# 14.4 1st Order Half Life Example Problem

A certain 1st order reaction has a half-life of 40 minutes. Calculate (a) the rate constant and (b) how much time is required for this reaction to be 75% complete? So if we remember from the previous video our half life equation for our first order integrative rate law will be 0.693 is going to equal Kt to the 1/2 where t1/2 is the half life, okay? We know that the half-life is going to be 40 minutes, so 0.693 is going to be K times 40 minutes.

If we go and do the math for K, we get 0.0173 minutes to the minus 1 power. So then for part B, we can use this value that we calculated here. To calculate now part B, how much time is required for this reaction to be 75 percent complete? Alright. In this case we’re not at the half life, we’re at the 75% completion. We need to use integrated first order rate law which is the LN of the concentration of a naught divided by a at any time the concentration of a at any time t. That’s gonna equal k which we just determined to be 0.173 minutes to the minus 1 power times t. We’re going to be solving for t which is the time required for the reaction to be 75% complete, okay? At time equals 0, our a naught, is going to be equal to 1 or we can say, 100%. If we go to 75% completion, that means 25% of A remains.

So our concentration of a at any time t is going to be 25 % of what we initially started with. If we write these in a decimal this will be 1 and this will be 0.25, okay. So now we’ll go and use the rate law are initial concentration of a is going to be 0. Our concentration at any time t is 0.25 in this case. That’s going to equal 0.0173 minutes to the -1 power times t. We solve for t which is going to be the LN of 4 divided by 0.0713 and that’s minutes to the -1 power. And if we do the math here we’re gonna get 80 minutes.

So in this particular question we’re asked for two things, okay. One, we’re given some half life Information which allows us to calculate the rate constant. This is very important because once we have that rate constant we can use this integrated rate law to figure out concentrations at any given time. Okay, this particular question asked us how much time requires to get 75% completion of this particular reaction.

75% completion means you’re gonna have 25% of your initial concentration remaining. So we’ve plugged those values into our concentrations, used the k from the half life expression, and then solved for t which in this case is going to be 80 minutes..