pH and pKa

Relationship between pH and pK_a

Before diving into pH and pKa, let's recall the definition of Bronsted-Lowry acids and bases, and also the meaning of conjugate acids and bases.

Bronsted-Lowry acids are proton (H⁺) donors, whereas Bronsted-Lowry bases are proton (H⁺) acceptors. Let's look at the reaction between ammonia and water.

Fig. 1: The reaction between ammonia and water, Isadora Santos - StudySmarter Originals.

Conjugate acids are bases that gained a proton H⁺. On the other hand, Conjugate bases are acids that lost a proton H⁺. For example, when HCl is added to H₂O, it dissociates to form H3O⁺ and Cl^-. Water will gain a proton, and HCl will lose a proton.

Fig. 2: Conjugate pairs in a reaction between HCl and Water, Isadora Santos - StudySmarter Originals.

Now that those definitions are fresh in our minds, let's look at how pH and pK_a are related. The first thing you need to know is that we can use pH and pKa to describe the relationship between weak acids in an aqueous solution.

The definition of pK_a can sound confusing, especially if you are not familiar with the acid dissociation constant, also known as K_a. So, let's talk about that!

When it comes to weak acids and pH calculation, we need an extra piece of information, the acid dissociation constant (K_a). K_a is used to determine the strength of an acid and its ability to stabilize its conjugate base. It measures how fully an acid is able to dissociate in water. In general, the higher the K_a of an acid, the stronger the acid will be.

The general formula for a monobasic acid can be written as:$\mathrm{HA}\left(\mathrm{aq}\right)\rightleftharpoons {\mathrm{H}}^{+}\left(\mathrm{aq}\right){\mathrm{A}}^{-}\left(\mathrm{aq}\right)$, where:

• HA is the weak acid.

• H⁺ is the hydrogen ions.

• A^- is the conjugate base.

We can use the following formula for K_a:

$$K_{a}=\frac{[products]}{[reactants]}=\frac{[H^{+}]\cdot [A^{-}]}{HA}=\frac{[H^{+}]^{2}}{HA}c$$

Keep in mind that solids (s) and pure liquids (l) like H₂O (l) should not be included when calculating K_a because they have constant concentrations. Let's look at an example!

Now that we know what K_a means, we can define pK_a. Don't worry about pK_a calculations right now - we will deal with it in a little bit!

Difference between pK_a and pH

The main difference between pH and pK_a is that pK_a is used to show the strength of an acid. On the other hand, pH is a measure of the acidity or alkalinity of an aqueous solution. Let's make a table comparing pH and pK_a.

pH and pK_a Equation

When we have a strong acid, such as HCl, it will completely dissociate into H⁺ and Cl^- ions. So, we can assume that the concentration of [H⁺] ions will be equal to the concentration of HCl.

$$HCl\rightarrow H^{+}+Cl^{-}$$

However, calculating the pH of weak acids is not as simple as with strong acids. To calculate the pH of weak acids, we need to use ICE charts to determine how many H⁺ ions we will have at equilibrium, and also use equilibrium expressions (K_a).

$$HA_{(aq)}\rightleftharpoons H^{+}_{(aq)}+A^{-}_{(aq)}$$

ICE Charts

The easiest way to learn about ICE tables is by looking at an example. So, let's use an ICE chart to find the pH of a 0.1 M solution of acetic acid (The K_a value for acetic acid is 1.76 x 10^-5).

Step 1: First, write down the generic equation for weak acids:

$$HA_{(aq)}\rightleftharpoons H^{+}_{(aq)}+A^{-}_{(aq)}$$

Step 2: Then, create an ICE chart. "I" stands for initial, "C" stands for change, and "E" stands for equilibrium. From the problem, we know that the initial concentration of acetic acid is equal to 0.1 M. So, we need to write that number on the ICE chart. Where? On the "I" row, under HA. Before dissociation, we don't have H⁺ or A^- ions. So, write a value of 0 under those ions.

