're gonna say here pOH equals 14 minus p h. So 14 minus 8.17. So that comes out to 5.83. Now that I know my pOH, I know Oh minus. Oh minus equals 10 to the negative pOH. This equals 1.479 times 10 to the negative 6 molar. That number represents x. Because I know what x is, I can plug it into here and into here. Again, remember, anytime you're looking for your equilibrium constant, whether it be Ka or Kb or any other equilibrium constant, we cannot ignore any of the x variables. We keep all of them there to get the most accurate number for our equilibrium constant. We're gonna plug that number in. It's 1.479 times 10 to the negative 6 squared on top divided by point 10 minuteus, again, that x variable here on the bottom. So here, when we plug all that in, that's gonna give me 2.1878 times 10 to the negative 11. Equilibrium constants Ka and Kb have no units involved with them. So here, based on our options, the correct answer would have to be b after rounding. Just remember, if they're talking about calculating k or k b, we know that the species itself is weak because strong acids and strong bases have ka and k b values that are much greater than 1. In fact, we just say that they don't have kabka or kb values that are useful to us. For weak acids and weak bases, their k a and k b values are less than 1 which is why we use a nice chart here.