Half Life and Activity

Questions from email…

How would you solve a problem like:

A certain radioactive isotope has a half life of 6.25 hours. What percent of the original isotope is still present after 22.9 hours have passed?

And 

A certain radioactive isotope is found to have an activity of 0.567Ci per gram. What is the activity of 0.807g of the isotope? 

For the first one...

Almost all half-life problems use some form of the relationship:

(amount remaining) = (original amount)(1/2)ⁿ

where n = the number of half lives.

If one half-life is 6.25 hours, then 22.9 hours represents (22.9/6.25 = 3.664) half-lives. There are a LOT of different ways we can think about "amount" in these problems... in this context, let's say the "original amount" is "100%". That means:

(amount remaining) = (100%)(1/2)^3.664

I'll let you plug those numbers in... This is a problem where you can probably estimate a reasonable answer. If 3 half-lives have passed, there should be 1/8 of the original amount (0.125, or 12.5%); if 4 half-lives have passed, there should be 1/16 of the original amount (0.0625, or 6.25%). When you solve the equation above, your answer should be something less than 12.5% but more than 6.25%.

For the second problem... My guess is that you're trying to make it a super-complex chemistry problem. This is a good example of being attentive to the units on numbers in a problem and using them to get an answer, even if you might not "know" how the problem really works. You're given "Ci per gram", so if you have 1 gram of sample, the activity would be 0.567Ci. But you don't have 1 gram, you have 0.807g. If I told you that a blueberry cake had 28.0 blueberries per pound of cake (mmmm... pound cake...), how many blueberries would you expect to find in a 0.395 pound piece of cake?

Beware of googling answers from multiple sources

 Different sources use some slightly different conventions for calculating colligative properties. Sometimes, if you try to google search for help, it can look like the exact same problems is set up 2 or 3 different ways because of these different sources using different methods. In many cases, the different set ups are all equally correct, they just rely upon some different assumptions. {By the way, this is true for lots of things besides colligative properties, but this is a good place to point this out.}

For example, freezing point depression has a mysterious negative sign that might show up in a few different places. One commonly found set of formulas for freezing point depression is:

ΔT_FP = k_FPD•m

T_FP,solution = T_FP,solvent - ΔT_FP

Because freezing point is depressed, the change in freezing point is negative, so in this version of the formula, the change is subtracted from the freezing point of the pure solvent. This is a pretty common way of treating the formula, but it's not the only way. Some sources use the formulas:

ΔT_FP = -k_FPD•m

T_FP,solution = T_FP,solvent + ΔT_FP

In this case, the negative sign is incorporated into the formula for freezing point change and the change is added to the freezing point of the pure solvent. To make it even more "interesting", some sources use the fomulas:

ΔT_FP = k_FPD•m

T_FP,solution = T_FP,solvent + ΔT_FP

In this case, the source is using a value of the freezing point depression constant that is negative, so the mysterious negative sign is buried in the constant.

These methods all work equally well, and are really doing the exact same calculation, BUT if you mix-and-match information from these 3 methods and just plug numbers into a formula, you might get an answer that's wrong... Fortunately, most of the "wrong" answers you might get are pretty obviously wrong if you stop and think about it for a moment. Remember that freezing point of a pure solvent is depressed when you add a solute. That means the freezing point of the solution must be lower than the freezing point of the pure solvent. If you incorrectly mix and match the above methods, you will likely get the correct magnitude of answer, but the wrong direction of the change. So as a last step in any colligative property question, always make sure your answer is moving in the right direction. If you're calculating a freezing point depression temperature of an aqueous solution and your final answer is "+1.63°C", you know that answer can't be correct because the freezing point is higher than the freezing point of the pure solvent. The freezing point might have changed by 1.63°C (the correct magnitude of change), but the freezing point of the solution is probably "-1.63°C".

It's always a good idea to do a "reasonable answer" check as a last step on every problem, but often it's difficult to know what's reasonable. For colligative properties, you always know the direction of the correct answer, so it's a good idea to at least check that.