Activity Coefficients: Videos & Practice Problems

To determine the activity of ions in a solution, the activity coefficient (γ) is used, calculated as $A=\left(C\right)\left(\gamma \right)$. The extended Debye-Hückel equation relates γ to ionic strength (I) and ion size (α). As ionic strength increases, γ decreases, indicating non-ideal behavior. Larger charges and smaller sizes further deviate from ideality. Understanding these relationships is crucial for accurately predicting solubility and reaction dynamics in ionic solutions.

Activity Coefficient

In order to express the ionic strength based on the concentrations of species, we calculate its activity with the use of an activity coefficient. Now, the activity coefficient uses the units of gamma. Remember, your ionic strength is given by ac=[c∗γ_c]. A here represents the activity of the compound, c represents the concentration, and this gamma represents our activity coefficient. The activity coefficient is a way of testing to see if our solution behaves ideally or not.

We're going to say here, if our solution is behaving ideally, for example, if we have A+B→C. If our solution is behaving ideally, that would mean that our activity coefficient equals 1. When I performed the equilibrium expression for this, it would become c, with whatever the coefficient, it becomes the power to the c divided, times a^a times b^b. That's if our solution is behaving ideally.

Under ideal conditions, it just means that all the species dissolved within my solution all behave in the same way and have the same level of effect. It ignores differences in size and charge. But in actuality, there are differently sized ions and compounds within a solution, each of which exerts a level of influence greater or smaller than the next ion or compound. That's when your activity coefficient does not equal 1.

The activity coefficient is used to see whether we have an ideal solution where all the ions are treated the same, or a non-ideal solution where they are not treated the same. The activity coefficient and ionic strength can be more closely connected and related to one another with the extended Debye-Hückel equation. This equation is log(γ)=−0.51∗z2∗ionic strength1+α, where your activity coefficient equals negative 0.51 times z² times the square root of your ionic strength divided by 1 plus alpha, with alpha representing the size of the ion, typically done in picometers, but other length units can be used.

We're going to say, as your ionic strength, which is μ, increases, that's going to cause your activity coefficient to decrease. There's an inverse relationship between the two. The greater your ionic strength, the lower your activity coefficient. As the activity coefficient approaches 1, which we call unity, then the ionic strength will approach 0. That's because, remember, when your activity coefficient is approaching 1, that means we're acting ideally. That would mean that all the ions are treated the same. Ionic strength is just looking at all the ions within a solution. If the ionic strength equals 0, that means that there is no way of differentiating the ions from one another. They all have the same level of influence. When your ionic strength is different from 0, that means that we have to take into account the different concentrations of each ion and the charges involved, which is why we've been using that formula from earlier. As the size of an ionic charge increases, the more the activity coefficient moves away from unity. So, the less likely it is going to be next to 1, equal to 1. That's because a larger charge has a greater impact and causes the solution to deviate from an ideal presence. Then finally, the smaller the ionic size or alpha, the greater the effects of the activity coefficient. That makes sense because the smaller your ionic size gets, the less of an impact and influence they have in terms of differentiating themselves within a given solution. So just realize that this is really just theoretical in terms of how ions interact with each other.

In reality, there are no ideal solutions. Ions are of different shapes and charges. Therefore, they have different influences on one another within a solution. The activity coefficient is just a way of talking about this deviation from the ideal solution where everything is treated the same, even though everything is not the same. So now that we've talked about activity coefficients, let's look at example 1 in the next video where we just simply write out the solubility product expression for the following compound.

So guys, this takes into account what we've learned thus far in terms of writing down the activity expression for a compound and relating KSP to that concept.

Activity Coefficient

So here it says for the following compounds, state the solubility product expression with its activity coefficient. Realize here when we say solubility product, that's referring to Ksp, our solubility product constant. With Ksp, we're dealing with an ionic solid and we talk about how it breaks up into its ions. Here, this would break up into 2 copper(III) ions plus 3 sulfate ions. With Ksp, we're going to ignore the reactant because it's a solid.

Ksp will just equal products. It'll be the concentration of copper(III) ion. Because the coefficient is 2, it's going to be squared. But now I also have to take into account its activity coefficient. This is going to be multiplied by the activity coefficient and because the concentration is squared because of the coefficient, this is also squared, times the concentration of sulfate ion.