Fig. 3: How to fill the "I" row on the ICE chart, Isadora Santos - StudySmarter Originals

Step 3: Now, we need to fill out the "C" (change) row. When dissociation occurs, change goes to the right. So, the change in HA will be $-x$, whereas the change in the ions will be $\mathit{+}x$.

Fig. 4: Filling out the "C" row on the ICE Chart. Isadora Santos - StudySmarter Originals.

Step 4: The equilibrium row shows the concentration at equilibrium. "E" can be filled by using the values of "I" and "C". So, HA will have a concentration of 0.1 - x at equilibrium and the ions will have a concentration of x at equilibrium.

Fig. 5: Filling the "E" row on the ICE chart, Isadora Santos - StudySmarter Originals.

Step 5: Now, we have to create an equilibrium expression using the values in the equilibrium row, which will then be used to solve for x.

• x is equal to the [H⁺] ion concentration. So, by finding x, we will be able to know [H⁺] and then calculate pH.

$$K_{a}=\frac{[H^{+}]\cdot [A^{-}]}{HA}=\frac{x^{2}}{0.1-x}$$

Step 6: Plugin all the known values to the K_a expression and solve for x. Since x will usually be a small number, we can ignore the x that's being subtracted from 0.1.

$$K_{a}=\frac{x^{2}}{0.1-x}\cdot 1.76\cdot 10^{-5}=\frac{x^{2}}{0.1}x=\sqrt{(1.76\cdot 10^{-5})}\cdot 0.1=0.0013M=[H^{+}]$$

Step 7: Use the [H⁺] value to calculate pH.

$$=-log_{10}[H^{+}]pH=-log_{10}[0.0013]pH=2.9$$

pH and pK_a Formulas

To calculate pH and pK_a, you should be familiar with the following formulas:

Fig. 6: Formulas relating pH and pKa, Isadora Santos - StudySmarter Originals.

Let's look at a problem!

That was pretty straightforward, right? But, let's amp up the difficulty a bit more!

Now, what if you were asked to calculate pKa from Ka? All you need to do is use the pK_a formula if you know the value for K_a.

For example, if you know that the K_a value for benzoic acid is 6.5x10^-5 mol dm^-3, you can use it to calculate pK_a:

$$pK_{a}=-log_{10}(K_{a})pK_{a}=-log_{10}(6.3\cdot 10^{-5})pKa=4.2$$

Calculating pK_a from pH and Concentration

We can use the pH and concentration of a weak acid to calculate the pK_a of the solution. Let's look at an example!

Finding Percent Ionization given pH and pK_a

Another way of measuring the strength of acids is through percent ionization. The formula to calculate percent ionization is given as:

$$%\ ionization=\frac{concentration\ of\ H^{+}\ ions\ in\ equilibrium}{initial\ concentration\ of\ the\ weak\ acid}=\frac{x}{[HA]}\cdot 100$$

Remember: the stronger the acid, the greater the % ionization. Let's go ahead and apply this formula to an example!

Now, you should have what it takes to find the pH and pK_a of weak acids!

_References:

_{Brown, T. L., Nelson, J. H., Stoltzfus, M., Kemp, K. C., Lufaso, M., & Brown, T. L. (2016). Chemistry: The central science. Harlow, Essex: Pearson Education Limited.}

_{Malone, L. J., & Dolter, T. (2013). Basic concepts of of Chemistry. Hoboken, NJ: John Wiley.}

_{Ryan, L., & Norris, R. (2015). Cambridge International as and A level chemistry. Cambridge: Cambridge University Press.}

_{Salazar, E., Sulzer, C., Yap, S., Hana, N., Batul, K., Chen, A., . . . Pasho, M. (n.d.). Chad's general chemistry Master course. Retrieved May 4, 2022, from https://courses.chadsprep.com/courses/general-chemistry-1-and-2}