7. Activity and the Systematic Treatment of Equilibrium

Activity Coefficients

7. Activity and the Systematic Treatment of Equilibrium

Activity Coefficients - Video Tutorials & Practice Problems

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Activity coefficients is factor used in describing the departure from ideal behavior for a reaction mixture.

Activity Coefficients

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concept

Activity Coefficient

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So in order to express the ionic strength 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. And remember your ionic strength is just mu here we have A. C equals um in brackets see times gamma C. A. He represents the activity of the compound. C. Represents the concentration and this gamma represents our activity coefficient. Now the activity coefficient is a way of testing missy does our solution behave ideally or not? We're gonna say here if our solution is behaving ideally. So for example if we have let's say we have A plus B. Large B gives us C. If our solution is behaving ideally that would mean that our activity coefficient equals one. So when I did the equilibrium expression for this it would become C. Remember whatever the coefficient is becomes the power to the C divided times A. To the A. Times B to the 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 they're different sized ions, different size compounds within a solution. Each of them exerts a level of influence, greater or smaller than the next ion or compound. That's when your activity coefficient does not equal one. Okay, so the activity coefficient is just used is just used to see do we have an ideal solution where all the ions are treated the same Or do we have a non ideal solution where they are not treated the same. Now we're gonna say the activity coefficient and ionic strength they can be more closely connected and related to one another with the extended Debbie huckle equation. Now here the equation is log of gamma. So your activity coefficient equals negative 0.51 times Z squared z is just the charge of your ion whether it be negative or positive times the square root of your ionic strength divided by one plus alpha. Here is just the size of the ion. Um Typically it's done in PICO meters but you can use any length really if you want. It's just that PICO meters tends to be the most basic unit used or nanometers. Times again the square root of your ionic strength divided by 305. Now here we're gonna say what is the effect of ionic strength? Ionic charge and ionic size on the activity coefficient? What we're gonna say here as your ionic strength with which is new increases That's gonna cause your activity coefficient to decrease. So there's an inverse relationship between the two. So the greater your ionic strength, the lower your activity coefficient and as the activity of coefficient it preaches approaches one which we call unity Then the ionic strength will approach zero. That's because remember when your activity coefficient is approaching one that means we're acting ideally. And that would mean that all the ions treated the same ionic strength is just looking at all the ions within a solution and if the ionic strength equal zero, 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 zero. That means that we have to take into account the different concentrations of each ion, we have to take into account the charges involved, which is why we've been using that formula from earlier. Now, as the size of an ionic charge increases, the more activity coefficient moves away from unity, so the less likely it is going to be uh next to one equal to one. That's because a larger charge has a greater impact and causes the solution to deviate from an ideal ideal presence. Then finally three, the smaller the ionic size of alpha, the greater the effects of the activity coefficient. That makes sense because the smaller ionic size gets the less of an impact and influence they can 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 is no ideal solutions, ions are different shapes, different 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 um even though everything is not the same. So now that we've talked about activity coefficients, let's look at example, one in the next video, where we just simply write out the Saudi ability 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 k S. P to that concept.

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example

Activity Coefficient

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So here it says for the following compounds state the saw liability product expression with its activity coefficient. Realize here when we say sigh ability product that's referring to K. S. P. R. C. Liability product constant with K. S. P. We're dealing with an ionic solid and we talk about how it breaks up into its ions here. This would break up into two copper three ions Plus three sulfate ions with K. S. P. We're gonna ignore the reactant because it's a solid. So K SP will just equal products. So to be the concentration of copper three ion Because the coefficient is two, it's gonna be squared but now I also have to take into account its activity coefficient. So this is gonna 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 coefficient is three. So this is cubed times the activity coefficient also cubed of sulfate ion so that there would represent the Saudi ability product expression which includes the activity coefficients. Now and remember if your activity coefficients deviate from unity which is one, then we're no longer dealing with an ideal solution. And in fact we're dealing with a real world non ideal solution. So now that you've seen that one, just try to do example to come back and see if your answer matches up with mine

