Unit 1: Atomic Theory

Unit Objectives:

1. Understand the scientific method and how it is used.
2. Classify matter by state, as substance or mixture, as homogeneous or heterogeneous.
3. Know the names and symbols of the elements on the periodic table.
4. Distinguish physical changes from chemical changes of matter.
5. Know and understand the law of conservation of mass.
6. Describe Dalton's atomic theory and its significance in the study of matter.
7. Describe the history of our understanding of the structure of the atom using the following scientists: Dalton, Thomson, Millikan,Chadwick, and Rutherford.
8. Know the structure of the atom including the location and size of electrons, protons, and neutrons.
9. Know the significance and how to determine the atomic number, mass number and atomic mass of an element.
10. Know how to write the correct isotope notation for specific atoms.
11. Explain the processes of radioactivity and radioactive decay.
12. Distinguish between isotopes and radioisotopes.
13. Describe the characteristics of alpha, beta, and gamma radiation and list their origins.
14. Write equations for nuclear reactions.
15. Explain the term half-life and perform half-life calculations.
16. Explain the operation of a Geiger counter.

Unit Terms:

Lesson 1: Scientific Method

Objective

• Understand the scientific method and how it is used.

The Scientific Method

The scientific method is a way of answering questions about the world we live in.  The scientific method is a process that all scientists use to investigate the observations that they make.  The steps of the scientific method are:

1. Observation
2. Question
3. Hypothesis
4. Experiment
5. Conclusion
6. Natural Law
7. Theory

A scientist makes an observation:  "Leaves on the trees are green."  The scientist takes that observation and forms a question:  "Why are the leaves on the tree green?"  A hypothesis is a possible explanation for the answer to the question.  The hypothesis must be able to be tested through experiments.  If you are unable to create an experiment then you need to develop a different hypothesis.  A possible hypothesis to our question is: "The green color of the leaves is caused by the brown soil surrounding the trees."  Now that we have an hypothesis, we must develop an experiment.  In the case of our trees, we could grow them in black soil, brown soil, red clay, white sand, and brown sand.  The independent variable, variable that you change, is the color or type of soil.  The dependent variable, variable that changes from the experiment, is the green color of the leaves.  You would have to grow at least three trees in each type of soil in order to verify our results using statistics.  Your experiment would either prove or disprove your hypothesis.  A conclusion is the result of our experiments.  We know that our experiment would disprove our hypothesis.  The color of the soil has nothing to do with the green color of leaves, rather, the color results from a chemical called chlorophyll.  Since our experiment disproved our hypothesis, we would have to form a different hypothesis and a new set of experiments.  Scientists often spend their entire life searching for the answer to their question. 

If a set of experiments prove your hypothesis correct, then you must develop a new set of experiments that could be used to prove your hypothesis.  If all of your experiments prove your hypothesis correct, then you may be able to develop a natural law, which describes how nature behaves but does not describe why nature behaves in that particular way.

After many experiments and many different approaches to the question, the scientist may be able to develop a theory.  The theory explains why nature behaves in the way described by the natural law.  It answers not only the original question, but also any other questions that were raised during the process.  The theory also predicts the results of further experiments, which is how it is checked.  Theories are not the end of the process.  Theories can continued to be proved or they can possibly be disproved.  Theories are not facts, but our best guess as to how the world works, based on the scientific evidence we have collected.  Many theories that we will discuss during the year could be disproved as our understanding of the universe continues to expand.

Lesson 1A: Lab: Six Solution Problem

Objective:

• Determine the identity of unknown metal solutions by comparing your results to a table of solubility.

Procedure

• Complete the lab assignment, identify your unknowns. Turn in your data table and a short paragraph describing how you have identified the unknown solutions.
  
  

Lesson 2: Classifying Matter

Objective

• Classify matter by state, as substance or mixture, as homogeneous or heterogeneous.
• Know the names and symbols of the elements on the periodic table.
• Distinguish physical changes from chemical changes of matter.
• Know and understand the law of conservation of mass.

Matter

When we look at the world around us we see that it is filled with "stuff" - tables, chairs, books, soil, plants, animals, and so on.  Matter is anything that has mass and volume.  This course is primarily concerned with studying the characteristics of matter.