K s p = [ Cu 3+ ] 2 γ × [ SO 4 2- ] 3 γ

Coefficient is 3, so this is cubed times the activity coefficient, the activity coefficient, also cubed of sulfate ion. That then would represent the solubility product expression which includes the activity coefficients now. Remember, if your activity coefficients deviate from unity which is 1, then we're no longer dealing with an ideal solution and in fact we're dealing with a real world non-ideal solution. Now that you've seen that one, just try to do example 2, come back and see if your answer matches up with mine.

Activity Coefficient

For here, we have to do the same exact process where we have to determine the solubility product expression for molybdenum 5 sulfide. Here, this is our solid which breaks up into molybdenum 5 ion plus, and remember there are 2 of them, plus 5 sulfide ions. When we take into account K_sp, it's going to be K_sp equals the concentration of molybdenum squared times the activity coefficient squared of molybdenum 5 ion times the concentration of sulfide ion to the 5^th times the activity coefficient to the 5^th for sulfide ion. Both example 1 and example 2 take into account that the activity coefficient is no longer equal to 1. Therefore, we're dealing with non-ideal solutions where the ionic strength will have some impact on the interactions of ions within my overall solution.

Now that we've seen how activity coefficients are used, we're now going to combine this idea with calculations dealing with ionic strength as well. So click on to the next video and see how we relate mathematically the idea of ionic strength and activity coefficients.

Activity Coefficient Table

Hey everyone. So here we're going to say that by calculating the ionic strength of a compound, the activity coefficient can be determined by the chart given below. So below in this chart, we have a grouping of ions, those that have charges that are plus or minus 1, plus or minus 2, plus or minus 3, as well as plus or minus 4. And besides that, we have ionic strengths that are connected to them. So, we have ionic strengths ranging from 0.001 all the way to 0.1, and with them, we can talk about, let's say, a lithium ion.

We can say that the lithium ion has an ionic size, which is represented by alpha in picometers and is 600 picometers. And these are the different ionic strengths involved, and then these values here represent the different activity coefficients associated with the lithium ion. So these are all your activity coefficients. Now, if we look at this chart, we're going to utilize this chart in order to solve this example question below. So we're going to say here in this example question below, it says, "find the activity coefficient for the ion specified in the following compound."

So here we need to find the activity coefficient of a sodium ion that's in 0.005 molar of sodium chloride. Alright. So the activity coefficient is represented by this formula, and we need it for the sodium ion. And we're going to say here our ionic strength equals half, and in brackets, we're going to have our first concentration, times the charge of the ion squared, plus the concentration of the second ion, times its charge squared. If we take a look here, we're dealing with sodium chloride which is composed of sodium ion which is plus 1, and the chlorine which is minus one.

What's important to understand here is that the concentration of sodium chloride is 0.005 molar, and since everything is a one-to-one ratio, that means that these ions are also 0.005 molar. With this information, we plug it into our equation for ionic strength. So, the concentration of my first ion, in this case, sodium, is 0.005, its charge is plus 1 and that's squared, plus the concentration of my second ion, which is the chlorine ion, which is also 0.005, times minus one squared. Plug that into your calculator; you'll get 0.005 as your ionic strength. So we have the ionic strength of the sodium ion, so we go up here to our chart.

So where is our sodium ion? If we look carefully, we see the sodium ion here, we found our ionic streak to be 0.005, which is right here, and we see where the two meet. So if we look, they both meet here at 0.928. So here we'd say that our activity coefficient of sodium ion equals 0.928. So that's how we'd utilize this chart of ionic strengths and activity coefficients for a plethora of different types of ions.

We first have to calculate the ionic strength and use it to find the value on the chart. Later on, we'll see that sometimes we'll find an ionic strength that doesn't quite match up with what we see here on the chart. We'll have to come up with a new way to figure out our activity coefficient. But for now, we've got an ionic strength of 0.005, which is on the chart, so look for sodium, line it up with the correct ionic strength, and you'll find your activity coefficient.

Activity Coefficient Table Calculations 1

Hey everyone. So here it says, Find the activity coefficient for the ion specified in the following compound. So here we need to find the activity coefficient for the cyanide ion in 1 millimolar of rubidium cyanide. Alright, so first of all, we're going to say rubidium cyanide breaks up into the rubidium ion. Since it's in Group 1a, it's plus 1 and cyanide is a polyatomic ion so it's minus 1.

