Fractional Compositions and Concentrations - Online Tutor, Practice Problems & Exam Prep

Understanding fractional compositions in acid-base chemistry is crucial for determining the concentrations of weak acids and their conjugate bases at specific pH levels. For monoprotic acids, the equilibrium expression is given by the acid dissociation constant, Ka = [H⁺][HA] . At pH = pK_a, the concentrations of the acid and conjugate base are equal. In diprotic systems, multiple forms exist, requiring consideration of both K_a1 and K_a2 for accurate calculations of each species' fraction as pH changes.

Fractional Composition

So fractional compositions are just a way of determining the amount of acid to base at a given or specific pH. As our pH decreases, our solution becomes more acidic because the amount of acid present is increasing. As we increase pH, it becomes more basic because we're increasing the amount of conjugate base. Now remember, with weak acids, we have the formation of an equilibrium because weak acids are weak electrolytes. So they don't completely ionize into their products. Here, our weak acid, which is just a simple generic formula, partially ionizes to form H⁺ion and A^- as our conjugate base.

Setting up the equilibrium expression gives us the acid dissociation constant Ka equals the ions as products on top divided by the initial concentration of my weak acid. Now the formal concentration of my solution equals the concentration of my weak acid plus the concentration of my conjugate base. Now when we talk about the fraction of acid molecules present, we use the variable α to represent HA here. If we take a look here, what this chart is showing me is at a pH, the lowest possible pH, the composition of my solution will be 100% in the acid form, and 0% of the basic form.

As the pH begins to increase as we move from left to right, we're going to have an increase in the amount of conjugate base and a decrease in the amount of weak acid. Here, they intersect at this point here. At that pH, we have 50% of the weak acid form and 50% of the conjugate base form. This is true when the pH is equal to my pKa. So when those two equal each other, it's because the amount of conjugate base and weak acid are equal in amount.

Remember this coincides with the whole Henderson-Hasselbalch equation, which is pH = pKa + log(conjugate baseweak acid). When they're both equal to each other, this all becomes equal to 1. The log of 1 is simply equal to 0, and that's why pH = pKa.

So when pH equals pKa, the amount of weak acid is equal to the amount of its conjugate base, and if we continue past this pH point, we're going to continually have a drop in my weak acid form and a steady rise in the amount of my conjugate base form, until we get to a pH where the weak acid form is basically near 0 and the conjugate base is near 100%.

Now here we'd say that the concentration of my weak acid is equal to the concentration of H⁺ times the formal concentration of my solution divided by H⁺+Ka. Here, if we want to find out the fraction that exists in the acid form, that just simply gets broken down into the concentration of H⁺ divided by the concentration of H⁺+Ka, so that there represents the fraction in the acid form and here the fraction in the conjugate base form or A^- is represented by α_a-. Here, that would just be equal to Ka / (H⁺+Ka).

So these are the two equations we can utilize in order to figure out the fraction of either my conjugate acid or my weak acid form, and remember together both of them would equal 1, and if you multiply that by 100, it would be 100%. Okay? So if you know one, you know what the other one is because together they represent 100% of all possible molecules and ions present within my solution at any given pH. Now that you see it in terms of a weak monoprotic acid, click on the next video to look at it in terms of a diprotic acid.

Fractional Composition

With diprotic systems, we need to take into account that there are more forms that exist based on pH. With a diprotic system, we have our acidic form where we still possess both of our acidic hydrogens. Once we lose that first acidic hydrogen, we now have the intermediate form. Then finally, we have our basic form. With these different forms for our diprotic system, we have to take into account more than one k value. So remember, losing the first acidic hydrogen means we're dealing with ka1, which then gives us this equilibrium expression. Rearranging it gives us our concentration of the intermediate form as the acid form times ka1 divided by the concentration of H⁺ ion produced.

When we lose that second acidic hydrogen, we have the formation of additional H⁺, as well as the basic form. Here, once we rearrange our first equilibrium expression, we can figure out what our concentration of A2^- is. Remember, HA^-, our intermediate form, equals this entire expression here, which you can then substitute in for this minus here. So, by substituting in that portion, we get this with only one H⁺ getting multiplied by ka2 plus one H⁺ on the bottom which is why this is squared here. From that we get our formal concentration as being equal to the concentration of our acid form, times this large expression here.

