Half Life: How Rapid the Approach to Infinity



Decay Relationship
Relevant Questions and Implications
Alternative Equation
There are many phenomena where the likelihood of a future situation depends on current properties.  The likelihood of a forest fire depends partly on the dryness of the forest.  And the amount of aspirin still in you blood stream an hour later depends on the amount now present. If the relationship is known, the future likelihood can be anticipated and appropriate plans made.

The simplest relationship is one that doesn’t change.  But a very common relationship has the likelihood of occurrence diminishing as some key concentration dwindles.  This is the relationship that governs the effect as a medication wears off.  It is also the relationship of radioactive material as they age.  The amount of radiation emitted depends on the amount of material present, and this declines with time (unless more material is added).

It is common to ask “how long will the benefit or danger last.”  But as we shall see, this is a question poorly suited for such relationships.  It asks a question of important concern, but fails to provide enough information to allow an answer to be formulated.  While the limited purpose of this page will be to understand radioactive decay, a broader goal will be to better understand cause and effect relationships so that we might pose questions about those relationships that supply information we find useful.


Improvements in vacuum pumps in the middle of the 19th Century made it possible to study the nature of electric sparks generated through near vacuums.  (These studies lead to the discovery of the electron.)  Sometimes cards coated with materials that fluoresced when struck with sparks were used to track a spark's path.  In November 1895 Wilhelm Röntgen noticed such a coated card glowing some distance  OUTSIDE a glass vacuum container when a spark passed inside.  Believing it impossible for the spark to pass through the glass and travel a yard's distance through the air to the card, Röntgen temporarily labeled the unknown cause as “X rays.”  That name remains today although Röntgen soon identified most of the properties of the X rays.

A few months later Henri Becquerel in Paris began to investigate if the phenomena, like many in nature, might be reversible.  (Such symmetries are a major key used by physicists to understand the universe.  And Emmy Noether showed that for each symmetry, there is a conservation law.)  Becquerel placed a mineral called pitchblende on a window sill to instead let the sun's ultraviolet light cause fluorescence.  He speculated that X rays might accompany the fluorescence.  He hoped to detect any X rays on an underlying photographic plate wrapped in black paper as protection it from the sun’s radiation.  But rainy weather common to Paris in February immediately caused him to suspend the exposure and place the mineral and wrapped plate in a drawer until the sun reappeared.  March 1st 1896, motivated by the brief sensitivity of the photographic emulsion on the glass and no sign of the rain ending, Becquerel developed the emulsion hoping for a faint image that might indicate that the experiment should be repeated when weather improved.  Surprised by a strong fogging of the emulsion, Becquerel confirmed that some kind of radiation was emitted by pitchblende even when it did NOT fluoresce.

Becquerel confirmed that the heaviest known element, uranium (pitchblende is about 80% U3O8), and all uranium’s compounds emitted such radiation.  A colleague at the Sorbonne, Pierre Curie developed equipment for measuring the radiation’s intensity.  Curie’s wife, Marie, used his electrometer to test most of the known elements, but found that thorium, the next heaviest known element, and its compounds were the only other emitters of radiation.  Then Marie found that pitchblende was four to five times as “radioactive” (a term she coined) as the uranium it contained.  Marie, an unemployed chemist, labored in a shed at her home to isolate several additional elements that were determined by spectra to be different from any previously known.  These they named Polonium (to honor Marie’s native country then dominated by Russia) and Radium (which glowed in the dark).

In England, Sir Ernest Rutherford distinguished by differences in penetration two types of radiation which he labeled by the Greek letters a (alpha) and b (beta).  P. Villard found more penetrating radiation hence called g (gamma).

The Curies found that the radioactivity of a sample depended on the quantity of radioactive atoms a sample contained.  Federick Soddy proposed that alpha and beta emission occurred when atoms split off the radiation, becoming smaller atoms.  (Gamma radiation is only energy.)  The energy for all kinds of radioactive decay comes from mass losses as given by Albert Einstein's famous E = mc2.

Decay Relationship:

There are several dificulties understanding, measuring, and predicting radioactive decay.  The decay of any particular atoms seems totally random and therfore impossible to predict.  (A number of nature's phenomena have similar difficulties.)  However large groups of a given element decay at a steady rate.  The solution is to use the mathematics from measurements of large numbers as probabilities for individual events.