Next, remember when looking at our activity coefficients table we were looking at concentrations connected to ionic strength in terms of molarity not millimolarity. So we're going to have to convert this, so we have 1 millimolar. We're going to say for every 1 milli it's 10-3. So this is 0.001 molar. Knowing this information we can now figure out our ionic strength.

Remember, your ionic strength equals half, and it's concentration 1 times charge 1, the charge will be squared, plus the concentration 2 of our second ion times its charge also squared. Here, we're going to say that rubidium cyanide has a concentration of 0.001 molar, and since the ions all break up in a one to one relationship, so it's all 1 to 1 to 1, these are also 0.001 molar. So with this information we're going to plug it in and figure out what our ionic strength is. We're going to say this equals half the concentration of rubidium ion is 0.001 times its charge squared plus the concentration of the cyanide ion, which is 0.001 also, times its charge squared and that comes out to be 0.001. We have our ionic strength, so take a look at your activity coefficients table.

Look for the cyanide ion. If you look carefully enough you'll find that its activity coefficient, when its ionic strength is 0.001 equals 0.964. So 0.964 would be our final answer for the activity coefficient of your cyanide ion.

Activity Coefficient Table Calculations 1

Here we have to find the activity coefficient for zirconium 4 ion. What we're going to do first is we're going to convert the millimolar to just molar. We have 5 millimolar. This time we'll say 1 millimolar is 10-3 molar. So that gives me 0.005 molar.

Now I have to break this ion up. It breaks up into zirconium 4 ion plus 4 nitrate ions. Here, the concentration of the zirconium 4 ion is 0.005 molar, and then this would be 4 times 0.005 molar. So the concentrations at the end are 0.005 molar and 0.020 molar. Now that we have that, we can calculate the ionic strength.

So ionic strength equals half the concentration of the first ion, which is 0.005 molar, times its charge squared, plus the concentration of the second ion, times its charge squared. When we plug that in, that gives me 0.05 as my ionic strength. Now that we have that, look on your activity coefficients table. Line up the activity, the strength, ionic strength with the activity coefficient for zirconium 4 ion. When you do that, you should get 0.10 as the activity coefficient for zirconium 4 ion.

That would be our answer for that one. Finally, for this practice one, we're asked to determine what the activity coefficient is of the hydrogen ion. Utilize this equation in order to determine what the activity coefficient is. Once you've attempted it and even if you didn't, click on to the next video and see how I approach that same problem. Remember, what units did we say were most preferable for the ionic size for any ion?

Keep that in mind when solving for alpha within this equation. And remember, mu here is just ionic strength, and z is the charge of that particular ion. Good luck, guys.

Calculate the activity coefficient of H⁺ using the extended Debye-Huckel equation for a solution comprised of H⁺ and I ^–. Given that H⁺ has a size of 9.00 x 10^-10 m and the molar concentration of the solution is 0.075. 
log γ = − 0.51 z ² √µ / 1 + (α √µ / 305)

• A. 0.917
• B. 0.982
• C. 0.837
• D. 1.49

Answer: (B)

Non-Ideal Ionic Strength

Up to this point, we've had to calculate our ionic strength, which would result in one of five numbers. From those numbers, we then look on the activity coefficient table to find the desired activity coefficient for a specific ion. Now, there are going to be times when we find an ionic strength and it won't be found on that chart. In those cases, we're going to use interpolation to find our missing activity coefficient. When it comes to interpolation, we're going to say the basic setup is our unknown activity coefficient interval divided by the change in our activity coefficient equals our known ionic strength interval divided by our change in ionic strength. We'll see how best to use this ratio in order to find our missing activity coefficient.

So click on to the next video and see how we approach the example right below.

Non-Ideal Ionic Strength

Here we're told to find the activity coefficient from the given ionic strength for the following ion. Here we have a barium ion, and the ionic strength equals 0.075. So, if we look here, here's our barium ion, and our ionic strength is 0.075. So, here if we look at the table given before us, we'd say that it would fit somewhere within here. Right?

So this is the missing, well, it'd be somewhere in here. This is our missing activity coefficient. Right? Because 0.075 will be in here. So we have to use interpolation in order to determine what our activity coefficient will be when this ionic strength is given to us.