Now the fractions of the three major diprotic forms can be seen as we have our fraction of the acid form, the fraction of our intermediate form, and the fraction of our basic form, each with their own equation. We can see here that this darker line represents the acid form. This dotted line represents the intermediate form, and then this gray line represents the basic form. As the pH is increasing, we're going from left to right, and the amount of the acid form is decreasing as well as the intermediate form, and the basic form is increasing. We're going to say here at this particular pH, we have the intersection between the acid form and the intermediate form. So we'd say here pH equals pKa1 at this intersection. At that intersection, we can also say that the concentration of my acid form is equal to the concentration of my intermediate form. Then we have this intersection here at a higher pH. At this point, we'd say that pH is equal to pKa2. So at that point, we'd say that the intermediate form concentration is equal to the basic form. So, again, we can utilize this chart here to determine what the fraction of two forms are in terms of one another at any given pH. Remember, when it comes to diprotic systems, there are more intermediate forms involved, which leads to more complex equations to represent the fractions of each of those forms. And remember, your fractional composition is all based on the pH. The higher the pH the more the basic forms exist and predominate, and at lower pHs the acid forms will predominate.

Fractional Composition Calculations 1

Here we're told we have a dibasic compound B that has a pK_b1 value of 5.5 and a pK_b2 value of 8. Find the fraction of the acidic form when the pH equals 9.00. Alright, so we're looking for the fraction of the acid form of a dibasic species. That means we're dealing also with a diprotic species looking at it from the opposite direction. So here, that would be the fraction of the acid form equals the concentration of H⁺ squared divided by the concentration of H⁺ squared plus the concentration of H⁺ times K_a1 plus K_a1 times K_a2.

What we have to do here is determine our concentration of H⁺ and determine what my acid dissociation constants K_a1 and K_a2 are. We're going to say here that H⁺ equals 10 to the negative pH. So H⁺ is equal to 10 to the negative 9.00. We have that figured out so far. We'll plug that in. Next, we need to determine what my K_a1 and K_a2 will be. But here we're dealing with pK_b1 and pK_b2. What we need to remember is the relationship between your acid dissociation constant and your base association constant is pK_a1 + pK_b2 equals 14 and pK_a2 + pK_b1 equals 14. So K_a1 and K_b2 are connected and K_a2 and K_b1 are connected.

We're going to say here that pK_a1 equals 14 minus pK_b2. So that'd be 14 minus 8 which gives us 6. Here, pK_a2 equals 14 minus pK_b1. So that's 14 minus 5.00, so that's 9. Now that we know pK_a1 and pK_b1, we know what K_a1 and K_a2 will be. That's because we're gonna say that K_a1 equals 10 to the negative pK_a1 and K_a2 equals 10 to the negative pK_a2. So plug those values in. So we're gonna get here 10 to the negative 6.00 and 10 to the negative 9.00. Take those values and plug them in. So that's gonna be 10 to the negative 18 divided by when we add up everything on the bottom, that gives us 2.001 times 10 to the negative 15.

So at the end, the fraction in my acid form would equal 0.0005. This makes sense because we'd say that at pK_a1, our pK_a1, we were told, is 6.00. When our pH was equal to our pK_a1, we'd have an equal amount of the acid form and the conjugate base form. But as the pH gets higher and higher, we're going to have less and less of the acid form. The pH is 3 units higher than my pK_a1 value of 6. So that means you're going to see a decrease in the amount of my acid form. Being 3 units different is a huge difference in magnitude concentrations. It makes sense that my acid form would be such a small value. But remember, dibasic alludes to diprotic. Here, we're talking about finding the acid form. If they're asking for the intermediate form or the basic form, all we have to do is recall the equations associated with those different forms of diprotic, dibasic species. Utilizing them helps us determine the concentration of any one of those in terms of their fractions. Now that you've seen this example, attempt to do the next one. We're talking about the fraction of Tyrosine. Remember the equations that we've seen on previous pages. Utilize them to find the fraction of that particular amino acid. Once you do, come back and see if your answer matches up with mine.

Fractional Composition Calculations 1

Here it states, what fraction of tyrosine, which has 2 pKa values, exists in all of its forms at pH equals 10.00. Here, because we're dealing with 2 pKa values, that means that tyrosine exists as a diprotic acid. Tyrosine, one of the amino acids, has 2 acidic Ka values. Remember, for diprotic species, we have the acid form, the intermediate form, and the basic form. Halfway here, we have pH equals pKa 1. At that point, we should have 50% of the acid form and 50% of the intermediate form actually. And then here, halfway, pH equals pKa 2. So we should have 50% of the intermediate form and 50% of the basic form. Now, in actuality, we won't really have a flat 50% of both of these at this pH equals pKa 1. There will be some weak acid in small minute amounts that we're dealing with. Realize here that our pKa 2 is 8.67 and our pH is above that. What that tells me is that we've gone beyond this line here, so the predominant form of tyrosine would be its basic form.