States of Matter

From everyday observations we know that matter exists in three states - solid, liquid and gas.  A fourth state of matter - plasma - is found inside stars.

Solid - A solid holds a particular shape and has a definite volume.  A solid has a orderly arrangement.

Liquid - A liquid does not hold its own shape but it does occupy a definite volume.  A liquid flows freely and takes the shape of its container.

Gas - A gas has no definite shape or volume.  It expands to fill the available volume of its container.

Plasma - Very hot ionized gas that is found in stars.

Properties of Matter

Physical Properties - The characteristics of a substance that can be observed without altering the identity of the substance are called physical properties.  Density, color, and melting point are examples of physical properties.

Chemical Properties - The characteristics of a substance that cannot be observed without altering the substance are called chemical properties.  Flammability, which is the tendency of a substance to burn in air, is a chemical property.

Changes in Matter

Physical Changes - Changes which do not alter the identity of the substance are called physical changes.  Crushing, tearing, and changes in state are examples of physical changes.

Chemical Changes - Changes which alter the identity of the substance are called chemical changes.  Burning, cooking, rusting are all examples of chemical changes.

Conservation of Matter

Matter, like energy, is neither created nor destroyed in any process.  Antoine Lavoisier (1743-1794) performed many experiments to prove that matter was conserved during a chemical reaction.  Lavoisier used a balance to measure the amount of matter before and after a reaction.  It was through his careful measurements that the Law of Conservation of matter came to be and it applies throughout the universe, in all branches of science.

Elements and Compounds

Early thinkers believed that the variety of substances we see around us is actually the result of combinations of just a few simple forms of matter.  These forms were called elements.  The thinkers had identified four elements: earth, wind, water and fire.  It was from these four elements that all matter on earth was made.

Elements

Our understanding of what defines a chemical element has changed radically since ancient times, but the basic concept of a fundamental kind of matter has endured.  An element is a substance that cannot be separated into simpler substances by a chemical change.  We have identified over 104 elements that make up all of the matter in the universe.

Elements are named after famous scientists, countries, states, and even planets.  Chemical elements have abbreviations, called element symbols.  Element symbols consist of one or two letters.  The first letter is always capitalized, and the second, if present, is lower case.  Some elements, such as hydrogen, have symbols straight from the English name.   

 

Names and Symbols of Selected Elements
Symbol	Element	Symbol	Element
Al	Aluminum	Mn	Manganese
Ar	Argon	Hg	Mercury (hydrargyrum)
As	Arsenic	Ne	Neon
Ba	Barium	Ni	Nickel
Be	Beryllium	N	Nitrogen
B	Boron	O	Oxygen
Br	Bromine	P	Phosphorus
Cd	Cadmium	Pt	Platinum
Ca	Calcium	K	Potassium (kalium)
C	Carbon	Ra	Radium
Cs	Cesium	Rn	Radon
Cl	Chlorine	Rb	Rubidium
Cr	Chromium	Se	Selenium
Co	Cobalt	Si	Silicon
Cu	Copper (cuprum)	Ag	Silver (argentum)
F	Fluorine	Na	Sodium (natrium)
Fr	Francium	Sr	Strontium
Au	Gold (aurum)	S	Sulfur
He	Helium	Te	Tellurium
H	Hydrogen	Th	Thorium
I	Iodine	Sn	Tin (stannum)
Fe	Iron (ferrum)	W	Tungsten (wolfram)
Kr	Krypton	U	Uranium
Pb	Lead (plumbum)	Xe	Xenon
Li	Lithium	Zn	Zinc
Mg	Magnesium

Scientists develop a table to organize the elements in a logical, orderly fashion.  This table is called the periodic table of elements and is essential to the study of chemistry.

Compounds

A compound forms from the combination of two or more elements in a fixed proportion.  The millions of compounds that exist in the universe are formed from the elements on the periodic table.  When magnesium burns in the presence of oxygen a compound called magnesium oxide forms.  Magnesium oxide is composed of 60.32 percent magnesium and 39.68 percent oxygen.  Magnesium oxide always forms in these fixed proportions.  Chemists do not usually write the names of the compounds, but write the symbols called formulas.  The chemical formula for magnesium oxide is MgO.

