A radioactive sample contains 3.25 1018 atoms of a nuclide that decays at a rate of 3.4 1013 disintegrations per 26 min.
(a) What percentage of the nuclide will have decayed after 159 d?
%
(b) How many atoms of the nuclide will remain in the sample?
atoms
(c) What is the half-life of the nuclide?
days
I have a general idea of how to solve this but keep getting the incorrect answer...I've tried a few different ways so I'm not sure where I'm going wrong. My initial attempt was to convert days into minutes for my calculations. I used the equation N'=Ne^-kt where I calculated k to be the rate given (over minutes). I don't need the answers per se...I think once I know how to correctly approach this I should be fine in figuring them out...Thanks
Oh and when I try to put this into my calculator I get 0... sorry if I am not catching on fast enough for some....I worked graveyard last night and Haven't slept in over 35 hours...
Another attempt I made gave me relatively "reasonable" answers...but still doing something wrong.
I converted the minutes to days and used that in the rate to find k via k= rate/N ; N being 3.25E18 atoms. this calculated k to be .00145.
I plugged this into N'=Ne^-kt....can someone please point out what I am doing wrong?
My recommendation is
a. go to bed for some rest.
b. take fewer classes next quarter/semester/year.
Sleep is a bit of a luxury.....unfortunately I'm one of those students that doesn't have an option of taking a lighter load or working less....luckily I graduate in a few weeks. In the meantime I need to finish these problems so that I can go to bed....but they are due at 11pm tonight and if I go to sleep before completing them I know I won't wake up.
Could you please just explain what I am doing wrong so that I can know how to approach these problems? I appreciate the advice and am well aware I need to go to bed...but I have to get these finished and my professor is out of town attending a memorial service and unable to answer emails until Sunday...
To correctly solve this problem, you need to use the radioactive decay equation and the concept of half-life. Let's break it down step by step:
Given information:
- Number of radioactive atoms (N) = 3.25 x 10^18 atoms
- Decay rate (dN/dt) = 3.4 x 10^13 disintegrations per 26 min
- Time period (t) = 159 days
Step 1: Convert the time period into minutes
To match the units of the decay rate given, you need to convert the time period from days to minutes:
1 day = 24 hours = 1440 minutes
So, 159 days = 159 x 1440 minutes = 228960 minutes.
Now let's move on to each part of the question:
(a) What percentage of the nuclide will have decayed after 159 days?
To determine the percentage of the nuclide that will have decayed, you need to find the fraction of remaining atoms.
First, calculate the decay constant (k) using the decay rate equation:
dN/dt = -kN
k = - (dN/dt) / N
k = - (3.4 x 10^13 disintegrations per 26 min) / (3.25 x 10^18 atoms)
k ≈ -1.046 x 10^-5 min^-1
Next, substitute the values into the exponential decay equation:
N(t) = N0 * e^(-kt)
N(228960) = 3.25 x 10^18 * e^(-1.046 x 10^-5 min^-1 * 228960 min)
Now, to find the percentage of the nuclide decayed, use the formula:
Percentage decayed = (1 - (N(t) / N0)) * 100
(b) How many atoms of the nuclide will remain in the sample?
To find the number of atoms that will remain in the sample after 159 days, substitute the time (t) into the exponential decay equation:
N(228960) = 3.25 x 10^18 * e^(-1.046 x 10^-5 min^-1 * 228960 min)
(c) What is the half-life of the nuclide?
The half-life (t1/2) is the time it takes for half of the radioactive substance to decay.
To find the half-life, we use the formula:
t1/2 = (ln(2)) / k
Substitute the value of k calculated above to find the half-life.