Radioactive Dice - Background Introduction

The following pages contain a Radioactive Dice - Background summary of the core concepts at play in Introduction The following pages contain a Radioactive Dice - Background Introduction

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Keyword STRUCTURE whereby each concept gets a keyword, an Illustration concepts may be relevant to your explanation, and illustration (or example). this exercise. Some or all of these classroom objectives. The pages of this guide have a three column structure, Explanation

STRUCTURE whereby each concept gets a keyword, an Illustration this exercise. Some or all of these concepts may be relevant to your explanation, and illustration (or example). classroom objectives. The pages of this guide have a three column structure, Explanation

STRUCTURE whereby each concept gets a keyword, an Illustration concepts may be relevant to your explanation, and illustration (or example). classroom objectives. The pages of this guide have a three column structure, Explanation

STRUCTURE whereby each concept gets a keyword, an Illustration explanation, and illustration (or example).

Table of Contents Keyword Page Page Title ATOM 1 CHARGE 1 Anatomy of an Atom ELEMENT 1 ISOTOPE 1 STABILITY 2 HALF-LIFE 2 Radioactive Decay EXPONENTIAL DECAY 2 PROBABILITY 3 P(e) 3 “OR” 3 Probability Theory “NOT” 4 “AND” 4 TRANSLATION 4 LAW OF LARGE NUMBERS (LLN) 5 DECAY OF A SINGLE DIE 5 N(r) = N0·(5/6)r 5 Dice & Exponential Decay FIVE-SIXTHS-LIFE 5 CONVERTING TO HALF-LIFE 6 RADIOMETRIC DATING 6 Radioactive Dice - Background 1 Anatomy of an Atom

Atoms have a nucleus - a core - This is an atom. It has made of some number of protons and 2 protons, - neutrons, and are surrounded by a cloud e 2 neutrons, and ATOM of electrons. + o p 2 electrons. n o p+ n Electrons have a negative charge, e- protons have an equally strong positive charge, and neutrons have no charge. If an atom has an unequal number of 2×(+1) 2×(0) electrons and protons, the atom itself This atom’s total charge is + 2×(-1) has a net charge. Charged atoms and zero, so it is not an ion. 0 molecules are called ions. Oppositely

CHARGE charged ions might come together like magnets to form ionic bonds, or they might flow from one place to another in an electromagnetic field.

The number of protons an atom contains determines essentially This atom has 2 protons, so it is everything about how that atom behaves classified as the element helium (He). chemically. For this reason, we classify atoms into elements based on this ELEMENT number.

Two atoms belonging to the same element might have a different number of neutrons, resulting in slightly different This atom has 4 total protons and versions of the same element. We call neutrons, so it is classified as the these versions “isotopes.” Different isotope of helium called helium-4, isotopes of the same element behave abbreviated 4He. mostly the same way chemically, but are ISOTOPE a little heavier or lighter. Radioactive Dice - Background 2 Radioactive Decay

The number and configuration of protons and neutrons in the nucleus of an atom also determines if that atom is stable. An unstable atom is said to be radioactive, and might spontaneously undergo radioactive decay, emitting Helium-4 is considered stable radiation and transforming into a

STABILITY because it has never been observed different isotope or even a different to undergo radioactive decay. element. 3He Radioactive decay is a random process, but it is more or less likely in 3H different isotopes. Radioactive isotopes pop! can be very unstable, lasting for only ν e - fractions of a second, or they can be e only a little unstable, lasting for billions Unlike helium-4, the isotope of years. We measure this instability in hydrogen-3 is radioactive. 3H decays terms of half-life. The half-life of a into 3He with a half-life of 12.32 radioactive isotope is an interval of time years. That means that if you watch HALF-LIFE within which an atom has a 50-50 chance an atom of 3H for 12.32 years, there’s of decaying. Very unstable isotopes have a 50-50 chance you’ll see it short half-lives. Less unstable ones have spontaneously transform into 3He. long half-lives. This happens when one of its neutrons turns into a proton by Of course, we almost never kicking out an electron and an consider one atom at a time, but the electron anti-neutrino. septillions of atoms that make up “stuff.” If you watch a large number of Radiogenic (radioactively produced) radioactive atoms over time, they will 3 ex po decay according to the exponential nen He tial 3 deca decay function: a function of time and H y curve half-life. 50% of the remaining atoms remaining years 12.32 24.64 36.96 49.28 61.60 half-lives EXPONENTIAL DECAY will decay every half-life. 1 2 3 4 5 Radioactive Dice - Background 3 Probability Theory

The probability of an event is the Consider the rolling of chance that it will occur. Probability a standard die. theory is the mathematical description There are six of probability. We encounter probability possible, equally every day, and have good intuition for likely outcomes. things like coin tosses, but probability theory lets us determine the chance of The die could land

PROBABILITY more complicated events like a coin on 1, 2, 3, 4, 5, or 6. landing heads-up three times in a row.