So remember, the way we're going to set it up, we're going to have two tables: one, we're going to have our activity coefficient, and here we're going to have our ionic strength. So here, we're going to say that our ionic strengths are 0.050. 075 is in between 0.1 and 0. So when it's 0.050, its activity coefficient is 0.465. We don't know what this new activity coefficient will be, so this is x, and then when it's 0.100, the activity coefficient is 0.380.

Alright. So we have to figure out what that x variable is. So, the way we're going to set it up, based on what we've seen in our previous video in terms of the formula, we're going to say we have 0.380 and then we're going to say, minus x divided by 0.380 - 0.465 equals. Alright, so now for this one, we're going to say it's going to be 0.100 - 0.075 divided by 0.100 - 0.050. So here, we just have to solve for the missing variable. We have all the numbers on this side, which means we can reduce it down to just a single value, so that's going to come out to be 0.05, which still equals 0.380 x.

Here we can subtract these two and have an actual number, so when we do that, we're going to get negative 0.085. We're going to cross-multiply these two here together. So when I do that, I'm going to get 0.380 - x equals -0.0425. We're going to subtract 0.380 from both sides. So when I do that, I'm going to bring the work over here, I'm going to get negative x equals -0.4225, divide both sides by -one, x equals 0.4225. So this will represent the activity coefficient for my barium ion when the ionic strength is 0.075.

So that would be our final answer.

Find the activity coefficient from the given ionic strength, µ, for the following ion.
F – when µ = 0.0080

• A. 0.8896
• B. 1.378
• C. 0.1040
• D. 0.9104

Answer: (D)

Here’s what students ask on this topic:

What is the activity coefficient and how is it used in chemistry?

The activity coefficient (γ) is a factor used in chemistry to account for deviations from ideal behavior in ionic solutions. It adjusts the concentration of ions to reflect their effective concentration or 'activity'. The activity (A) of an ion is given by the product of its concentration (C) and its activity coefficient (γ), expressed as A = Cγ. This coefficient helps in understanding and predicting the behavior of ions in non-ideal solutions, where interactions between ions affect their activity. The extended Debye-Hückel equation is often used to calculate γ, considering factors like ionic strength and ion size.

How does ionic strength affect the activity coefficient?

Ionic strength (I) has an inverse relationship with the activity coefficient (γ). As ionic strength increases, the activity coefficient decreases, indicating more significant deviations from ideal behavior. This relationship is described by the extended Debye-Hückel equation. When the ionic strength is high, ions interact more strongly with each other, reducing their effective concentration or activity. Conversely, when the ionic strength approaches zero, the activity coefficient approaches one, indicating ideal behavior where ions do not significantly interact with each other.

What is the extended Debye-Hückel equation and how is it used?

The extended Debye-Hückel equation is used to calculate the activity coefficient (γ) of ions in a solution. It is given by:

$\log \left(\gamma \right)=-0.51\times (z^2\times \sqrt{I})/(1+\alpha \times \sqrt{I}/305)$

Here, z is the charge of the ion, I is the ionic strength, and α is the ion size. This equation helps in understanding how ionic strength and ion size affect the activity coefficient, which in turn affects the behavior of ions in a solution.

Why is the activity coefficient important in predicting solubility and reaction dynamics?

The activity coefficient (γ) is crucial for predicting solubility and reaction dynamics because it accounts for non-ideal behavior in ionic solutions. In real solutions, ions interact with each other, affecting their effective concentration or activity. By using the activity coefficient, chemists can more accurately determine the solubility of compounds and the equilibrium constants of reactions. This leads to better predictions of how reactions will proceed in various conditions, which is essential for applications in fields like pharmaceuticals, environmental science, and chemical engineering.

How do you interpolate to find a missing activity coefficient?

Interpolation is used to estimate a missing activity coefficient (γ) when the ionic strength (I) is not listed in a table. The basic setup involves using the known intervals of γ and I. The formula is:

$\frac{\mathrm{\Delta \gamma }}{\mathrm{\Delta I}}=\frac{\mathrm{\Delta \gamma }}{}$_knownΔI_known

Here, Δγ is the change in the activity coefficient, and ΔI is the change in ionic strength. By setting up this ratio, you can solve for the unknown γ. This method ensures that you can accurately estimate the activity coefficient even when direct data is unavailable.