Isotope Half-Life and Sample Size Calculation

In summary, the isotope 239Pu has a half-life of 24,100 years and after 10,000 years, a sample of the isotope is reduced by 1.6 grams. To find the initial size of the sample, the equation Ao-1.6=Aoer10,000 is used. However, when dividing by Ao and taking the Ln of both sides, a negative number is obtained which is not possible. To solve this, the exponential part can be solved first and then the equation can be rearranged to gather A0 together on one side.
  • #1
AryRezvani
67
0

Homework Statement



The isotope 239Pu has a half-life of 24,100 years. After 10,000 years, a sample of the isotope is reduced 1.6 grams. What was the intial size of the same (in grams)? How large was the sample after the first 1,000 years. Round your answer to four decimal places.

Homework Equations



A=Aoert

The Attempt at a Solution



Intially, for the value of A, I subbed in "Ao-1.6" and then did the following:

Ao-1.6=Aoer10,000

Divided each side by Ao, and then take the Ln of both sides.

That's where I get stuck. My calculator is indicating that it's not possible to take the Ln of a negative number.
 
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  • #2
How did you get a negative number after your algebraic manipulation?
 
  • #3
SteamKing said:
How did you get a negative number after your algebraic manipulation?

The final isotope is the initial mass - 1.6.

When you divide by Ao, the negative stays there, right?
 
  • #4
AryRezvani said:
The final isotope is the initial mass - 1.6.

When you divide by Ao, the negative stays there, right?

You'd better show your algebra, exactly. You seem to be making an elementary mistake.
 
  • #5
It's not clear (to me) what you're solving for. What is r? Since you're given a half-life, shouldn't there be a power of two involved? And a negative power of two at that.
 
  • #6
You might find it easier to solve for the exponential part, THEN just gather the A0 together on one side (and yes, don't forget that the equation should be A0-1.6 = A0e-rt/ln2
 

What is isotope decay?

Isotope decay is the process by which an unstable isotope of an element spontaneously breaks down and transforms into a different element or isotope, releasing energy in the form of radiation.

What causes isotope decay?

Isotope decay is caused by the instability of an isotope's nucleus, which may have too many or too few neutrons compared to the number of protons in the nucleus. This imbalance leads to a process of radioactive decay in order to achieve a more stable state.

How is isotope decay measured?

The rate of isotope decay is measured using a unit called half-life. This is the amount of time it takes for half of the original amount of a radioactive substance to decay into a more stable form.

What are some common types of isotope decay?

The most common types of isotope decay include alpha decay, beta decay, and gamma decay. In alpha decay, an alpha particle (two protons and two neutrons) is emitted from the nucleus. In beta decay, either a beta particle (an electron) or a positron (a positively charged electron) is emitted. Gamma decay involves the release of high-energy photons.

What are the practical applications of isotope decay?

Isotope decay has several practical applications, including radiometric dating to determine the age of rocks and fossils, medical imaging using radioactive isotopes, and nuclear power generation. It is also used in agriculture, archeology, and environmental studies to track the movement of substances and study their effects.

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