Calculate Half Life
The half-life of a substance undergoing decay is the time it takes for the amount of the substance to decrease by half. It was originally used to describe the decay of radioactive elements like uranium or plutonium, but it can be used for any substance which undergoes decay along a set, or exponential, rate. You can calculate the half-life of any substance, given the rate of decay, which is the initial quantity of the substance and the quantity remaining after a measured period of time.
Steps
Understanding Half-Life
- Understand exponential decay. Exponential decay occurs in a general exponential function <math>f(x) = a^{x},</math> where <math>|a| < 1.</math>
- In other words, as <math>x</math> increases, <math>f(x)</math> decreases and approaches zero. This is exactly the type of relation we want to describe half-life. In this case, we want <math>a = \frac{1}{2},</math> so that we have the relationship <math>f(x+1) = \frac{1}{2}f(x).</math>
- Rewrite in terms of half-life. Of course, our function does not depend on generic variable <math>x,</math> but time <math>t.</math>
- <math>f(t) = \left(\frac{1}{2}\right)^{t}</math>
- Simply replacing the variable doesn't tell us everything, though. We still have to account for the actual half-life, which is, for our purposes, a constant.
- We could then add the half-life <math>t_{1/2}</math> into the exponent, but we need to be careful about how we do this. Another property of exponential functions in physics is that the exponent must be dimensionless. Since we know that the amount of substance depends on time, we must then divide by the half-life, which is measured in units of time as well, to obtain a dimensionless quantity.
- Doing so also implies that <math>t_{1/2}</math> and <math>t</math> be measured in the same units as well. As such, we obtain the function below.
- <math>f(t) = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}</math>
- Incorporate initial amount. Of course, our function <math>f(t)</math> as it stands is only a relative function that measures the amount of substance left after a given time as a percentage of the initial amount. All we need to do is to add the initial quantity <math>N_{0}.</math> Now, we have the formula for the half-life of a substance.
- <math>N(t) = N_{0}\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}</math>
- Solve for half-life. In principle, the above formula describes all the variables we need. But suppose we encountered an unknown radioactive substance. It is easy to directly measure the mass before and after an elapsed time, but not its half-life. So let's express half-life in terms of the other measured (known) variables. Nothing new is being expressed by doing this; rather, it is a matter of convenience. Below, we walk through the process one step at a time.
- Divide both sides by initial amount <math>N_{0}.</math>
- <math>\frac{N(t)}{N_{0}} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}</math>
- Take the logarithm, base <math>\frac{1}{2},</math> of both sides. This brings down the exponent.
- <math>\log_{1/2}\left(\frac{N(t)}{N_{0}}\right) = \frac{t}{t_{1/2}}</math>
- Multiply both sides by <math>t_{1/2}</math> and divide both sides by the entire left side to solve for half-life. Since there are logarithms in the final expression, you'll probably need a calculator to solve half-life problems.
- <math>t_{1/2} = \frac{t}{\log_{1/2}\left(\frac{N(t)}{N_{0}}\right)}</math>
- Divide both sides by initial amount <math>N_{0}.</math>
Example Problems
- Problem 1. 300 g of an unknown radioactive substance decays to 112 g after 180 seconds. What is the half life of this substance?
- Solution: we know the initial amount <math>N_{0} = 300 {\rm \ g },</math> final amount <math>N = 112 {\rm \ g },</math> and elapsed time <math>t = 180 {\rm \ s }.</math>
- Recall the half-life formula <math>t_{1/2} = \frac{t}{\log_{1/2}\left(\frac{N(t)}{N_{0}}\right)}.</math> Half-life is already isolate, so simply substitute and evaluate.
- <math>\begin{align}t_{1/2} &= \frac{180 {\rm \ s }}{\log_{1/2}\left(\frac{112 {\rm \ g }}{300 {\rm \ g }}\right)} \\
&\approx 127 {\rm \ s }\end{align}</math>
- Check to see if the solution makes sense. Since 112 g is less than half of 300 g, at least one half-life must have elapsed. Our answer checks out.
- Problem 2. A nuclear reactor produces 20 kg of uranium-232. If the half-life of uranium-232 is about 70 years, how long will it take to decay to 0.1 kg?
- Solution: We know the initial amount <math>N_{0} = 20 {\rm \ kg },</math> final amount <math>N = 0.1 {\rm \ kg },</math> and the the half-life of uranium-232 <math>t_{1/2} = 70 {\rm \ years }.</math>
- Rewrite the half-life formula to solve for time.
- <math>t = (t_{1/2})\log_{1/2}\left(\frac{N(t)}{N_{0}}\right)</math>
- Substitute and evaluate.
- <math>\begin{align}t &= (70 {\rm \ years })\log_{1/2}\left(\frac{0.1 {\rm \ kg }}{20 {\rm \ kg }}\right) \\
&\approx 535 {\rm \ years }\end{align}</math>
- Remember to check your solution intuitively to see if it makes sense.
Tips
- An alternative formulation for half-life makes use of an integer base. Note that this flips the <math>N(t)</math> and <math>N_{0}</math> in the logarithm expression.
- <math>t_{1/2} = \frac{t}{\log_{2}\left(\frac{N_{0}}{N(t)}\right)}</math>
- Half-life is a probabilistic estimate of the amount of time required for half of the remaining substance to decay rather than an exact calculation. For instance, if there is only one atom left of the substance, there won't be only half an atom left after the half-life time expires, but either one or zero atoms left. The greater the amount of the remaining substance, the more accurate the half-life calculation will be due to the law of large numbers.
