Half life problems

Answer 1

It's a process involving experimentation and mathematical modelling.

Method #1:

One way to solve for half-life is to use the following equation:

t1/2 = (t ln 1/2)/(ln mf/mi)

where:

t1/2 = half-life

t = time that has passed

mf = the final or remaining mass of undecayed sample

mi = the initial or original mass of undecayed sample

(The fraction mf / mi is of course equivalent to the fraction or percentage of undecayed sample remaining, in case you are given the fraction remaining rather than specific masses.)

Note: You can also use base-10 logarithms instead of natural logarithms.

For instance, you are told that after 2.00 hours a sample decays such that 80.0% remains undecayed. Substituting these values into the formula allows us to find the half-life of the substance in essentially one step:

t1/2 = (2.00*ln(0.5))/(ln(0.800)) = 6.21 hours

Method #2:

Half-life can alternatively be found in a two-step process using the related model:

At = A0e-Bt

where:

At = Amount at time t

A0 = Initial amount

e = exponential

B = a constant

t = time

However, before you can determine a half-life, first you need to determine what the constant, B, is. This can be done via experimentation. For example, imagine you are observing the decay of a radioactive substance. After 2.00 hours you determine that you only have 80.0% left of the initial amount...

That is, A2 = 0.800A0

So, 0.800A0 = A0e-2.00B

Rearrange to get B = -ln(0.800)/2.00 = 0.1116

So now you have what you need to determine the half life. That is, how many hours will it take before you only have 50.0% left of the decaying substance?

As above, 0.500A0 = A0e-Bt

Solving for t this time, t = -ln(0.5)/B = -ln(0.5)/0.1116 = 6.21 hours.

Note: As in the other method, you could also have used base-10 logarithms instead of base-e (natural) logarithms. Just be sure to use the same base in all your calculations.

As you can see, both these methods yield the same answer, a half-life for the substance of 6.21 hours.

Another Answer:

First you have to find the rate of decay for whatever "half-life" problem you're trying to solve. Try this formula to find the half-life:

-ln(2)=tk

Where t stands for years and k stands for the rate of decay. You'll always be given one of these, just plug in either the years to find the rate of decay or plug in the rate to find the # of years.

Another Answer:

In order to solve half-life equations, consider the equation ...

AT = A0 2(-T/H)

... where A0 is the initial radioisotope's activity, AT is the decayed activity after time T, and H is the half-life. Given 3 of these parameters, you can solve for the fourth. For instance, given starting activity, ending activity, and time, you can solve for half-life as follows ...

AT = A0 2(-T/H)

AT/A0 = 2(-T/H)

ln2(AT/A0) = -T/H

H = -T/ln2(AT/A0)

... and don't forget that ln2(x) = log(x)/log(2)

Answer 2

In order to solve half-life questions, consider the equation ...

AT = A0 2(-T/H)

... where A0 is the initial radioisotope's activity, AT is the decayed activity after time T, and H is the half-life. Given 3 of these parameters, you can solve for the fourth. For instance, given starting activity, ending activity, and time, you can solve for half-life as follows ...

AT = A0 2(-T/H)

AT/A0 = 2(-T/H)

ln2(AT/A0) = -T/H

H = -T/ln2(AT/A0)

... and don't forget that ln2(x) = log(x)/log(2)

Answer 3

The concept of a half-life is only meaningful in regards to an exponential decay process, ie. a decay process that is dependent on the amount of stuff that is decaying. This idea is expressed by the following differential equation:

dN/dt=-λΝ, where N is the amount of stuff, t is time, and λ is a proportionality constant called the decay constant. Solving this equation for N is extremely easy, yielding the equation: N(t)=N0e-λt where N0 is the initial quantity. The half-life of that equation is defined to be when N(t)=t1/2=(1/2)N0. In other words, when (1/2)N0=N0e-λt. Solving that equation (cancel N0, lake the natural log, divide by -λ) gives: t1/2=ln(2)/λ

Answer 4

Half-life refers to the time it takes for half of a substance to decay or be eliminated. In a half-life problem, you typically need to determine the amount of substance remaining after a certain number of half-lives have passed. This involves using the formula A = A₀(1/2)^n, where A is the final amount, A₀ is the initial amount, and n is the number of half-lives that have passed. Remember to keep track of the units of time and make sure you're consistent with your calculations.