### Video Transcript

What is the pH value for a 0.1-molar ammonia solution? K b for NH₃ is equal to four times 10 to the minus five.

The pH value of a solution is a measure of the concentration of protons. However, ammonia is a base and will therefore generate OH⁻ ions in solution. We can go from the concentration of hydroxide ions to pOH and from pOH to pH. The question includes the equilibrium constant for ammonia acting as a base in water. This is the chemical equation for the corresponding process, ammonia plus a molecule of water in equilibrium with the ammonium ion and the hydroxide ion. Now, let’s consider the concentration of each of these species in solution and then construct the equilibrium constant expression.

The initial concentration of ammonia is 0.1 molar, as per the question. Water is the solvent. Therefore, in the equilibrium constant expression, its concentration can be ignored. And since the reaction has yet to occur, there are no ammonium or hydroxide ions. At equilibrium, we can call the change in the concentration of ammonia minus 𝑥. Ammonia is in a one-to-one ratio with both the ammonium ion and the hydroxide ion. Therefore, the change in concentration of these species is plus 𝑥. And we complete the ICE table by summing the initial and the change in concentration to produce the equilibrium concentration for each species.

Now, we can move on to the equilibrium constant expression. Here, the equilibrium constant expression is equal to the concentration of ammonium ion multiplied by the concentration of the hydroxide ion divided by the concentration of ammonia. Here, we ignore the concentration of the water because it’s the solvent. And its concentration is assumed to be constant. If we substitute the values from our ICE table into the equilibrium constant expression, we get 𝑥 multiplied by 𝑥 divided by 0.1 minus 𝑥, which is equal to 𝑥 squared divided by 0.1 minus 𝑥.

Ammonia is a weak base, as shown by the low value of its equilibrium constant. Therefore, relative to the initial concentration of ammonia, the change in the concentration of ammonia is going to be relatively small. So we can approximate 0.1 minus 𝑥 as 0.1. Therefore, the equilibrium constant is equal to 𝑥 squared divided by 0.1. We can rearrange this expression in terms of 𝑥 to equal the square root of 0.1 multiplied by the equilibrium constant. Substituting the value of the equilibrium constant into this expression gives the square root of 0.1 multiplied by four times 10 to the minus five. This gives a value of 𝑥 of 0.002 molar. 𝑥 is equal to the concentration of the hydroxide ion.

So now we can move on to stage two, calculating the value of pOH. pOH is equal to the negative log of the hydroxide ion concentration. Therefore, pOH is equal to the negative log of 0.002. This is equal to 2.69897. pH and pOH are complementary measures. pH is equal to 14 minus pOH. Therefore, pH is equal to 14 minus 2.69897, which is equal to 11.30103, which is equal to 11.3 to three significant figures. If you’re curious, if we hadn’t made the approximation that 0.1 minus 𝑥 was equal to 0.1, we would’ve calculated a pH value of 11.29669. This again rounds to 11.3 to three significant figures. Therefore, our approximation is appropriate.

So the pH value for a 0.1-molar ammonia solution is 11.3.