 In this video, we are going to explore what we mean by the average rate and the instantaneous rate of a chemical reaction and we will see how we can calculate these values for a given reaction. Let's say you are driving a car and you go from point A to point B, which is like a hundred kilometers apart, let's say you do that in two hours. So how do you calculate how fast you are moving? So you can say that the speed of your car is the distance that you cover that is 100 kilometers in two hours. So this is going to be 50 kilometers per hour. But where are you moving with this speed throughout your journey? Of course not. When you drive your car, sometimes you drive fast while at other times you drive slow and finally you reach your destination. So this speed that we have calculated is kind of the average of all the speeds that we are moving. So this is called the average speed. Now if you wanted to check your speeds at any particular instant, all you need to do is look at the speedometer. So out here say you are moving at 80 kilometers per hour, sometime later maybe at 40 kilometers per hour. So these speeds that are shown in your speedometer, these are your speeds at that instant. So these are called your instantaneous speed. Similarly, in a chemical reaction the conversion of the reactants into the products do not happen at the same speed but it keeps changing continuously throughout the course of the reaction. Typically, when the concentration of the reactants is high, the rate of conversion into products is also high. But as the concentration of the reactants keeps decreasing, the rate of conversion into the products also keeps decreasing. This is because a typical chemical reaction happens because of collisions between the reactant molecules. When the reactant molecules collide against each other, some of the old bonds get broken down and some new bonds can get formed and this leads to the formation of new products. So initially when the concentration of the reactant molecules is high, wherever they move around they will always face other reactant molecules. So they are bound to collide and form the products. However, as the reaction proceeds, the concentration of the reactant molecules decreases and so the probability of one reactant molecule colliding with another reactant molecule decreases. So it takes longer time for successful collisions to happen and therefore a longer time for the products to be formed. So for a typical chemical reaction, if we check the concentration of the reactant with respect to the time of the reaction, then we will get a graph that looks like this. Initially there is a sharp decrease in the concentration of the reactants, but as time progresses, the change in the concentration of the reactants also slows down. This is because initially the rate of conversion of the reactants to the products was fast, but with time this rate slows down and so the concentration of the reactants does not change by much. So now that we have established that a chemical reaction proceeds at different speeds, let us see if we can figure out the average and instantaneous rates for this particular reaction. So how do we calculate the rate of a reaction? Well, out here we can think of the rate of the reaction in terms of rate of disappearance of air and the rate of disappearance of air as we have seen in the previous videos will be equal to the change in concentration of air by the change in time. We put a negative sign when it comes to the rate of reactants as during the course of a chemical reaction, the concentration of the reactant will always decrease. So the change in the concentration of the reactant will be a negative quantity, but since rate of disappearance of the reactants is reported as a positive value, so to make it positive we multiply it with a negative sign. So coming back to this reaction, initially at time t equal to zero, the concentration of air was 10 molar, so at t equal to zero the concentration of air was 10 molar and it dropped down to almost zero in 50 minutes, so at time t equal to 50 the concentration of air is almost zero molar. So the rate of disappearance of air out here will be the change in concentration of air by the change in time and change is always final minus of initial, so zero minus of 10 and change in time will be 50 minus of zero. So this is going to come out to be plus of 10 by 50, 10 molar in 50 minutes. So this will be plus of 0.2 molar per minute. So 10 molar of air disappears in 50 minutes. So the rate of disappearance of air is 10 molar by 50 which is 0.2 molar per minute. Now is this reaction always happening at 0.2 molar per minute? Of course not, sometimes the reaction is fast while at other times it is slow, so what we have calculated out here is actually the average rate of disappearance of air over 50 minutes. Similarly instead of 0 to 50, I can calculate the average rate between any time interval. Here for example if I wanted to calculate the average rate between 10 minutes, let's assume that the concentration at 10 minutes is 2.5 molar and say 20 minutes, the concentration at 20 minutes is 1 molar. So if I wanted to calculate the average rate between 10 minutes and 20 minutes, what would the answer be? We can pause the video and try to come up with the answer. Well out here at time 10 minutes the concentration of air was 2.5 molar which dropped down to 1 molar in 20 minutes, so at 20 minutes the concentration of air drops down to 1 molar. So out here 1.5 molar of air dropped down in 10 minutes. So the rate of disappearance of air will be plus of 1.5 molar in 10 minutes which is going to be equal to 0.15 molar per minute. So 0.15 molar per minute is the average rate of disappearance of air between 10 and 20 minutes. Similarly if I wanted to calculate the average rate between say 10 minutes and 15 minutes, I can easily do that all I need is the concentration at 10 minutes and the concentration at 15 minutes, right? However, if I wanted to calculate the rate of disappearance of air at exactly 10 minutes, how do I do that? What is the change in concentration by the change in time that is happening at this instant? Can we even figure that out? Well, it turns out there is an interesting way in which we can do that. To understand let's look again at how we calculated the average rates between 10 and 20 minutes. To calculate the average rate all we needed was the change in the concentration of air that happened in this time interval. So this is the change in concentration of air that happened over this time interval, right? Now if we join these two points, what we now have out here is a right angle triangle with sides delta C and delta T. Now if we extend this line, it will cut the y-axis and the x-axis out here and will have another right angle triangle out here, right? Now if we compare these two right triangles, if we compare these two triangles, we will realize that they are actually similar triangles, right? Now in similar triangles, the ratio of the sides is equal. So the ratio of delta C by delta T out here will be the same as these values which is the y-intercept and the x-intercept. Similarly if I had to calculate the average rate between 10 minutes and say 15 minutes, I can easily figure that out by checking for the change in the concentration that happened in this time interval, right? Even out here we can draw a line connecting these points and if I extend this line, I will again get a similar triangle out here and the ratio of delta C by delta T will again be in the same ratio as the y-intercept and the x-intercept, right? Now to calculate the instantaneous rate at time t equal to 10 minutes, we need to figure out the change in concentration that happened when delta T was almost equal to 0. So if delta T is almost equal to 0, then both these points are almost going to coincide. And now if we join these two points and extend it, we will get a line that barely touches this curve at this point and we call it the tangent to the curve at this point. Now the change in concentration by the change in time that's happening at this instant, this ratio also has to be equal to the ratio of the y-intercept to the x-intercept that's made by this tangent to the curve at this point, right? So if you look into the data, the y-intercept out here is 5 molar while the x-intercept is almost 20 minutes. So the instantaneous rate, the rate instantaneous, the rate instantaneous at 10 minutes, this will be equal to the y-intercept by the x-intercept, which is going to be 5 molar by 20 minutes, which is going to be equal to 0.25 molar per minute. So to calculate the instantaneous rate at any given instant, we need to draw a tangent to that point and then calculate the ratio of the y-intercept to the x-intercept. Now this value is also called the slope of the graph at that point because if this value is higher, then we'll have a steeper incline or a steeper slope. And if this value is low, that is if the y-intercept is low and the x-intercept is high, then we'll have a less inclined line and so a lower slope. So for most chemical reactions, if we draw the concentration versus time graph, initially the rate of the reaction is high as represented by this steep slope and slowly as the reaction keeps proceeding, the slope of the graph keeps decreasing and so the rate of the reaction keeps slowly decreasing. Now this is what happens generally and there may be certain chemical reactions which do not follow this trend and we'll talk more about how the rate exactly varies during the course of a chemical reaction in this series of videos.