What we're going to do now is use the equations for the fractions of the acid form, intermediate form, and basic form to figure out how much we really have of these 3 different forms. For a diprotic species, the fraction in the acidic form can be expressed using MathML: H+ H+ + Ka1 + Ka1*Ka2 .

The fraction of the intermediate form is: Ka1*H+ H+ + Ka1 + Ka1*Ka2 .

Finally, for the fraction of the basic form, it would be: Ka1*Ka2 H+ + Ka1 + Ka1*Ka2 .

All calculations involve the same denominator, therefore, once these values are input into the equations above, we'll get the expected fraction for each form of tyrosine. So, when these values are plugged into a calculator, we find that the fraction in the acid form would be 1.05*10-19, which makes sense because of how small this value is. The predominant form would indeed be the basic form because the pH is above the second pKa value.

As calculated, the fractions would be approximately 0.047 for the intermediate form and around 0.955 for the basic form, showing the basic form is the largest in amount as expected because the pH is at 10. If these values were to be converted to percentages, they would indicate 95.5% in the basic form.

Now that we've seen this example, attempt to do the practice question that's left on the bottom of the page. Once you do, come back and see if your answer matches up with mine.

Calculate the fraction of the intermediate for sulfurous acid, H ₂SO₃, at pH = 11.00? pK_a1 = 1.80, pK_a2 = 7.19. 

• A. 6.31x10^-11
• B. 9.77x10^-14
• C. 0.000155
• D. 0.00977
• E. 1.00

Fractional Concentrations

So recall when it comes to monoproduct systems, that the fraction of the acid molecules and the conjugate base molecules were represented by αH and αA-. Here, when we're talking about the fraction of acid molecules, it would equal the H+ concentration divided by both H+ and Ka, and the fraction of the base concentration was equal to Ka divided by H++Ka. As we look at this chart here on the left, as our pH increased going from left to right, the amount of the acid molecules would decrease as the amount of conjugate base molecules would increase. Here at this intersection is where both the concentration of the HA molecules and A- molecules would be equal to one another. Also, remember that that happens when pH is equal to the pKa.

Now, when we try to determine what our concentrations are for these fractions of these molecules, we can restructure our equations to now represent the concentration of my acid is equal to the fraction of the acid molecules times the formal concentration of the acid molecules. Using this expression now inputs the formal concentration of my weak acid solution. And if we're looking for the overall concentration of my conjugate base, it's equal to the fraction of my conjugate base molecules times the formal concentration of my solution, acid solution, so that'd be Ka times formal concentration of acid again divided by H++Ka. So remember the fraction can be determined by comparing the pH of the solution to the pKa value to see which species is greater in amount. We can utilize these equations themselves to find either the exact fraction of the weak acid or the exact fraction of the conjugate base. From there, we can attach formal concentration to find the new concentrations of the weak acid and the conjugate base. Click on to the next video and see what we do when dealing with diprotic systems.

Fractional Concentrations

So as we stated before in the past, when it comes to diprotic systems we have 3 possible forms that our compound can take. We have the acidic form, we have the intermediate form after it's lost its first acidic hydrogen, then we have the basic form. Here with these 3 different forms, we have the introduction of Ka1 and Ka2. When removing the first acidic hydrogen, we produce your hydronium ion plus some intermediate. K1 would be involved, which would give us this expression of products over reactants. By rearranging it, we'd be able to isolate the concentration of our intermediate and the concentration of your acid form times Ka1 divided by H⁺. Here from the intermediate form, we could lose our second acidic hydrogen to give us more hydronium ion created, plus now the basic form of our compound. By substituting in this expression for the intermediate, we can get the concentration of our basic form as being H2AKa1Ka2, over H⁺². Now our formal concentration comes about through the use of our mass balance. Here, we'd have our formal concentration equal to the concentration of my acid form times the expression within here. Now remember the full concentrations now of the 3 major diprotic forms can be restructured as the following. We've seen the equations related to the fractions of the acid form, the intermediate form, and the basic form. With the introduction of our formal concentration here, we'll be able to calculate the new concentrations of each one of these compounds. By doing that, we just have the insertion of the formal concentration into each one of these compounds, and remember all three forms, although different, possess the same denominator on the bottom. Also recall that when we're taking a look at the fraction versus pH chart, that this line here represents the amount or concentration of my acid form. This here represents the concentration of my intermediate form, and then this one here represents the concentration of my basic form. At this intersection, we have pH equal to pKa1 because the concentration of H2A would be equal to the concentration of HA-. And here, at our second intersection, we'd have pH equal to pKa2 because the intermediate form would be equal to the basic form. This just shows us as the pH is increasing going from left to right, the proportion of the conjugate or the basic form is increasing, and remember even at high enough pHs, we're still going to have some amount of our acidic form remaining. It'll just be a very very very small amount the higher the pH gets, and remember all three forms exist in some proportion the higher the pH gets. So we've seen these equations in terms of the fractions, now we're introducing our formal concentrations to find our new conditional concentration of our acid form, our intermediate form, and the basic form.