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example

Activity Coefficient

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So for here we have to do the same exact process where we have to determine the soy Billy product expression from all Denham five sulfide. So here this is our solid which breaks up into molybdenum five ion Plus. And remember there's two of them plus five sulfide ions. So when we take into into account K. S. P. It's gonna be K. S. P. Equals the concentration of molybdenum squared times the co if it the co activity coefficient squared of molybdenum five ion times the concentration of sulfide ion to the fifth times the activity coefficient to the fifth for sulfide ion. So both example, one example to take into account that the activity coefficient is no longer equal to one. 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

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example

Activity Coefficient Table

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So we know that when our activity coefficient is approaching unity or one that are ionic strength will be equal to zero and therefore we have an ideal solution. But when our ionic strength is not equal to zero, we will no longer have an ideal solution. And therefore we have to determine what our new activity coefficient will be. Now we're gonna say by calculating the ionic strength of a compound, the activity coefficient can be determined by the chart given below. So here we have our chart our activity coefficients chart which is separated into charges. Here we have our plus or minus one charges, r plus or minus two charges plus or minus three and plus Or -4 charges. Here we have the basic ionic size or alpha in PICO meters for each one of these ions. And then here we have our ionic strengths. Now when we calculate an ionic strength and it happens to be one of these five values, we just have to look to see where exactly there's our ion fall at that particular ionic strength and for that particular ion. So for example, let's say that we were calculating ionic strength and we found that at equal 50.1 and we had to look up the calcium ion. So 0.1 is here And then we look for calcium which is plus two. So we'd see calcium is here and then we would just line them up here. So we'd say when the ionic strength is 0.1, the activity coefficient for calcium ion is 0.675. Again it's away from unity. So we're not dealing with an ideal solution and therefore the ions charge and size would have some effect in terms of the overall distribution of ions within the solution and the overall activity within the solution. Alright, so now that we've seen that, let's do this example down here, it says find the activity coefficient for the ion specified in the following compound. So we have to find the activity coefficient For sodium ion to be able to do that, we need to determine what the ionic strength of .005 molar of sodium chloride is. So remember, ionic strength here equals half the concentration of my first ion times its charge squared plus the concentration of my second ion times its charge squared. So here we're dealing with sodium ion and chloride ion. So we'd say ionic strength here equals half, there's only it's a 1-1 relationship. So the concentrations are the same as the overall concentration for the ionic compound. So it would be concentration of the first ion times it's charged squared plus concentration of the second ion times its charge squared. So that will give me .005 as the ionic strength for this compound. So now I have my ionic strength, I have my eye on. So I come up here, My ionic strength is .005. And then I have to look for sodium and sodium is right here and they meet here. So we'd say that the activity coefficient for sodium ion on those conditions after finding the ionic strength is .928. So that's the approach we do. I'm in answering these types of questions. We find the ionic strength. We look for the particular ionic strength on the chart and then we find where that ion is located on the chart. Now there will be instances where we find our ionic strength and it doesn't match up with any of the ionic strengths given there. In those situations, we'll have to use a different approach in order to determine what our activity coefficient will be for that particular ion. So we'll see how to approach that in a later on video. So guys just remember some of the fundamentals we learned in terms of relating ionic strength and activity coefficient together mathematically.

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example

Activity Coefficient Table Calculations 1

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So here it says find the activity coefficient for the ion specified. So here we're looking for cyanide ion in its activity coefficient. First realize that we want our concentration to be in malaria. T not just millie Mueller. So we're gonna convert our our millie moller into polarity. So we have one million Mueller times for everyone moller it's 1000 millimeter or one million Mueller equals 10 to the negative three Mohler. Doesn't matter which one you use to get the same answer of 10.1 molar. Here we have our rubidium Sinai breaks up into our B plus one & CN -1. It's a 1-1 relationship. So their concentrations equal the concentration of the entire ionic compound. We're gonna figure out what the ionic strength is. So that ionic strength equals half the concentration of my first ion times its charge squared plus the concentration of my second ion times its charge squared. When we do that, we're gonna get a concentration of .001 molar. Actually .001 has the ionic strength. Now we have to review our chart and see where exactly um with cyanide fall in terms of our activity coefficient chart. So we have 0.1 molar. And then we have to find our B. And we see our our BCN cyanide ion. So we see cyanide ion is right here And so they meet up here at .964. So .964 represents the activity coefficient for my cyanide ion. So now that you've seen this move on to example to keep in mind you want your polarity, you want polarity, not millie moller. So first do that. Then determine what the ionic strength is to figure out what the activity coefficient is for the specified ion. Once you do that, click onto the next video and see what I give as the correct answer.

