 In this video, we'll be looking at how we can use Kohlrausch's law to calculate the limiting molar conductivity of weak electrolytes. Now this plot here is a variation of molar conductivity with concentration and we have seen these curves before. The green one is the variation of molar conductivity for a weak electrolyte and this white line is the variation of molar conductivity for a strong electrolyte. And in one of the earlier videos, we have discussed how to find the limiting molar conductivity or the molar conductivity at infinite dilution for a strong electrolyte. If you don't remember, I'll just quickly remind you. So what we did was we extrapolated this curve to the point where it cut the y-axis. So here if I extend this line, it intersects the y-axis at this point and this value will be the limiting molar conductivity or the molar conductivity at infinite dilution for a strong electrolyte. So here it was only a matter of extending this line and seeing where it intersects the y-axis. But can we carry out the same exercise for the weak electrolyte? So if we look at this green line, as we go to smaller and smaller concentrations like say at this point, a curve approaches the axis but we don't know where it will intersect the axis. Like if we extend the axis as well, we can see that the green curve is getting closer and closer to the axis but we cannot say for sure where it will intersect the axis. So in such a case, when we want to find out the limiting molar conductivity of the weak electrolyte, we cannot do that by just extrapolating this curve. And in such situations, we can use Kohl-Rosch's law of independent migration to determine the limiting molar conductivity of a weak electrolyte. And one more thing. Now because of my bad drawing skills, it may look like this green line will intersect this white line but we know that that will not happen because earlier we had derived the equation for this curve which was a hyperbola and we know that this hyperbola will be an asymptote to this line. So it will never intersect this line but it will keep getting closer to this line as we go to smaller concentration values. And the other thing is, let's say you really want to find out where this is going to intersect and to do that, you measure the conductivity at smaller and smaller values of concentration. So when you are at these very small values of concentration, you run into experimental limitations. Like for example, we know that the least count of a regular scale or ruler is 1 mm. So we cannot measure any length which is less than 1 mm using that scale. So just like that, when we try to measure the concentrations at very small values and the corresponding conductivities which are very large, we could hit these experimental constraints. And again, one way to get around this would be to use Kohl-Rosch's law of independent migration. So let's see what that is. Now when Kohl-Rosch was looking at the data for molar conductivities of different electrolytes, a pattern was observed. So let's say if you have two electrolytes, one in the form of NaX and one in the form of KX, where X can be any anion. So basically we have the same anion, but different cations for both of these electrolytes. Now for these, if we look at the data for the limiting molar conductivities of these electrolytes for different choices of X, we see a pattern. So let's say first we take the case where X is Cl negative and we calculate this value, that is the limiting molar conductivity of NaCl minus the limiting molar conductivity of KCl or the difference between these two values when X is chloride ion. So this is the first case. Now similarly, we calculate this difference again, but this time let's say we take this X to be the bromide ion or Br negative. And again, we calculate the limiting molar conductivity for NaBr minus the limiting molar conductivity for KBr. And let's say we take one more case, which is the similar exercise carried out for an iodide ion. So we have the limiting molar conductivity of NaI minus the limiting molar conductivity of Ki. Now what we see is in all these cases, these values come out to be approximately equal. What this could mean is that the value of this expression seems to be independent of the choice of the anion because this difference turns out to be approximately equal irrespective of which anion we choose. And a similar trend is also observed when we take the same cation, but we use different anions. So based on these observations, we can make some calculations. Let's see what those are. So if we go back to the examples that we used because we noticed that the difference seemed to be independent of the choice of anion or the cation. Let's take a simple assumption. Let's say that the limiting molar conductivity of NaX will be equal to the sum of the molar conductivity of the individual ions. So in this case, the limiting molar conductivity of NaX will be equal to the limiting molar conductivity of the Na plus ion plus the limiting molar conductivity of the X negative anion. And we can write a similar expression for this KX as well. So these lowercase lambdas are the limiting molar conductivities of the individual ions. And now if we look at the difference between the limiting molar conductivity of NaX and that of KX, we can see that this value is common to both of them. So if we subtract this value from this value, the limiting molar conductivity of the anion gets cancelled out. So we can write lambda naught m of NaX minus lambda naught m of KX will be equal to this value minus this value. So we have this expression and we can see that the lambda naught X negative gets cancelled here. And so this difference is equal to the difference of the molar conductivities of the individual ions. And this explains why we got similar values for different choices of X. So let's say if X is chlorine, we know that the limiting molar conductivity of NaCl will be equal to the sum of the limiting molar conductivities of the Na plus ion and that of the Cl minus ion. But what if we were looking at something like MgCl2? In that case, how can we write this expression? So it was observed that in such cases, we can write it as lambda naught for the Mg2 plus ion and two times the lambda naught for chlorine negative ion. So we have multiplied the limiting molar conductivity of this ion by the number of such ions when MgCl2 dissociates. So based on this, we can write a generalized expression for the limiting molar conductivity of any electrolyte as equal to the sum of the limiting molar conductivity of the individual ions multiplied by the number of cations or the number of anions. So this is the generalized expression of Kohlrausch's law of independent migration. Now let's see how we can use this expression to find out the limiting molar conductivity of a weak electrolyte. Let's say we want to find the limiting molar conductivity of CH3COOH and we are given these three values which are the limiting molar conductivity values of CH3COONA, HCl and NACL. And if you notice CH3COOH is a weak electrolyte and all these three are relatively strong electrolytes. So we can calculate this value by applying Kohlrausch's law of independent migration on this data which is given. So if you want you can pause the video and give it a try before we continue. Okay. So let's see how we can find out this value. Now the first thing is that since we are given these three values let's apply Kohlrausch's law for CH3COONA, HCl and NACL which will look something like this. So we know that the limiting molar conductivity of CH3COONA will be the sum of the individual limiting molar conductivities of CH3COO negative and NA plus and similarly for NACL it will be the sum of the limiting molar conductivities of NA plus and CL minus. And similarly for HCl it will be the sum of the limiting molar conductivities of H plus and CL negative. And I have labeled these equations as 1, 2 and 3. So now looking at this let's say we subtract equation 2 from equation 1. So in this case we get 1 minus 2 and this lambda naught NA plus is common to both of them. So this will get subtracted out and we get lambda naught CH3COO negative minus lambda naught of CL negative. Now what if we add this value to equation number 3 here? So if we do that we have this 1 minus 2 added to equation number 3 and again here we have this positive value and this negative value which will get cancelled out. So when we add equation 3 to this equation on the right hand side we only have lambda naught CH3COO negative plus lambda naught H plus and if we look at this value we just now saw how from Kohlrausch's law this value that is the addition of the individual molar conductivities of these ions will be equal to the molar conductivity of CH3COH and so because we know the values of equation 1, 2 and 3 that is these values the left hand side values we can simply substitute them and based on these values we can find out the limiting molar conductivity of CH3COH which is a v-character light.