What is the concentration?

Question

One type of insecticide decomposes according to second-order kinetics. The rate law of this decomposition is 1.45 yr-1. A sample of this insecticide washed into a lake on July 1st and produced an insecticide concentration (in the lake) of 5.0 x 10-6g/mL. What is the concentration of the insecticide on July 1st of the following year?

Answer 1

If this is in fact a 1st order reaction (not 2nd order as stated), then a rate constant of 1.45 yr^-1 would make sense. And we can then find the half life.

t_1/2 for a 1st order reaction = 0.693 / k

t_1/2 = 0.693 / 1.45 yr^-1

t_1/2 = 0.478 yrs

Time elapsed = 1 year (July1 to July 1)

Fraction remaining = 0.5ⁿ where n = number of half lives

n = 1 yr x 1 half life / 0.478 yrs

n = 2.09 half lives

Fraction remaining = 0.5^2.09 = 0.2349

Concentration remaining after 1 year = (0.2349)(5.0x10^-6 g/ml) = 1.2x10^-6 g / ml

Answer 2

First, this problem has at least two big issues with it that make it insoluble as given. I'm going to show how we might solve a similar problem:

A second order reaction must have a rate law with its exponents adding up to 2 (that's where the second comes from). In this case, there's only one reactant, so it must have an exponent of 2 on its own. Therefore, a second order reaction with a single reactant (in this case it's insecticide) has this general formula:

Rate = k[reactant]²

"k" is the rate constant. "Rate" should be measured in density (in this case the problem uses g/ml, which I'll fill in for density at the end) per year. The pesticide itself is also measured in terms of density. We don't yet know what units "k" should have, so we'll fill in the other units to find that out. I'm going to rewrite the variables whose units we know in the rate law I gave above.

Density/year = k*Density²

Solving for k's units, we get:

(Density/year)/Density²

1/(Density*year)

So k should have units of 1/(Density*year). This is where we come to what I believe is a mistake in your question: your question mentions that the "rate law" of the decomposition is 1.45/year. However, this doesn't mean much, since as I showed above, a rate law always has Rate=k, and k may be multiplied by some concentrations like the one I replaced with density above. So the 1.45/year isn't a rate law, it's probably meant to be the rate constant "k" itself. Yet, the units I found for what k should have (1/(Density*year) do not match the 1.45/year given in the problem.

There are two possible explanations for this: the question was meant to say first order, not second order, in which case the given units for k would be appropriate, or the question meant to give "1.45/Density*year" as the rate constant.

I'll solve it assuming the second explanation is correct.

We can fill in our rate law somewhat now:

Rate = (1.45/(year*Density))*Density²

And now I'll fill in the given initial concentration:

Rate = (1.45/(year*Density))*(5*10^-6 Density)²

Solving for Rate gives 7.25*10^-6 Density/year.

If 1 year passed (as was said in the problem), the density would have dropped from the initial 5*10^-6 by 7.25*10^-6. The result is less than 0, so the insecticide would have completely decomposed before July came around again!

I have two further comments:

1. The problem asked for concentration but gave only density and no way of converting between the two. We needed the atomic weight of the pesticide in order to do so. The math above isn't actually valid for density, we would've converted to concentration in the first place if we were given sufficient info to do so, and
2. This is probably not the answer that was expected given the multiple errors in the problem that was asked