 In an earlier video we looked at molar conductivity represented by this lambda m and we saw how it varies with concentration. So in this video what we're going to do is we want to plot this variation on a graph and try to graphically understand how molar conductivity varies with concentration. So before we go to graph this let me give you a quick recap of how these two are related. So the first thing is when we talk of concentration we know that decreasing the concentration is the same as increasing the dilution and when we think of molar conductivity you can think of it as basically the conductivity of one mole of an electrolyte. So how molar conductivity varies with concentration or dilution depends on the type of the electrolyte as well and we've seen before that an electrolyte can be a strong electrolyte or a weak electrolyte depending on how they dissociate. So if you have some electrolyte HA which is dissociating into some H plus and some A negative. Now if this is a weak electrolyte the extent of this dissociation will be very less and if it's a strong electrolyte this dissociation will be 100%. So what we saw before was that in case of a strong electrolyte the dissociation is 100% and when you increase the dilution or when you decrease the concentration the molar conductivity increases and for weak electrolytes where the dissociation is not 100% even then when you increase the dilution the molar conductivity increases but in the case of weak electrolytes on dilution the increase of molar conductivity is much higher because what happens during dilution is that on dilution of a weak electrolyte the degree of dissociation which is this alpha increases. How do we know that? Let's take this example. So if HA is a weak acid let's write down the equilibrium constant for this reaction it will be given as this concentration of H plus into concentration of A minus divided by the concentration of HA. Now let's say you have reduced the concentration by a factor of 10 or you could say that you have increased the dilution by a factor of 10. So we have seen when we deal with equilibrium constants that if the reaction conditions change we can calculate the reaction quotient or Q and in this case I am saying that the concentration has reduced by a factor of 10 for all three. So now this will be H plus divided by 10 times A negative divided by 10 divided by HA divided by 10. I can cancel off these tens and if I rearrange this I can write this Q as 1 by 10 times concentration of H plus times concentration of A negative divided by concentration of HA. So if you look at this the Q or the reaction quotient is one tenth the equilibrium constant or the Q is less than K and we know that when the reaction quotient is less than the equilibrium constant the reaction shifts to the right and that means we would have more of H plus and A negative forming which means that the dissociation of this weak electrolyte HA will increase on dilution and again this will not happen with a strong electrolyte because the definition of a strong electrolyte is that it is already dissociated completely but in case of weak electrolyte which does not dissociate fully on dilution the dissociation increases. So when the dissociation increases the number of ions also increase and we know that when we talk of molar conductivity it is the ions that are actually conducting the electricity in the solution. So because now there are more ions in the solution the molar conductivity increases much more than in the case of a strong electrolyte and so the increase in molar conductivity is much higher in case of a weak electrolyte but the question is how will this change or how will this variation look graphically. Let's see how we can plot this. So I want to plot the variation of molar conductivity with concentration and again we will take two cases one for a strong electrolyte and one for a weak electrolyte. So on the y-axis I have the molar conductivity here which is this lambda m and on the x-axis I have this under root C which is the square root of the molar concentration and one reason why we are not directly plotting this against C is that when we look at the experimental data it is easier to fit the data points into a straight line as you will see in a second. So first let's take the case of a strong electrolyte. So for a strong electrolyte if we calculate the molar conductivity at different values of concentration and if we plot this on this graph we see a trend that looks like this. So these blue dots are data points for a chosen set of concentrations so to see the trend if you try to fit these points you will see that they will fall on a straight line. So let's check this trend with what we know from before. So we know that in case of a strong electrolyte when we increase the dilution the molar conductivity increases and we know that increasing the dilution is the same as decreasing the concentration. So if we look at this plot as we go from right to left the concentration is decreasing because this is the origin and this is the positive x-axis. So as we go to the left the concentration is decreasing or the dilution is increasing and accordingly we can see that the points along the y-axis increase. So from the plot also we can see that when the dilution increases or the concentration decreases the corresponding molar conductivity increases. Now this was the case for a strong electrolyte. For a weak electrolyte the trend is again the same that is on increasing the dilution the number of ions increase and therefore the molar conductivity increases. So here also if we plot the experimental data we see that they follow this hyperbolic type curve and here also as we move from right to left the concentration is decreasing which means the dilution is increasing and the corresponding lambda m points which are these marked with the pink dot here increase as we go from right to left. So even for weak electrolytes the effect of dilution that we saw before is reflected in this trend and there is one more thing to note here. We discussed how in the case of weak electrolyte since the number of ions increase the increase in molar conductivity is way higher and you can