Distinguishing Between Elements and Compounds

Elements and compounds are pure substances.  Every pure substance has a unique set of physical and chemical properties.  Elements react to form compounds and compounds can be divided into individual elements.  Dividing a compound into elements require it to be torn apart through a process.  Electrolysis uses electricity to divide compounds into elements.  Water can be divided into hydrogen and oxygen through electrolysis.

Mixtures

A mixture is a blend of two or more pure substances.  A heterogeneous mixture is one in which the parts are clearly visible.  A piece of granite is an example of a heterogeneous mixture.  A homogeneous mixture is one in which the parts are not clearly visible.  Air is an example of a homogeneous mixture because the oxygen, nitrogen and carbon dioxide are all colorless and are indistinguishable.  To determine whether a substance is a mixture, you first have to separate it into two or more pure substances.

Separating the Components of a Mixture

Special equipment and techniques have been developed to separate mixtures into their pure substances.

1. Filtration:  A piece of paper, or other porous solid, is used to separate liquids from the solids.  The liquid part of the mixture passes through the paper, while the solids are collected on the paper.  This method is used to separate heterogeneous mixtures.
2. Distillation:  Distillation is used to separate homogeneous mixtures.  The mixture is separated based on the boiling points of the components.  The water and salt in sea water are separated by boiling the water.  The clean water is collected and the salt, which has a much lower boiling point, is left behind.  Crude oil is separated into its components by a process called fractional distillation.
3. Crystallization:  This method is used to produce solids of very high purity.  Gemstones are crystals that formed as our young planet slowly cooled.
4. Chromatography:  A solution can be separated by allowing it to flow slowly over a stationary surface.  The different components flow at different rates because they interact with the stationary surface differently.  Mixtures of gases, liquids, and solids can be separated by chromatography.  Chromatography is the most powerful tool chemists have to separate a homogeneous mixture into its pure substances.

Lesson 3: History of the Atom

Objective

• Describe Dalton's atomic theory and its significance in the study of matter.
• Describe the history of our understanding of the structure of the atom using the following scientists: Dalton, Thomson, Millikan, Chadwick, and Rutherford.
• Know the structure of the atom including the location and size of electrons, protons, and neutrons.

Pre-Boyle

The history of our theory of the atom begins in Greek about 450 B.C.  A philosopher named Leucippus began thinking about whether matter could be divided in half indefinitely.  He thought that at some point matter could not be divided any more.  A pupil of Leucippus, named Democritus, took his idea further and said that matter was made up of "atomos" or atoms which mean "unbreakable."  Epicurus (341 - 270 B.C.) took up the idea of atomism and wrote books on the subject.  These books did not survive, but a Roman named Lucretius (96 - 55 B.C.) wrote a long poem "On the Nature of Things" which described the ideas of Epicurus in great detail.  Lucretius' poem was very popular, but when Christianity became prevalent in Europe, he was deemed an atheist and his writings were lost.  One copy of Lucretius' poem was found in 1417 and was one of the first pieces of work that was widely published after the invention of the printing press.  

Post-Boyle

Robert Boyle

Robert Boyle (1627 - 1691) was influenced by the writings of Gassendi (1592 - 1655) who had written about Lucretius and atomism.  Robert Boyle did experiments to prove the atomism theory.  Boyle worked with gas pressures and explained the compressibility of gases on the existance of atoms.  If gases were made of atoms that were far apart and the gas was put under pressure causing the atoms to move close together then the volume would decrease.  Boyles experiments were hard to argue with because other scientists could repeat his experiments and make the same observations that Boyle had made.

John Dalton

A meaningful atomic theory was finally published during the period 1803-1807 by John Dalton, an English schoolteacher.  Dalton designed his theory to explain several experimental observations.  His efforts were so insightful that his theory has remained basically intact up to the present.

The essence of Dalton's atomic theory of matter is summarized in the following postulates:

1. Each element is composed of extremely small particles called atoms.
2. All atoms of a given element are identical; the atoms of different elements are different and have different properties (including different masses).
3. Atoms of an element are not changed into different types of atoms by chemical reactions; atoms are neither created nor destroyed in chemical reactions.
4. Compounds are formed when atoms of more than one element combine; a given compound always has the same relative number and kind of atoms.

