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# Exponential decay and semi-log plots

## Video transcript

here we have a graph of exponential decay where n refers to the number of radioactive nuclei all right as a function of time and so this write this equation describes our graph so this would be the number of radioactive nuclei at any time T is equal to n naught the initial number of radioactive nuclei e to the negative lambda T and lambda right lambda is equal to the decay constant so this is just some constant number here and you could also call this K if you wanted to you call it K if you're thinking about the rate constant so that's just some constant and then we're going to multiply by the time here so let's say we wanted to let's say we wanted to find what this point right here represents on our graph well that's when time is equal to zero so let's plug in let's plug in time is equal to zero into our equation so this would be the number of radioactive nuclei when time is equal to zero is equal to n naught times e to the negative lambda times T which is zero so that's that is equal to this would be n naught times e to the zero each of the zero is one so the number of radioactive nuclei at time is equal to zero is equal to n naught so this represents and not the initial number of radioactive nuclei on our graph and you could do this for any time T right you could just pick a time let's say that's our time that we wanted right and go up to here and find what this value is on the graph and so at any time T right this would represent the number of radioactive nuclei let's do it for half-life so remember when half-life so when time is equal to a half-life write the number of radioactive nuclei this would be let's do one half lives this would be the initial divided by two so half of it remains so let's look at that on our graph right so if we take and naught we divide that by two that's approximately here so let's say this is not over - and we go over - we go over to here and we drop down to our time so this should represent our half-life that time should represent our half-life so that's what it looks like graphically let's take let's take these numbers let's take let's take the half-life and and what the number of radioactive nuclei would be and let's plug it into this equation alright so let's go ahead and do that let's get some more room so I'm just going to rewrite that equation here so we had we had the number of radioactive nuclei as a function of time is equal to the initial number of radioactive nuclei times e to the negative lambda T so let's plug in those let's plug in those so when we're going to talk about the half-life we're going to plug that in here for the time and then the number of radioactive nuclei would be N naught divided by 2 so let's plug those in so we would have n naught divided by 2 is equal to n naught e to the negative lambda times the half-life all right well that that cancels out the N knots all right so that gives us on the left side 1/2 is equal to e to the negative lambda times the half-life so next let's get rid of the e and we could do that by taking the natural log of both sides right so if I take the natural log of 1/2 on the left side I take the natural log of e to the negative lambda T 1/2 all right on the left side natural log of 1/2 is equal to negative point 6 9 3 so this is just plug it in your calculator you'll get negative 0.693 and the write that takes care of this and all that's equal to just this over here right so now we would have this would be negative lambda T 1/2 and so we don't have to worry about the negative signs right so this is just point 6 9 3 is equal to lambda times T 1/2 so we could solve for the half life right so if we solve for T 1/2 so T 1/2 would be equal to 0.693 divided by lambda divided by the decay constant and so this is one of those equations right that you see for half-life so what if you wanted to go ahead and solve for lambda the decay constant right so that's obviously really simple we just do we just do the decay constant is equal to 0.693 divided by divided by the half-life and so obviously I'm just rearranging this equation here so you could solve for half-life or you could solve for the decay constant you go back and forth between the half-life and the decay constant so if you know one you you can figure out the other one all right so that's that's thinking about the exponential decay graph let's talk about semi-log plot sneck switch is another way of looking at the data and so let's get some room here I'm going to rewrite our equation right so the number of radioactive nuclei is equal to the initial number times e to the negative lambda T all right so let's convert this into into a linear into a straight line so we have to do we have to divide by n not so we have n divided by and naught so we divide both sides by and naught and we get e to the negative lambda T here all right now to get rid of this e once again we just take the natural log so we can take the natural log of both sides so natural log of n over N naught is equal to the natural log of e to the negative lambda T all right so on the left side we have a log property so natural log of n over N naught is equal to natural log of n minus natural log of n naught and then on the right alright this goes away and we're left with this so negative lambda T so we have negative lambda T on the right side here and if we just rearrange this right so let's just do natural log of N is equal to negative lambda t so we're going to add Ln of n not to this side and now we have now we have a very interesting very interesting form if you look at it closely you'll see this is the same thing as y is equal to MX plus B which is the equation for a straight line so y y would be equal to natural log of n M would be equal to negative lambda all right so we're talking about T right as being X here and then natural log of n naught is equal to B so y is equal to MX plus B if you remember equation four it's the graph of a straight line alright where where m m is the slope all right so we could say the slope of this is equal to negative lambda and remember that this is your vertical intercept all right so so B is your vertical intercept so if we if we graph this our vertical intercept should be natural log of n not so let's let's go ahead and sketch this out really quickly so I won't be too concerned with with details here but if we're graphing it so if you think about this as being your y-axis I'll just put this in parentheses that's not really what we're doing and this being your x-axis all right let's look at those again so for my Y I would be graphing natural log of n so let me go ahead and use different colors so we can see it here so natural log of n on the y axis on the x axis here that would be time so we have time over here alright and we know that the vertical intercept is going to be natural log of and not so this vertical intercept here is going to be natural log of and not and we could prove that really quickly all right we could we could say when time is equal to zero all right so when time is equal to zero let's plug that in we'd have natural log of n which is equal to negative lambda times zero plus natural log of and not all right so this would go away and you can see that we would have natural log of and not would be equal to this point here so that's our vertical intercept and we know this is the graph of a straight line alright and we're going to have a negative slope here so if I go ahead and draw this in it would look something like well just pretend like that's a straight line I didn't do a very good job but you can use your imagination there and the slope of this alright so the slope remember what slope is that's change in Y over change in X all right so that would be that would be the change in this axis over the change in this axis that's equal to alright that is equal to negative lambda we talked about over here and so if you do a semi-log plot right so it's semi-log because we have this this natural log over here versus here it tells us some information right it's just another way to look at the data you could find the slope of this straight line I take the negative of it and get your decay constant and then from your decay constant you could get your half-life and so and so once again sometimes you'll see you'll see semi-log plot done as as just a different way of looking at the data