# Half life Question

#### TheRealAngeleno

Hey guys can someone go through the thought process in solving this question? I have it down to C & D but I don't know where to go from here. Thanks!

What is the half-life of a material that obeys first-order decay kinetics if 40% of its original quantity remains after one hour?

A. 30 min
B. 46 min
C. 50 min
D. 64 min

#### docntrainin

You need to know that because it follows first order kinetics, the half-life is constant. If only forty percent remains after one hour, then you know that the half life is less than one hour (100 --> 50 should take less time than going from 100 -->40, which takes one hour). This narrows down your answers to A,B, or C. A is too much of a difference, so the answer choice should be either B or C. Someone else want to try?

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#### docntrainin

Alright, wow! This question was bugging me because I wanted to leave you with an answer. So I did some digging and found this equation:

Time of half life = time (ln 2) / log (initial concentration / final concentration)

Time of half life = 60 min (ln 2) / log (100 / 40 ) = approx. 45 min

Your answer is B. I would have guessed B just because 50 min, C seemed "too perfect" #### indianjatt

10+ Year Member
Alright, wow! This question was bugging me because I wanted to leave you with an answer. So I did some digging and found this equation:

Time of half life = time (ln 2) / log (initial concentration / final concentration)

Time of half life = 60 min (ln 2) / log (100 / 40 ) = approx. 45 min

Your answer is B. I would have guessed B just because 50 min, C seemed "too perfect" Exactly, but if this was on the mcat.... how would u approximate the lns....

#### geeyouknit

7+ Year Member
Hey guys can someone go through the thought process in solving this question? I have it down to C & D but I don't know where to go from here. Thanks!

What is the half-life of a material that obeys first-order decay kinetics if 40% of its original quantity remains after one hour?

A. 30 min
B. 46 min
C. 50 min
D. 64 min
First order mean's it's constant. That means in X min, amount decreases by half.
It can't be A, because if it was 30 min, 1 hour would be two half lives so it would be at 25%.
If it's C, then at the 50 min mark, there's 50 % left. In another 10 min, 1/5 of amount would have decreased ( if its constant ), so you'd be left with 80% (4/5) of the amount that's left at the 50 min mark (50% left).
80% of 50% is 48%.

That means you'd have 48% of original amount left. So it has to be B.
EDIT: 80% of 50% is 40%. My bad. It's C
Hope that helps. I know the second paragraph can get confusing.

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#### Kent Money

Umm, isn't 80% of 50% equal 40%? Not 48%

I think that would make C the correct answer...

#### geeyouknit

7+ Year Member
Umm, isn't 80% of 50% equal 40%? Not 48%

I think that would make C the correct answer...

Yea, sorry, I don't know what I was multiplying. I seem to be making a lot of those silly mistakes.

#### Swagster

7+ Year Member
Hey guys can someone go through the thought process in solving this question? I have it down to C & D but I don't know where to go from here. Thanks!

What is the half-life of a material that obeys first-order decay kinetics if 40% of its original quantity remains after one hour?

A. 30 min
B. 46 min
C. 50 min
D. 64 min
I did it backwards.

If 30 min, then there would be 25% after one hour.
If 46 min, then there would be 50% after 46 min, 25% after 1 hour 32 min, so about 40% after one hour.
If 50 min, then there would be 50% after 50 min, 25% after 1 hour 40 min, so about 45% after one hour.
If 64 min, then there would be over 50% after one hour.

This is a TBR question, right? I remember not liking their solution as much as mine.

#### neurodoc

##### Neurologist
15+ Year Member
Here's the general equation for exponential decay half-life (from the definition):

N(t)=N(0)(1/2)^t/t1/2, where N(0)=initial quantity at t=0, N(t)=quantity remaining at time t, and t1/2=the "half life."

So, N(0)=1, t=1 hour, and N(t)=0.4 and 0.4=1x(1/2)^1/t1/2. Taking ln of both sides and simplifying yields: t1/2=ln0.5/ln0.4=0.7565hours, or 45.3882 minutes, which means 46 minutes is the closest answer.

Of course you could guesstimate this answer without using a calculator, as previously described. #### Class1P

10+ Year Member
5+ Year Member
if there is 40% left after 60 minutes wouldn't the half life be just under an hour since 60% is gone in an hour.? I'd say 50min

#### MedPR

Removed
It's 46 minutes.

In a half life of 46 minutes you would have 40 left at 57.5 minutes.

In a half life of 50 minutes you would have 40 left at 62.5 minutes.

Since only 60 minutes has passed, your half life must be less than 50 minutes long.

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#### MrNeuro

7+ Year Member
It's 46 minutes.

In a half life of 46 minutes you would have 40 left at 57.5 minutes.

In a half life of 50 minutes you would have 40 left at 62.5 minutes.

Since only 60 minutes has passed, your half life must be less than 50 minutes long.
how'd you do that?

#### MedPR

Removed
how'd you do that?
At the first half-life you have 50% of the original. In this case you only have 40%.

So if you setup a ratio: 50/40 = 1.25, you can use that ratio to find out how much time would pass for X amount to be remaining for any half-life you want.

This method really only works since there are only 4 possible answers (actually 2 since you can easily rule out 30min and 60min). It's a guess and check type of thing.

So anyway, if the half-life is 46 minutes, 46*1.25 = 57.5 Same for if the half-life is 50minutes. 50*1.25 = 62.5 or whatever I said earlier.

#### MrNeuro

7+ Year Member
At the first half-life you have 50% of the original. In this case you only have 40%.

So if you setup a ratio: 50/40 = 1.25, you can use that ratio to find out how much time would pass for X amount to be remaining for any half-life you want.

This method really only works since there are only 4 possible answers (actually 2 since you can easily rule out 30min and 60min). It's a guess and check type of thing.

So anyway, if the half-life is 46 minutes, 46*1.25 = 57.5 Same for if the half-life is 50minutes. 50*1.25 = 62.5 or whatever I said earlier.
It's 46 minutes.

In a half life of 46 minutes you would have 40 left at 57.5 minutes.

In a half life of 50 minutes you would have 40 left at 62.5 minutes.

Since only 60 minutes has passed, your half life must be less than 50 minutes long.
huhhh?? isn't the margin of error for both the same

and maybe im just tired but can you explain the ration again?

#### MedPR

Removed
huhhh?? isn't the margin of error for both the same

and maybe im just tired but can you explain the ration again?
Maybe I made up something that doesn't work. I'm not really sure.

Anyway, I did the math the long way (with the logs and all that crap) and with a calculator and it came out to be something like 45.7.

#### MrNeuro

7+ Year Member
Maybe I made up something that doesn't work. I'm not really sure.

Anyway, I did the math the long way (with the logs and all that crap) and with a calculator and it came out to be something like 45.7.
yeah same i just don't have ln memorized..... 