Dalton's theory explains several simple laws of chemical combination that were known at the time.  One of these was the law of constant composition.  In a given compound, the relative number and kind of atoms are constant.  Another fundamental chemical law was the law of conservation of mass (also known as the law of conservation of matter):  the total mass of materials present after a chemical reaction is the same as the total mass before the reaction.

Dalton used his theory to deduce the law of multiple proportions:  If two elements A and B combine to form more than one compound, then the mass of B that can combine with a given mass of A are in the ratio of small whole numbers.  This law can be illustrated by considering the substances of water and hydrogen peroxide, both of which contain only hydrogen and oxygen.  We find that in forming water, 8.0 g of oxygen reacts with 1.0 g of hydrogen.  In hydrogen peroxide, there are 16.0g of oxygen per 1.0 g of hydrogen.  In other words, the ratio of the mass of oxygen per gram of hydrogen in the two compounds is 2:1.

Since the atom was the smallest particle it was viewed as a very small ball.  The model of the atom became known as the "Billiard Ball Model".

Joseph John (J.J.) Thomson

During the middle to late 1800's scientist began to work with a new device that was called a cathode ray tube.  A cathode ray tube, originally called a Crookes Tube in honor of its inventory, consists of a vacuum tube with two pieces of metal inside.  If an electric current was applied to the tube, a ray could be seen being emitted from the cathode and traveling the length of the tube to the anode.  The model of the atom did not change from Dalton's atomic theory until 1897 when British physicist J.J. Thomson determined that cathode-rays consisted of particles each carrying a negative charge.  Thomson called this particle an electron.  Thomson used electric fields and magnets to deflect cathode-ray particles.  He used these deflections to calculate the mass of an electron.  The mass was determined to be 1/1837 that of the smallest atoms.  The idea that there was a particle smaller that an atom changed the way that we look at the atom.  Thomson came up with a new model that would replace Dalton's billiard ball model.  Thomson thought that the atom had a positive body with negative particles dispersed throughout.  He called his model the "Plum Pudding" model.

Ernest Rutherford

In 1899, the New Zealand-born physicist Ernest Rutherford (1871 - 1937) studied the manner in which radioactive radiations penetrated sheets of aluminum.  He found that some of the radiation could be stopped by 1/500 of a centimeter of aluminum, while the rest required a considerably thicker sheet to be stopped.  Rutherford called the first type of radiation alpha rays, from the first letter of the Greek alphabet, and the second type beta rays, from the second letter.  A third type of radiation, which was the most penetrating of all, was discovered in 1900 by the French physicist Paul Ulrich Villard (1860 - 1934), and was called gamma rays, from the third letter of the Greek alphabet.

Rutherford thought that the alpha particle, which had a mass 7000 times greater than the electron, would be good for studying the structure of the atom.  In what has now become the famous Gold Foil experiment, Rutherford allowed alpha particles to bombard a very thin piece of gold foil.  He found that most of the particle went through the foil but that every now and then an alpha particle would be deflected.  Rutherford concluded from his experiment that the atom had to be mostly empty space with a very small nucleus of great mass.  Rutherford proposed that the atom now had a positive nucleus with electrons moving around the the nucleus in the empty space.  Rutherford could not account for all of the mass of an atom and thought that there must be another particle within the nucleus.

Henry Moseley

Henry Moseley (1887 - 1915), one of Rutherford's students, studied X-ray diffraction of crystals.  Moseley found that if he went up the list of elements in the periodic table, the wavelength of the X rays produced decreased regularly.  The greater the atomic weight of the atoms, the shorter the wavelength of the X rays.  The change in wavelength was more regular than the change in atomic weight.  Physicists were sure that the deceleration of electrons was brought about chiefly by the size of the positive charge on the atomic nucleus, which was an indication that the size of the charge as one went up the periodic table increased more regularly than did the mass of the atomic nucleus.  

Moseley suggested that the size of the charge increased by one with each step up the table.  Hydrogen, the first element, had as its nucleus the proton, which had a charge of +1.  Helium, the second element, had a nucleus with a charge of +2.  Lithium, the third element, had a nuclear charge of +3, and so on up to uranium, with the most massive atom then known, which had a nuclear charge of +92.

Moseley called the size of the nuclear charge the atomic number of the element, and this proved to be more fundamental than the atomic weight.

