Nuclear Chemistry

Curie

(symbol Ci) is a non-SI unit of radioactivity originally defined in 1910. Older but still often used. 3.7 * 10^(10) disintegrations per second, which is the rate of decay of 1g of radium. Thus a 4.0 mCi sample of cobalt-60 undergoes 4.0 * 10^(-3) Ci * ((3.7 * 10^(10) disintegrations/s)/1 Ci) = 1.5 * 10^(8) disintegrations/s = 1.5 * 10^(8) Bq

Relative Biological Effectiveness (RBE)

1 rad of alpha radiation is more dangerous than 1 rad of beta radiation. To correct for these differences radiation absorption is multiplied by the RBE to give the effective dose. For gamma and x-rays the RBE is approximately 1, for alpha rays it is 10 (I assume if ingested). RBE varies with dose rate, total dose, and type of tissue affected.

Radioactive Decay Rules

1. Nuclei above the belt of stability (high neutron to proton ratios). These neutron rich nuclei can lower their ratio and move to the belt of stability by emitting a beta particle. This decreases the number of neutrons and increases the number of protons. 2. Nuclei below the belt of stability (low neutron to proton ratios). These proton rich nuclei move to the belt of stability through positron emission or electron capture because both decays increase the number of neutrons and decrease the number of protons. 3. Nuclei with atomic numbers >= 84. These heavy nuclei tend to undergo alpha decay, which decreases the number of protons and neutrons moving it closer to the belt of stability. There are a few exceptions to these rules.

More Observations to Help Predict Stable Nuclei

1. Nuclei with magic numbers of protons or neutrons are generally more stable than those that do not contain these number of nucleons. Magic numbers are 2, 8, 20, 28, 50, or 82 protons and 2, 8, 20, 28, 50, 82, or 126 neutrons 2. Nuclei with even numbers of protons, neutrons, or both are more likely to be stable than those with odd numbers of protons and/or neutrons. Approximately 60% of stable nuclei have an even number of both protons and neutrons, whereas less than 2% have odd numbers of both These observations can be understood in terms of the shell model of the nucleus, in which nucleons are described as residing in shells analogous to the shell structure for electrons in atoms. Just as certain numbers of electrons correspond to stable filled-shell electron configurations, so too do certain numbers (known as magic numbers) of nucleons represent filled shells in nuclei.

Nuclear Equations Examples

1. Write the nuclear equations for a) Mercury-201 undergoing electron capture. b) thorium-231 decaying to protactinium-231 a) Hg(201, 80) + e(0, -1) → Au(201, 79) b) Th(231, 90) → Pa(231, 91) + B⁻

Mass of Neutron

1.00866 amu

Mass of a Proton

1.67 × 10⁻²⁷ kg 1.0073 amu

Speed of Light

2.9979 * 10^(8) m/s

Clinical Effects of Radiation Exposure

600 rem is fatal for humans. To put this in perspective a typical dental x-ray leads to a 0.5 mrem exposure. Average exposure in a year to ionizing radiation is about 360 mrem.

Mass of an Electron

9.11x10⁻³¹ kg 5.4858 * 10^(-4) amu

Alpha Particle

A Helium-4 nucleus that is emitted from a radioactive nucleus. A cluster of 2 protons and 2 neutrons.

Radioactive Decay Example 3

A freshly prepared sample of curium-243 undergoes 3312 disintegrations per second. After 4.00 years, the activity of the sample declines to 2930 disintegrations per second. The half life of curium-243 is? Substitute initial disintegrations per second for N₀ and disintegrations per second after time t for N. Then use the formula: N(t) = N₀e^(-kt) to find k. After we find k we can use the formula tsub(1/2) = -ln(1/2)/k to find the half life. 2930 = 3312*e^(-k*4.00 years) 2930/3312 = e^(-k*4.00 years) ln(2930/3312) = -k*4.00 years -(ln(2930/3312)/4.00) = k k = 0.0306 yrs⁻¹ Now tsub(1/2) = -ln(1/2)/0.0306 yrs⁻¹ = 22.6 years

Other Types of Reactors

A heavy water reactor uses deuterium as moderator and coolant. A gas cooled reactor uses gas, typically CO2, as primary coolant and graphite as moderator. Deuterium and graphite as moderator absorb more neutrons meaning the uranium does not need to be as enriched.

Nuclear Transmutations

A nuclear reaction that occurs when a nucleus is hit by a neutron or another nucleus, rather than occurring spontaneously to achieve the belt of stability.

Boiling Water Reactor

A nuclear reactor that turns the primary coolant into steam that turns a turbine, instead of transferring the heat to a secondary coolant first.

Fast Breeder Reactor

A reactor that could reduce radioactive waste. "Breeds" more fissionable material than it consumes. Operates without a moderator, which means the neutrons are not slowed down. In order to catch the fast neutrons the fuel must be heavily enriched with Uranium-235 and Plutonium-239. Water cannot be used because it would moderate the neutrons, so a liquid metal sodium is used. The core is surrounded by Uranium-238 (which is not a fissile material) that catches the neutrons and produces Plutonium-239 as a result. The Plutonium can then be processed and used as fissionable material. The waste is less radioactive than other reactors but these reactors have higher operational costs, and incur more nuclear proliferation concerns because of all the plutonium-239 created.