A second problem is that the decay or large numbers of atoms requires an infinite period of time, a number difficult to measure, compare, and compute.   The solution is to measure not the period required for total decay, but the period required for say half of the atoms to decay.  Then the rate at which each radioactive element decays (and approaches zero) is both measurable and characteristic.

As a result of this technique, the material remaining, N, after a “decay” is related to the initial number, No, decrease by reduced by 1/2 to an exponential power: the equation is

N = No(1/2)p
 where p = t/t1/2
That exponent, p, is determined by the time that transpires, t, divided by the time required to reduce the number in half (called the half life), t1/2.1  Any consistent set of units is acceptable.  For example, N could be a number in dozens, moles, or even Molar concentration.


Determine how much of 100g of carbon 14 would be would be left after 20,000 years if about 5000 years is needed to reduce the amount in half.


p = t/t1/2
p = 20,000yr/5000yr
p = 4 1

N = No (1/2)p
N = 100g (1/2)4
N = 100g (1/16)
N = 6g

Of the original 100g of carbon-14, only 6% remains after 20,000 years.  94% has decayed to nitrogen-14.  Such information is valuable in determining how long ago a fossil was alive.

Relevant Questions and Implications:

The remaining amount is reduced in half every "half life."  It should be noted that while the amount remaining reduces with time, it NEVER reaches zero.  Reaching zero takes to infinity.  But half life is a way of using finite numbers to describe how fast zero is approached.

As a corollary, it is meaningless to ask when a material will be entirely gone.  (Opponents to nuclear power sometimes express the fear that radioactive wastes remain forever.)  Such a concern fails to recognize that benefits or risks decline even though they haven’t totally decayed.  For example, pain medication in a body will decay in a few hours to levels that are inadequate for treating pain even though some small amount remains.  A similar decline occurs in the risks due to decaying hazards both chemical and nuclear.  (To reject nuclear power because of the infinite time required for complete decay of wastes is akin to refusing a person in pain more medication because the previous medication is not totally gone.)  The important but sometimes difficult question is whether the needs and benefits are greater than the harm and risks.

Alternate Equation:

Sometimes the rate of change is easier information to assimilate than the amount of material present.  For example, it is difficult to comprehend the hazard of 100 grams of uranium.  It is easier to appreciate the hazard of the amount of radiation the 100 grams of uranium will emit each minute.  Using the calculus tool of differential equations to transform the equation above, rate of change, R, can be describes as
R = 0.693 N / t1/2
where the N is the number present, t1/2 is the half life, and the 0.693 is a constant that comes from the exponential nature of the original function.


How much radiation should be expected from 100g of uranium 238?

Using Avogadro’s number and the atomic mass of uranium 238, the number of uranium atoms can be determined:

moles = mass/molecular mass
moles = 100g/238g/mole
moles = 0.42mole =0.42mole x 6.02 x 1023atoms/mole = 2.5 x 1023atoms

t1/2 = 4.5 x 109yr = 4.5 x 109yr x 365day/yr x 24hr/day x 60min/hr = 2.4 x 1015min

R = 0.693 N / t1/2
R = 0.693 (2.5 x 1023atoms) / 2.4 x 1015min
R = 4.2 x 108atoms/min 2


1 Calculations of the number remaining, N, can be determined for any transpired time.  Calculations with a exponential power, p,  that is not an integer can be done easily with hand held calculators, but is moderately difficult to do "in one's head!"

2 A half billion radiation emissions per minute at first seems hazardous.  But uranium decays primarily by alpha emission which cannot penetrate out of the uranium!  Only alpha rays emitted outwards by surface atoms escape the metal.  So the risk due to alpha radiation is very, very low compared to the total R calculated above.  However 23% of the energy is released in the form of low energy (0.05MeV) gamma rays which are more penetrating, but also largely absorbed inside the metal.  Uranium 238  also may spontaneously fission, but even more rarely.  Because very little radiation is actually emitted, uranium is generally considered safe for humans to handle without any shielding.


back to the mathematics menu
© by D.Trapp