SOLUTION: An isotope of thorium, 227Th, has a half-life of 18.4 days. How long will it take for 85% of the sample to decompose? Please round the answer to the nearest tenth.

Question 613573: An isotope of thorium, 227Th, has a half-life of 18.4 days. How long will it take for 85% of the sample to decompose?
Please round the answer to the nearest tenth.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
A general formula for exponential growth or decay is:
Amount = (original amount)*(1 + percent change)^(number of time units)
where
"percent change" is the percent change in decimal or fraction form. For decay/decrease problems this should be negative.
"number of time units" is the how many time units that have elapsed since the time of the original amount.
"time units" are whatever amount of time it takes for the percent change to occur.

In your problem,
"percent change" = -50% = -0.5
"time units" = 18.4 days since that is how long it takes for one 50% decrease to occur.

We are asked to find an amount of time. So it should be no surprise that we do not yet know what "number of time units" is.

But what about the amounts? We don't know them either one of them. As you may have learned, in order to find an unknown with just one equation you need to know everything but that one unknown. The fact that the problem does not ask about a specific amount of thorium, like: "how long does it take for 100g of thorium..." should be a a clue. The answer is the same for any amount of thorium! We can see this by writing expressions for the amounts. First,
Let A = the original amount
then, if 85% of the thorium decomposes, what percent is left? Answer: 15%. And how do we express 15% of A? Answer: 0.15*A

Let's see what happens if we use our number in for percent change and our expressions in for "original amount" and "amount" and use "n" for "number of time units" into our general equation:
Amount = (original amount)*(1 + percent change)^(number of time units)

If we divide both sides of this by A, the A's cancel!

Since the A's disappeared, it shows that an 85% decay of thorium will take the same amount of time no matter how much thorium there is!

We now have an equation with just one unknown, n. We can solve this. First we simplify:

Next, since the unknown is in an exponent, we use logarithms. Any base of logarithms can be used. But there are advantages to choosing certain bases:

Using a base for the logarithm that matches the base of the exponent will result in the simplest exact expression for the answer.
Using a base for the logarithm that your calculator "knows" (base 10 or base e) will result in a less simple expression. But the expression will be easier to convert to a decimal approximation.

I'll show you both of these. First we'll use base 0.5 logs:

Then we use a property of logarithms, , to move the exponent of the argument out in front. (It is this very property of logarithms that is the reason we use logarithms on equations like this. It allows us to move an exponent, where the variable is, out fin front. With the variable no longer being in an exponent we can use "regular" algebra to solve for the variable.) Using this property on the logarihtm on the right side we get:

And since for all bases, this becomes:

This is an exact expression for the number of time units it takes for an 85% decay. Since a time unit is 18.4 days, it takes days for an 85% decay.

Now we'll try again with logarithms we can easily convert to decimals. Using base e (aka ln):

Using the property to move the exponent:

The ln(0.5) is not a 1 so this expression does not simplify as much as earlier. Dividing both sides by ln(0.5) we get:

This is another (not as simple) exact expression for the number of time units it takes for an 85% decay. Like before, the number of days this is would be:
18.4*(ln(0.15)/ln(0.5))
For a decimal approximation, which your problem clearly requests, use your calculator. If your calculator has buttons for parentheses then just type what you see above. If not, then be sure to find the two logs before you divide. Do not divide 0.15 by 0.5! If your calculator does not have a button for ln, just use the button for log, instead. (Remember, I mentioned earlier that any base of log can be used. The answer will be the same no matter which base of logarithm is used:
, etc.
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