okay so let's continue with half-life as we go over into chemical kinetics the half-life is a time taken for the reaction so the concentration reacted decreased to half of its initial value for the first order reaction the half-life does not depends on the starting concentration which means for the same amount of time half-life the concentration reactant will decrease to half of the original so this we can prove mathematically and so here just briefly write down how should i approve it you can spend more time to work it out yourself based on the integrated rate law of the first order kinetics we will have logarithm of the concentrated reactant at time t it equals natural log of the initial concentration subtract rate constant times t so if we subtract t with a half-life then the 80 would equals half of the initial concentration so by substituting the a t with this solving for time t will have this equation which shows the half-life of the reactant first order reaction is independent of the concentration as a matter of fact is natural log of 2 divided by the rate constant or 0.0.693 divided by k so this shows you as a reaction proceed very fast with very high rate constant k the half-life will be smaller so that's about the first order reaction the rate constant graphically with the first order reaction if we graph the natural log of the concentration reactant versus time we'll find out if we start from certain concentration the time it needed for the concentrated reactant decrease to half this time is half-life for this reaction for the concentration reactor that keeps decreasing to half of this point which is 0.015 times the time needed is also equals the half-life and so forth so for the first order reaction as the concentration reactant decreased to half the time needed is one half-life after another half-life the concentration reacted will further decrease to half of this half which means after two half-lives the concentration reacted will decrease to one quarter of the original concentration after three half-lifes the concentration reacted will further decrease to half of the quarter so this concentration will be one eighth of the orange dot concentration so as we goes by we have a general formula which shows the concentration of reactant at time t equals concentration of initial reactant times how many half-lives as a power of the one half so this is a general formatter how we correlate of the concentration reactant to the initial concentration so like like we said before the first order reaction the half-life does not depends on the initial concentration and this is the equation we calculate half-life equals the natural log of 2 divided by rate constant for the signal order reaction we can prove mathematically that a half-life is inversely proportional to the initial concentration for the zero-order reaction half-life is directly proportional to the initial concentration so let's see some examples how we solve with half-lives this is a reaction we have seen it before at a thousand degrees celsius the rearrangement reaction of the cycle propane has rate constant of 9.2 uh inverse of second what's the half life of the reaction and how long does it take for the concentration of propane to reach one tenth of the initial value so for this reaction first what is the half the laptop mentioned the reaction kinetics is a first order reaction so therefore the half-life of the reaction equals natural log of 2 divided by k or 0.693 divided by the rate constant okay so substitute k with value that we have and now we keep the unit which is inverse of second we'll get the answer for this question the final answer keep two sig figs because the rate constant is two sig figs natural log of two is exact number infinite sig figs even though the 0.693 shows on the three digits significant digits so let's find out 0.075 the unit this is the inverse of a second and inverse second and the inverse and that'll give me second at the time second question how long does it take for the concentration propane to reach one tenth of the initial value so this question we can use what we learned in the earlier part solving problems with integrated law to solve and if we solve for the time t this equation if you combine that will give me this formula as we've seen that before the concentration of the reactant reaches one tenth of the initial value so the a t over a zero will equals 1 10 which is 0.1 data should equals negative times rate constant times time then substitute k with a value of 9.2 we can solve for time t and k equals nine point two final answer is point two five seconds keep only two sig figs so here's a summary of the half-life in regard to the order of the reaction zero's order the rate is constant independent of concentration but the half-life depends on the concentration reactant first order reaction as we go over the half-life is independent of concentration for the second order reaction half-life also depends on the concentration but it inversely depends on the concentration in reality the question in our exercise or quiz or test the first order reaction kinetics is the focus so make sure i understand and remember how we calculate the half-life from the rate constant