In probability theory, the Events are denoted by variables. probability of an event “e” occurring is t: “the die lands on 3” symbolized P(e). When e is impossible, s: “the die lands on 6” we say P(e)=0. When e is certain, we say P(e)=1. When an event is neither certain, P(e) nor impossible, the value of its “P(e)” reads “the probability of e.” probability is between 0 and 1. = 1/6 = 16.6% = .16 Probabilities are often expressed as P(t) decimals, fractions, or percentages. P(s)= 1/6 = 16.6% = .16

A pair of events can be mutually t and s are mutually exclusive. exclusive, meaning they can’t happen at There is no chance they will both the same time. In these cases, the happen. combined probability of one event or “OR” the other happening is the sum of their So, P(t or s) = 1/6 + 1/6 = 1/3 individual probabilities. Radioactive Dice - Background 4 Probability Theory

The complement of an event The complement of t is comprises the set of all possible symbolized t, which you could read outcomes other than that event. The as “not-t”. It stands for the event probability of one of these outcomes “the die lands on 1, 2, 4, 5, or 6”

“NOT” occurring is 1-P(e). The complement of e could be thought of as “not-e,” and is P(not t) = P(t) = 1-P(t) = 5/6 often symbolized “ē”. A pair of events can be Imagine a series of n dice rolls. independent, meaning the occurrence of t : “The die lands on 3 on the nth roll.” one does not affect the probability of n the other. For example, the results of t1 and t2 are independent, so the series of dice rolls are independent, chance of rolling two 3’s in a row ND” because past rolls don’t affect future is:

“A rolls. In these cases, the overall probability of both events (i.e. the first

event and the second event) occurring is P(t1 and t2) = 1/6 × 1/6 = 1/36 the product of their probabilities.

Knowing only how to say “or,” a: “The die lands on 3 at least once “not,” and “and,” we can calculate in a series of three rolls.” chances that appear to be beyond our abilities to describe if we can be clever What is the value of P(a)? translators. For example, we can’t We don’t know how to directly symbolize... symbolize this, but we can do it if we manage to say “at least once” in “two coins land on the same side,” terms of “not,” “and,” and “or.” but we can symbolize... One way to do this is...

TRANSLATION “[coin 1 lands on heads and coin 2 lands P(a) = 1 - P(t and t and t ) on heads] or [coin 1 does not land on 1 2 3 heads and coin 2 does not land on which might read “The probability heads]” of a is the probability of not Which might be symbolized: rolling not-three on the first and second and third roll.” [P(h1)×P(h2)]+[(P(h1)×P(h2)] = .5 × .5 + .5 × .5 = .5 1 - (5/6 × 5/6 × 5/6) = 42% Radioactive Dice - Background 5 Dice & Exponential Decay

The Law of Large Numbers An LLN Experiment expected value (3.5) (LLN) is a principle of statistics that with a standard die running average states that while small numbers of 6 probabilistic entities like dice or 5

radioactive atoms behave unpredictably, 4 larger groups behave more predictably - 3 tending to converge as a group on an Individual Roll Result Roll Individual 2 LAW OF LARGE LAW NUMBERS (LLN) expected behavior. 1 A single “radioactive die” like the A Series of 50 Rolls one pictured to the right has a 1/6 chance of “decaying” on a given roll. In order to Consider the rolling of a this survive two rolls, it must survive one roll special die: and then another roll - an event with a It has one face probability of 5/6 × 5/6 = 25/36. marked☢, and five The chance that such a die will last blank faces. SINGLE DIE DECAY OF A DECAY “r” rolls is 5/6 × 5/6 ×.... 5/6 repeated r We will say that it times, or (5/6)r. “decays” when it A single such die behaves lands on ☢. unpredictably, but because of the LLN, r Expected Decay of a Large Number of Dice about 5/6 of a large population of N radioactive dice will survive a given roll. A 0 ∙(5/6) 0 mathematical description of this decay might look like: r N(r) = N0∙(5/6) , read: “The population (after so many rolls) is N(20) N(r) = N = N(r) the initial population multiplied by (5/6) A Series of 20 Mass Rolls once for every roll.” r “five-sixths-life” N(r) = N0∙(5/6) is an exponential in terms of rolls decay function. Functions of this same form govern the decay of radioactive rolls elapsed atoms. However, those functions tend to N(r) = N ∙(5/6)r / 1 roll be written in terms of half-life, which 0 allows us to compare, at a glance, the relative stabilities of different N(t) = N ∙(1/2)t /12.32 yr radioisotopes. Our equation, written in 0 terms of “five-sixths-lives,” is not as years elapsed half-life in

FIVE-SIXTHS-LIFE useful. We’ll need logarithms to convert it. terms of years Radioactive Dice - Background 6 Dice & Exponential Decay We want to know the half-life (in Math English terms of rolls) of a population of radioactive dice. We can see on the red (5/6)r = .5 “5/6 to the power r is .5.” graph that it takes between 3 and 4 rolls “The power that turns Log5/6.5 = r for N(r) to reach 50% of N0, but to 5/6 into .5 is r.” solve for the exact value, we need to r half-life in solve N(r)/No = (5/6) = 50% for r. Log5/6.5 = r = 3.8 terms of rolls! Basically, we are asking: “What power r would turn 5/6 into 1/2?” By solving for r, we find that half-life So, when 1 roll N(r)/No = 50% when r = 3.8. In other of 3H words, the half-life of a six-sided die is represents 3.24 12.32 yr. 3.24 yr. years elapsing, a 3.8 rolls. Now, by coming up with a = fictitious conversion factor between 3.8 rolls 1 roll six-sided die behaves just like

CONVERTING TO HALF-LIFE CONVERTING rolls and years, you can produce half-life of a an atom of 3H! populations of radioactive dice that six-sided die behave just like any given radioisotope.

r / 3.8 rolls Now that we are capable of setting Consider N(r)/No = (1/2) up an environment in which a group of

dice faithfully represents a group of / 3.8 rolls 14/(36+14) = (1/2)r radioactive atoms, we can do more than

just watch those dice decay. We can r / 3.8 rolls examine a partially decayed group and .28 = (1/2) determine its “age” based on how many Log .28= r / 3.8 and (1/2) dice have decayed and how many dice remain. This rearranging of the ≈ 7 rolls old, or exponential decay function is the basis r for . 7×3.24 = 22.68 years old

RADIOMETRIC DATING radiometric dating