Moseley would almost certainly have been awarded a Nobel prize for his work in this regard within a few years but, in 1915, he was killed in action in World War I at Gallipoli in Turkey.

James Chadwick

The British physicist James Chadwick (1891 - 1974) discovered the neutron and was awarded the Nobel prize in 1935 for his discovery.

Lesson 4: Isotope

Objective

• Know the significance and how to determine the atomic number, mass number and atomic mass of an element.
• Know how to write the correct isotope notation for specific atoms.

Definition of an Isotope

Atoms of an element are not all the same.  All atoms of an element have the same number of protons or atomic number, but their number of neutrons can be different.  The sum of the protons and neutrons for an atom gives the atoms mass number.  The mass number is what is used to identify different atoms or isotopes.  Isotopes are atoms of the same element with different mass numbers.

In order to identify the different isotopes we use a symbol called isotope notation.  Isotope notation has four parts:  X = element symbol, A = mass number, Z = atomic number, e = charge on the atom.

A = mass number (protons + neutrons)
Z = atomic number (number of protons)
e = atomic charge (protons - electrons)

The symbol for the isotope notation looks like this:

A	X	e
Z

 

Example 1:

The isotope notation for an atom of uranium with 146 neutrons would be written as:

238	U	0
92

An alternative notation for the above isotope is to write the name or symbol of the element followed by the mass of the isotope:  Uranium-238 or U-238.  Note that if the charge on the atom is zero, then the zero is not written.  If the number of electrons is not mentioned then it is assumed to be the same as the number of protons.

Example 2:

What is the isotope notation for an atom with 11 protons, 13 neutrons, and 10 electrons.

The number of protons tells us that we have a sodium atom with a mass number of 24 (11 + 13) and a charge of +1 (11 - 10).  The isotope notation would be:

24	Na	+1
11

The alternative isotope notation cannot be written because the atomic charge is not zero.

The isotope written in example 2 is an ion.  An ion is a an atom with a positive or negative charge.  An atom with a negative charge is called an anion.  An atom with a positive charge is called a cation.

Lesson 5: Lab: The Half-life of a Skittle

Objective:

• You are going to determine the rate of decay for a Skittle and then calculate the half-life of the decay reaction.

Procedure

• Complete the lab assignment and then write a formal laboratory. You will need to research half-life, reaction rates, the rate order for a reaction, the equations that are used to determine the half-life and the rate order. The report is due 14 calendar days after the completion of the lab.
  
  
  

Lesson 6: Nuclear Chemistry

Objective

• Explain the processes of radioactivity and radioactive decay.
• Distinguish between isotopes and radioisotopes.
• Describe the characteristics of alpha, beta, and gamma radiation and list their origins.
• Write equations for nuclear reactions.
• Explain the term half-life and perform half-life calculations.
• Explain the operation of a Geiger counter.

Radioactivity

Radioactivity is the spontaneous disintegration of an unstable nucleus with accompanying emission of radiation.  When an unstable nucleus disintegrates it produces several different kinds of particles and also produces pure energy.

 

Summary of the Properties of Alpha, Beta, and Gamma

	Type of Radiation
Property	Alpha	Beta	Gamma
Charge	+2	-1	0
Mass	6.64 x 10-24 g	9.11 x 10-28 g	0
Relative Penetrating Power	1	100	1000
Nature of radiation	nuclei	Electrons	High-energy photons

 

Particles common in Radioactive Decay and Nuclear Transformations

Particle	Symbol
Neutron
Proton	OR
Electron
Alpha Particle	OR
Beta Particle	OR
Positron

 

Radioactive Decay 

Radioactive Decay means that a particle is lost or in other words a particle is produced.  In algebra x - 1 = y can be rewritten as x = y + 1.  In chemistry we use the same properties to write a chemical equation.  If Iodine-131 produces (loses) a beta particle to produce xenon-131.  The equation could be written as , but since we do not write equations with negatives, we rearrange it algebraically to get  .  Keep in mind that all chemical equations are written without the use of negatives, regardless of what occurs in the reaction.

Alpha Decay

In alpha decay, an atom will produce an alpha particle.  When you write a nuclear equation, mass must be conserved.  The total mass on the left of the arrow, and the total mass on the right of the arrow are equal.  In addition to mass being conserved, the total number of protons on the left must equal the total number of protons on the right.