Fractional Concentrations Calculations 1

Here it states a dibasic compound b has a pKb1 value of 4.00 and a pKb2 value of 6.00. We're asked to find the concentration of the intermediate form when the formal or total concentration of the diprotic acid equals 0.150 molar and the pH equals 8.00. Alright. So here we need to determine the concentration of our intermediate.

We're going to say the concentration of our intermediate form equals the fraction of the intermediate form times the formal concentration of the diprotic acid. When we expand this out, it gives me Ka_1⁢⁢H⁺⁢⁢[diprotic acid]⁢⁢⁢/⁢⁢H⁺⁢⁢⁢²+H⁺⁢⁢Ka_1+Ka_1⁢⁢Ka_2. Alright.

So we know what H⁺ is equal to because [H⁺]=10-pH. In this case, we're dealing with a pH of 8.00. Next, we need to figure out what Ka_1 and Ka_2 will be. We're gonna say that pKa_1+pKb_2=14, and that pKa_2+pKb_1=14. So pKa_1 here equals 14-pKb_2, which is 6. So this comes out to being 8. Because we now know what pKa_1 is, we know what Ka_1 is. Ka_1=10-pKa_1. So that's 10-8.00. pKa_2 will equal 14-pKb_1. So it equals 10. So Ka_2=10-pKa_2. So it's 10-10.

We now have all the units that we need, so we're just gonna plug them into the formula above. So Ka_1=10-8⁢⁢0.150 molar, dividing now by 10-8.002+H⁺⁢⁢⁢Ka_1+Ka_1⁢⁢Ka_2. When we figure out what's on the top and on the bottom, we're gonna get, 1.5⁢⁢10-17 divided by 2.01⁢⁢10-16. So at a pH of 8.0, the amount that exists in the intermediate form concentration-wise equals 0.075 molar. So that would be our amount of the intermediate form.

Remember, when we're dealing with our diprotic or dibasic above 6. Our pH is 8, which means that the majority of this formal concentration of my diprotic acid would actually exist in the basic form. The majority of it exists in that form and a portion of it exists in the intermediate form and the acidic form. What we just found was the amount that exists in the intermediate form from, again, from the initial formal concentration of 0.150 molar.

As the pH would increase, this number would decrease even more because the higher the pH gets, the more the basic form will become the predominant concentration involved in my solution. And if we were to find the concentration of my acidic form, it would be even less than this value here.

Now that you've seen this example in terms of determining the concentration, look to see if you can answer example 2 that's given down below. Once you do, come back and see if your answer matches up with mine.

Fractional Concentrations Calculations 1

Here it tells us to calculate the concentration of the acidic form for 0.230 molar tartaric acid when the pH is 6.00. We're given 2 pKa values: pKa₁ equals 3 and pKa₂ equals 4.34. Since they're telling us to calculate the concentration, we'll assume that this number here must be the full concentration of my diprotic acid. We're going to say here that the concentration of my diprotic acid equals the fraction of my diprotic acid times the formal concentration of my diprotic acid. This translates to:

H + 2 times the formal concentration H + 2 + H + times ka 1 + ka 2 times k a one times k a 2

