Under certain water conditions, the free chlorine...

Under certain water conditions, the free chlorine (hypochlorous acid, HOCl) in a swimming pool decomposes according to the law of uninhibited decay. After shocking his pool, Ben tested the water and found the amount of free chlorine to be 2.5 parts per million (ppm). Twenty-four hours later, Ben tested the water again and found the amount of free chlorine to be 2.2 ppm. What will be the reading after 3 days (that is, 72 hours)? When the chlorine level reaches 1.0 ppm, Ben must shock the pool again. How long can Ben go before he must shock the pool again?

Key Concepts

Logarithmic Calculations for Time Estimation

Logarithmic calculations are used to solve for time in exponential decay equations. By taking the logarithm of both sides of the decay equation, one can isolate the time variable to determine how long it takes for the concentration of a substance to decay to a certain percentage of its initial amount.

Rate Constant

The rate constant is a parameter in first-order kinetics that quantifies the speed of the decay process. It is determined experimentally and integrated into the exponential decay formula to predict the concentration of a substance at any given time, allowing for the calculation of time intervals for specific concentration thresholds.

Exponential Decay

Exponential decay refers to a process where the amount of a substance decreases at a rate proportional to its current value. This kind of decay is characterized by a constant half-life or decay rate, leading to a continuously decreasing quantity over time following an exponential function.

First-order Reaction Kinetics

First-order kinetics describes a reaction in which the rate of decay of a substance is directly proportional to its concentration. This model, applicable to uninhibited decay processes, implies that the rate does not depend on other factors, making the mathematical treatment straightforward using the exponential decay equation.

Transcript

00:01 Okay, so first thing we're gonna look at in this problem is towards the beginning where it says law of uninhibited decay with this we have a formula that we're just gonna write down right away.

00:14 So we have a of t equals a sub zero e to the kt all right so a of t here is going to be our new amount a sub zero is gonna be our initial amount basically what we start with.

00:47 All right, k is just gonna be our constant which we are going to have to find at first and then t is our time which is gonna be in hours here alright, so we're gonna start by finding the constant looks like it starts out with saying he found the amount of free chlorine right away to be 2 .5 parts per million.

01:13 So since our initial amount is a zero we are going to plug in that 2 .5 for a zero.

01:24 Okay, and another note on our k constant will end up being negative since this is decay we leave e as e.

01:36 We don't know k so let's do this k and then we know 24 hours later.

01:42 He tested the water again so we're gonna put 24 hours in for time and at that 24 hours found the new amount of chlorine in there to be 2 .2 ppm, so we're gonna put that over here.

02:00 Okay.

02:03 So now what we have to do is solve for k here so what we're gonna do is try to get that e to the 24 k by itself.

02:11 So divide both sides by 2 .5 5 let's cross out.

02:18 Okay, when we take 2 .2 divided by 2 .5 we get 0 .88 equals e to the 24 k.

02:30 All right now what we need to do somehow get rid of that e if you remember what we do with this is we take natural log of both sides and that is going to get rid of both the natural log and the e okay so basically what we're left with is natural log of 0 .88 equals e to the 24 k excuse me just 24 k.

03:05 I read that e.

03:07 Now our last step is to divide both sides by 24 if you put natural log of 0 .88 divided by 24 into your calculator you should get your constant make sure you're putting that 0 .88 in parentheses let's round to four decimal places here.

03:29 So point zero zero five three...

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