Radium-222 undergoes alpha decay to produce radon-218. 

The total mass on the left is 222 and the total mass on the right is 222 (4 + 218).  The total number of protons on the left is 88 and the total number of protons on the right is 88 (2 + 86).

During alpha decay, the mass of the parent atom decreases by 4 and the atomic number decreases by 2.

 

Beta Decay

In beta decay, an atom will produce a beta particle (electron).  Please keep in mind that this is different from losing an electron to become an ion.  In beta decay a neutron becomes a proton by producing an electron.  When an atom loses an electron to become an ion, the atom is simply losing an electron from outside the nucleus.  Thorium-234 will become protactinium-234 by producing a beta particle.

During beta decay, the mass does not change but the atomic number increases by 1.

Positron Decay

In positron decay, an atom will produce a particle that has the same mass as an electron but with a positive charge.  Sodium-22 will undergo positron emission to produce neon-22.

During positron decay, the mass does not change but the atomic number decreases by 1.

 

Nuclear Transformations

It is possible, under the right conditions, for us to transform one element into another.  This is done by "slamming" a particle into a nucleus, causing the nucleus to change and therefore the identity or the mass of the atom.  Alpha particles can be used to transform one nucleus into another.  Since the nucleus of the alpha particle is positive and the nucleus of the atom being bombarded is also positive, the particles will naturally repel each other.  In order to over come this repulsion, the reaction must be performed at very high speeds (speed of light).  These speeds are achieved using particle accelerators.

Examples of Nuclear Transformations

Nitrogen can be transformed into hydrogen by combining its nucleus with an alpha particle. An atom of hydrogen (proton) is produced as part of the transformation.

Aluminum-27 can be transformed into phosphorus-30 by combining its nucleus with an alpha particle.  A neutron is produced as part of the transformation.

 

Other small nuclei, such as Carbon-12 and nitrogen-15 can be used to bombard heavier nuclei.

Synthesis of Some of the Transuranium Elements

Uranium is the heaviest natural element.  In 1940, neptunium was produced by neutron bombardment of uranium-238.  The process initially give uranium-239, which decays to neptunium-239 by the production of a beta particle.

Neutron Bombardment	neptunium (Z = 93)
	americium (Z = 95)
Positive-ion Bombardment	curium (Z = 96)
	californium (Z = 98)	OR
	rutherfordium (Z = 104)
	dubnium (Z = 105)
	seaborgium (Z = 106)

Nuclear Fission

Nuclear fission is the process of taking a large nucleus and dividing it into smaller nuclei.  Commercial power plants and nucleuar weapons depend on the fission process.  Uranium-235, Uranium-233 and plutonium-239 undergo fission when struck by a slow moving neutron.  A heavy nucleus can split in many different ways.  Two different ways that the uranium-235 nucleus splits are shown below:

The extra neutrons that are produced by the fission of uranium-235 can then cause additional uranium-235 to go through the fission process.  This is called a chain reaction.  If the reaction is allowed to proceed unchecked, then a nuclear explosion can result.  If the process is moderated or slowed down then the process can be used as a fuel source in a nuclear reactor.  In a nuclear reactor, control rods are lowered between fuel rods to slow the fission process by capturing excess neutrons. 

Half-Life

Atoms decay by transforming protons and neutrons into other particles.  This decay process is predictable in that we can determine how many atoms will decay over a given period of time, we just don't know which atoms will decay.  Half-life is the time it takes for half of the original sample of radioactive nuclei to decay.  Half-life is a statistical model.  The number of decay is not completely regular, but on average the number of decays for a particular isotope is consistent.

Calculating the Rate Constant

Let's look at thorium-234.  It has been determined that thorium-234 has a half-life of 24.1 days, or in other words, every 24.1 days half of the sample will decompose by beta emission into protactinium-234.  Suppose that we start with a 50-g sample and measure the amount of thorium-234 that is present after every 24.1 day period.  The table and chart below show the results of the measurement.