see that here in the case of a strong electrolyte when we go from point say 1 to 2 and 3 and 4 the increase in molar conductivity is very gradual but at the same time if we look at the curve for the weak electrolyte as we go from these points from 1 to 4 you can see that the change in these values is much higher which is the same as we saw before. So this is what a graphical representation of the trends look like. Now there is some more information that we can get from these curves. Now since this is on a straight line if we just extrapolate this towards the y-axis let's say it is intersecting the y-axis at this point. So what would this mean? So if we look at the equation that we use for molar conductivity it was given as kappa divided by c where c is the concentration. So as we go towards the origin the value of c is decreasing. So this value of lambda m is the value that we get when c goes almost to zero. So if c will tend to go to a very small value which is next to zero the corresponding value of lambda m will be called lambda m infinity which is basically the molar conductivity at infinite dilution and if you are wondering where this infinite dilution is coming from. So we saw before how when we decrease the concentration the dilution increases. So when the concentration is reduced to a very very small number the corresponding dilution will be very large which is why we are calling this value the molar conductivity at infinite dilution. So now based on this information I want to write the equation of this straight line. I know that it will be in the form of y is equal to mx plus c and m is the slope and I can see here that the slope is negative. So let's say this slope is some value which is minus b and the c is the intercept which is basically this length and we define this length or this point to be the molar conductivity at infinite dilution. So the value of c here is this lambda m infinity and we already know the y and the x which are this lambda m and this under root c. So based on this we can write the equation of this straight line as lambda m is minus b under root of c plus lambda m infinity or we can rearrange this to write lambda m is lambda m infinity minus b under root c. So this is the equation of this straight line and the important point to note is that this straight line trend is only for a strong electrolyte. So just like we did for the strong electrolyte let's now try to find the equation for this line which is that of a weak electrolyte and we want to find an equation that represents this. Now sometimes you will not find this equation derived in introductory textbooks but it's an easy fun little derivation and you can already see that this looks like a hyperbola. So we know that the final equation could be that of a hyperbola between this y that is molar conductivity and this x which is square root of concentration. Let's start by writing down the dissociation of a weak electrolyte. So I have this electrolyte hb which is a weak electrolyte which dissociates to give this h plus and this b negative and we know that this is a weak electrolyte. So we know that if we started with 1 mole of hb and at the initial time we had no moles of h plus and no moles of b negative ion at equilibrium if we assume that the dissociation is alpha we will have 1 minus alpha moles of hb alpha moles of h plus and alpha moles of b negative or if we write the entire thing in terms of concentration if we start with the concentration c at time t is equal to 0 at equilibrium the concentration of hb will be 1 minus alpha times c and that of h plus will be alpha c and that of b negative will be alpha c. So if we write down the equilibrium constant for this reaction at equilibrium we have the k which is equal to alpha c times alpha c divided by 1 minus alpha times c we can cancel out the c and so we have alpha square c upon 1 minus alpha. Now at this point we are going to plug in the value of alpha and for that we are going to use this definition of alpha which is given by alpha is equal to the molar conductivity at some concentration divided by the molar conductivity at infinite dilution and to give you some intuition about this definition alpha is the degree of dissociation and when the dissociation takes place we get these ions. So you can think of the degree of dissociation at any concentration c as the number of ions at that concentration divided by the maximum number of ions possible and because the molar conductivity is proportional to the number of ions we can relate the degree of dissociation with the molar conductivities like this. So now if we take this definition of alpha and plug it into this relation we can rearrange this to write it like this and then by plugging in alpha we get a relation that looks like this. Now if we rearrange these terms we can write the equation like this. You can solve this and try it for yourself by rearranging this equation that you get this form but how do we know that this equation corresponds to this hyperbola? So to check that let us plot this equation. So we know that we have this molar conductivity on the y-axis and this square root of c on the x-axis. So we can rewrite this equation as 1 over y is equal to some constant c1 plus lambda m which is y times c which is x squared. So our x is under root of c so x squared will be c. So we can write this as x squared divided by some other constant c2 and we are assuming that we know the values of this molar conductivity at infinite dilution and the equilibrium constant which is why we are taking these values to be constants. So now all that is left is to plot this equation and to check whether we get a hyperbola. So I have used this online graphing calculator and I have plotted the equation here and we can see that for some selected values of p and q which were our two constants we do get something like a hyperbola and since both of our x and y cannot be negative we can only focus on the first quadrant and we can see that the plot will look something like this which is of course a hyperbola and so we know that this equation represents this curve which is the variation of molar conductivity with concentration for a weak electrolyte.