Remember here that H+ is equal to 10PH-. So, it equals 10-6. Ka₁ equals 10-pKA₁, so it equals 10-3. And Ka₂ equals 10-pKa₂, so it equals 10-4.34. All we do now is insert our values. So H+2-6, so that's 10-12 times the formal concentration divided by H+2-3+10-7.34. When we figure out the values for top and bottom, we get 2.3×10-13 divided by 4.67098×10-8. So that gives me a fraction or concentration of 4.92×10-6 molar. So, our concentration here is incredibly small, since, with our diprotic acid, we'd have the acidic form, the intermediate form, and the basic form. Here, pH would equal pKa₁, and here pH would equal pKa₂. Here at pKa₂, it is equal to 4.34. That's when we would have approximately 50% of the basic form and fifty percent of the intermediate form. Our pH is even higher than that. It's at 6, so we'd have a majority of my solution composed of the basic form. The intermediate form will form the second most amount, and because the acidic form is at 50% around a pH of 3, we'd expect very little of it when the pH is all the way up to 6. What this shows us is that although the pH is incredibly high, we never totally get rid of all our acidic form. There'll still be small amounts of it present within our solution. They'll just be incredibly small amounts. That explains why our concentration here is times 10 to the negative 6 molar. The other 2 would be much greater in amount because the pH again is at 6 and at pH 4.34, we'd have 50% of both. Since the pH is greater than 6, the majority of our solution exists in the basic form. Remember the formulas that are associated with each one of the forms for a diprotic system. Remember what our pH is telling us. It's giving us basically the fraction distribution between the 3 different forms for my diprotic species.

Here’s what students ask on this topic:

What is the relationship between pH and pKa in determining the fractional composition of weak acids and their conjugate bases?

The relationship between pH and pKa is crucial in determining the fractional composition of weak acids and their conjugate bases. When pH equals pKa, the concentrations of the weak acid (HA) and its conjugate base (A⁻) are equal. This is derived from the Henderson-Hasselbalch equation: ${\mathrm{pH}}_{}={\mathrm{pKa}}_{}+\log \left(\frac{{\mathrm{[A⁻]}}_{}}{{\mathrm{[HA]}}_{}}\right)$. At this point, the log term becomes zero, making pH equal to pKa. As pH increases above pKa, the fraction of the conjugate base increases, and the fraction of the weak acid decreases.

How do you calculate the fractional composition of a diprotic acid system?

In a diprotic acid system, you need to consider both dissociation constants, Ka1 and Ka2. The fractional compositions of the acidic form (H₂A), intermediate form (HA⁻), and basic form (A²⁻) are calculated using specific equations. For the acidic form: ${\alpha }_{=}$. For the intermediate form: ${\alpha }_{=}$. For the basic form: ${\alpha }_{=}$. These equations help determine the fraction of each form at any given pH.

What is the significance of the Henderson-Hasselbalch equation in fractional composition calculations?

The Henderson-Hasselbalch equation is significant in fractional composition calculations because it relates pH, pKa, and the ratio of the concentrations of the conjugate base and weak acid. The equation is: ${\mathrm{pH}}_{}={\mathrm{pKa}}_{}+\log \left(\frac{{\mathrm{[A⁻]}}_{}}{{\mathrm{[HA]}}_{}}\right)$. This equation allows you to determine the pH of a solution if the concentrations of the acid and its conjugate base are known, or vice versa. It is particularly useful for buffer solutions, where the pH is maintained within a narrow range. When pH equals pKa, the concentrations of the acid and conjugate base are equal, which is a key point in understanding fractional compositions.

How do you determine the concentration of a weak acid and its conjugate base at a given pH?

To determine the concentration of a weak acid (HA) and its conjugate base (A⁻) at a given pH, you can use the fractional composition equations. For the weak acid: ${\mathrm{[HA]}}_{}=\frac{{\mathrm{[H⁺]}}_{}{F}_{}}{{\mathrm{[H⁺]}}_{}+{\mathrm{Ka}}_{}}$. For the conjugate base: ${\mathrm{[A⁻]}}_{}=\frac{{\mathrm{Ka}}_{}{F}_{}}{{\mathrm{[H⁺]}}_{}+{\mathrm{Ka}}_{}}$. Here, F is the formal concentration of the solution. These equations allow you to calculate the exact concentrations of the weak acid and its conjugate base at any given pH, helping you understand the solution's composition.

What are the key differences between monoprotic and diprotic acid systems in terms of fractional compositions?

The key differences between monoprotic and diprotic acid systems in terms of fractional compositions lie in the number of dissociation steps and the complexity of the equations. Monoprotic acids have one dissociation constant (Ka) and two forms: the acid (HA) and its conjugate base (A⁻). The fractional compositions are simpler and can be calculated using: ${\alpha }_{=}$ for the acid and ${\alpha }_{=}$ for the conjugate base. Diprotic acids have two dissociation constants (Ka1 and Ka2) and three forms: the acid (H₂A), the intermediate (HA⁻), and the base (A²⁻). The fractional compositions are more complex and require additional equations to account for the intermediate form. This makes diprotic systems more intricate to analyze compared to monoprotic systems.