Time (t)	Amount in grams (N)
0	50.000
24.1	25.000
48.2	12.500
72.3	6.250
96.4	3.125
120.5	1.563
144.6	0.781

Notice that the graph is not linear.  It is difficult to calculate the half-life or to determine how much sample would remain after a given time.  If we take the natural log of the amount (N) and then re-plot the graph we get:

Time (t) Days	ln(N)	N
0	3.912023	50
24.1	3.218876	25
48.2	2.525729	12.5
72.3	1.832581	6.25
96.4	1.139434	3.125
120.5	0.446287	1.5625
144.6	-0.24686	0.78125

Notice that this new plot results in a linear graph.  This tells us that this reaction is a first-order reaction.  All first-order reactions result in a straight line when the natural log (amount) is plotted against time.  All half-life reactions are first order and can be plotted in this manner.  Now that we have a straight line we can write an algebraic equation that describes the linear relationship between ln(N) and time(t).   

the equation is based on the equation of a line (y = mx + b)

Nt = amount at time(t)

k = slope of the line or the rate constant

t = time

N0 = amount at time(0) (this is also the intercept of the line)

If we use the equation for a straight line and then solve for k we get:

ln[N]_t = natural log of the amount at time t.

k = rate constant

t = time interval

ln[N]₀ = natural log of the amount at time zero (initial amount).

If we use the data from the table and make the appropriate substitutions we get

Notice that the rate constant that we calculated has the same value as the slope of the line.  If we know the starting amount, the ending amount and the time interval, we can calculate the rate constant for the decay process.  We can then use this information to calculate the amount of radioactive isotope at any given time.

Determining the half-life

In order to determine the half-life (t_½) for an isotope we need to know the rate constant (k) or measure the time that it takes for exactly half of the sample to decay.  It is easier to simply collect data, plot it and calculate the rate constant than to try to determine the point at which half of the sample remains.  Let us use the information from the data collected for thorium-234.  The slope of the line for ln[N] vs time(t) was determined to be 0.0288, therefore the rate constant (k) is equal to 0.0288.  To make our calculations easier we are going to start with 1.0-g of the sample and calculate the time it takes for half of the sample to decay.  If we rearrange the equation for the line to solve for the time it takes for half of the sample to decay or t_1/2 we get:

The natural log of the starting amount N₀ (ln[1.0g]) is zero, therefore we can drop it from the equation.  

The half-life for thorium-234 is 24.1 days.  The negative from the calculation is a direction, meaning that the sample is decaying.

Radioactive Decay of U-238 to Pb-206

The radioactive isotope, uranium-238, undergoes a series of decays until all of the uranium has decayed to the stable isotope lead-206.  We can compare the amount of U-238 to the amount of Pb-206 in a rock sample to calculate the age of the sample.  The table below show the particle, the type of decay, the half-life and the rate constant for that isotope.

Isotope	Particle	Half-life		k
U-238	alpha	4.51E+09	years	1.53659E-10
Th-234	beta	24.1	days	2.87552E-02
Pa-234	beta	6.75	hours	1.02667E-01
U-234	alpha	2.48E+05	years	2.79435E-06
Th-230	alpha	8.00E+04	years	8.66250E-06
Ra-266	alpha	1.62E+03	years	4.27778E-04
Rn-222	alpha	3.82E+00	days	1.81414E-01
Po-218	alpha	3.10E+00	minutes	2.23548E-01
Pb-214	beta	2.68E+01	minutes	2.58582E-02
Bi-214	beta	1.97E+01	minutes	3.51777E-02
Po-214	alpha	1.60E-04	seconds	4.33125E+03
Pb-210	beta	2.04E+01	years	3.39706E-02
Bi-210	beta	5.00E+00	days	1.38600E-01
Po-210	alpha	1.38E+02	days	5.00723E-03
Pb-206		stable

 

Nuclear disintegration series for uranium-238.  The arrows that point left to right correspond to the loss of a beta particle, and each arrow pointing right to left corresponds to the loss of an alpha particle.

Radioactively "Hot"

Radium-222 has a half-life of 12 days.  If you have 50 million atoms of radium-222, then in 12 days 25 million would have undergone decay.  That would be an average of 2 million disintegrations per day.

Radium-226 has a half-life of 1600 years.  If you have 50 million atoms then in 1600 years or 584,000 days 25 million would have undergone decay.  That would be an average of 43 disintegrations per day.

The changes of detecting radium-222 at any given time would be 46,000 times greater then the chance of detecting radium-226.  The longer the half-life the less active the material.  Material with a very short half-life is said to be "hot".