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example

Activity Coefficient Table Calculations 1

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So here we have to find the activity coefficient for zirconium for ion. So what we're gonna do first is we're gonna convert the mili moller to just moller. So we have five million Mueller. This time we'll say one million moller is 10 to the -3 molar. So that gives me .005 molar. Now I have to break this ion up. It breaks up into zirconium for ion plus four nitrate ions. Here the concentration of the Zirconium four ion is .005 molar. And then this would be four times .005 moller. So the concentrations at the end are 0.5 moller and 0.0 to zero moller. Now that we have that we can calculate the ionic strength. So ionic strength equals half the concentration of the first ion which is .005 moller times its charge squared plus the concentration of the second ion times its charge squared. So when we plug that in that gives me .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 for ion when you do that, you should get 0.10 as the activity coefficient for zirconium for ion. So that would be our answer for that one. Finally, for this practice, one were asked to determine what the activity coefficient is of the hydrogen ion. So 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 approached that same problem, remember what units that we say we're most preferable, preferable for the ionic size for any ion. Keep that in mind when solving for ALPHA within this equation. And remember new here is just ionic strength and Z is the charge of that particular ion. Good luck, guys.

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Problem

Problem

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

-0.0772

C

0.837

D

1.49

E

0.671

Interpolation: Non-ideal Ionic Strength

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Non-Ideal Ionic Strength

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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 specific ion. Now there's gonna come times when we're gonna find an ionic strength and it won't be found on that chart. In those cases we're gonna use interpolation in order to find our missing activity coefficient. And when it comes to interpretation we're gonna say the basic setup is our unknown activity coefficient interval divided by the change in our activity coefficient equals are 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.

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example

Non-Ideal Ionic Strength

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So here for this example we're gonna rely on the interpolation method in order to figure out what the activity coefficient of barium ion is. When the ionic strength is 0.75. So remember interpolation is unknown activity coefficient interval divided by the change and activity coefficient equals the known ionic strength interval divided by the change in the ionic strength. So how exactly do we set this up? Well, first of all here are our ionic strengths, these five values here. And if we look we're gonna say that this number of .075, where exactly does it fall in terms of these ionic strengths? Well it falls in between these two particular ionic strengths, that's where .075 resides. Now that we know that will set up basically two tables on this table here we'll talk about the different activity coefficients involved. We'll start out with um the largest activity coefficient um proportionately really doesn't matter. But we'll start with the um the largest activity coefficient which is this .465. Now we know that there's an activity coefficient in here. Between these two numbers, we just don't know what it is. So that's gonna be our x or unknown. And then our smallest activity coefficient is .38 On this side here we have our ionic strengths associated with this activity coefficient of .465. Is this ionic strength here and then associated with this unknown activity coefficient is this ionic strength given to us. And then finally this activity coefficient was associated with this number now that we have that we're going to set up our unknown activity coefficient interval. So that is the smallest activity coefficient minus X divided by the change in my activity coefficient, which is smallest minus the largest here. So that's what we got on this side. On the other side, we do the same exact thing. We're gonna have this value here first minus this one and then this bottom value here minus this one. So we're repeating the same basic setup now for this breakdown, we have all the values we want here So we can break that down to one no and here we have numbers here so we can break that down to one number. So simplifying this, we're gonna have .38 -1 Divided by negative .085 equals .5. So we have that there. Now we're gonna have to isolate X. So we're gonna multiply both sides now by negative .085. So that's gonna give me a 0.38 minus X equals multiply these two numbers together gives me negative 20.4 to 5. Subtract .38 from both sides. So negative x equals negative .4225. Drop the negative sign. So X. Which is my unknown activity coefficient is .4225 which is a reasonable value because it's in between these known activity coefficients. So that's how we approach interpolation. When we don't have the use of the extended Debbie Hucles equation, we can rely on interpretation in order to figure out what are unknown activity coefficient is for any desired ion. Now that you've seen how to basically set this up and following the steps, attempt to do the practice question left here on the bottom. Don't worry, just come back and see how I approach that same question. So, good luck, guys.

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Problem

Problem

Find the activity coefficient from the given ionic strength, µ, for the following ion.