Radiometric Dating Example

A rock contains 0.257 mg of lead-206 for every milligram of Uranium-238. The half life of the decay of Uranium-238 to lead-206 is 4.5 * 10^(9) year. How old is the rock? tsub(1/2) = -ln(1/2)/k k = -ln(1/2)/4.5 * 10^(9) k = 1.5403271 *10^(-10) Remember: ln(N/N₀) = -kt We must consider the mass number when calculating the original mass of Uranium. 0.257 mg of lead-206/1.000 mg of Uranium-238. Therefore, before any Uranium-238 decayed there was 1.000 + (238/206)*0.257 mg = 1.297 mg of Uranium-238 Thus: ln(1.000 mg/1.297) = -kt We get: t = -ln(1.000/1.297)/(1.5403271 *10^(-10)) = 1688303122.98 years = 1,688,303,122.98 years which is 1 billion, 688 million, 303 thousand, 122.98 years.

Radiocarbon Dating Example

A wooden object from an archaeological site is subjected to radiocarbon dating. The activity due to C(14, 6) is measured to be 11.6 disintegrations per second. The activity of a carbon sample of equal mass from fresh wood is 15.2 disintegrations per second. The half life of carbon-14 is 5700 years. What is the age of the archaeological sample. ln(N/N₀) = -kt Since fresh wood has brand new carbon-14 it is the initial disintegrations per second. Since the carbon from the wood from the archaeological site is old it is the carbon at time t. Thus: ln(11.6/15.2) = -kt Now we calculate k from the half life formula. tsub(1/2) = -ln(1/2)/k k = -ln(1/2)/tsub(1/2) = -ln(1/2)/5700 yrs k = 0.000121605 yrs⁻¹ Now we substitute k in: ln(11.6/15.2) = -0.000121605 yrs⁻¹ *t yrs ln(11.6/15.2)/-0.000121605 yrs⁻¹ = 2222.69 years

Three Most Common Type of Radioactive Emission

Alpha (α), Beta (β), and Gamma (γ) radiation

Accelerating Charged Particles

Alpha particles and other positively charged particles must travel very fast to overcome the electrostatic repulsion between it and another atomic nucleus (neutrons do not need to be accelerated). The higher the charge on the bombarding particle or the target nucleus, the harder it is to make it happen. Particle accelerators accelerate the bombarding particles to make this happen. In a particle accelerated charged particles are manipulated in vacuum by electric and magnetic fields to accelerate the particle.

Mass Defect Example

An alpha particle He(4,2) has a nucleus with a mass of 4.00150 amu. The mass of 2 neutrons and 2 protons is 2(1.00728 amu) + 2(1.00866 amu) = 4.03188 amu. This is more than the mass of a He(4,2) nucleus. This change in mass can be used to calculate the nuclear binding energy, how much energy must be input to separate the protons and neutrons in the nucleus. The mass difference is 4.03188 amu - 4.00150 amu = 0.03038 amu Now using Einstein's equation E = ((2.9979 * 10^(8) m/s)^2)*(0.03038 amu) * (1g/6.022 * 10^(23) amu) * (1 kg/1000g) = 4.53 * 10^(-12) J This is the energy required to separate the protons and neutrons from a single He(4,2) atom.

Electron Capture

An electron is captured by the nucleus (reactants), which causes it to convert a proton into a neutron (product) like in positron emission. The difference in a nuclear equation is that it is on the reactant sides not product side of the equation and it has a -1 charge not a +1 charge.

Positron Emission

Another type of radioactive emission. A positron is written in nuclear equations as e(0, +1) or as β⁺. It is a particle that has the same mass as an electron but a positive charge.

Radioisotopes

Atoms containing radionuclide

Isotope

Atoms with the same atomic number but different mass numbers

Radiometric Dating

Because the half life of any particular nuclide is constant, the half life can serve as a nuclear clock to date objects.

Positron Emission Example

C(11, 6) → B(11, 5) + e(0, +1) With a positron emission the atomic number of the element emitting the positron decreases by 1 (the opposite of beta emission), and the mass number stays the same. Equivalent to a proton being converted into a neutron

Use of Radioisotopes

Can help determine the steps in a chemical reaction. Because isotopes have almost identical chemical properties to the typical elements, they go through the same reactions but can be followed by detecting which products are radioactive. When radioisotopes are used to follow the path of an element it is called a radiotracer.

Radiation Therapy Cancer

Cancer cells are actually more susceptible to damage from radiation than healthy cells (even though cancer cells are often caused by radiation). As a result radiation (typically gamma rays) can be used to treat cancer. Radionuclides are often packaged in platinum to block the alpha and beta rays. The gamma rays, however, will penetrate the platinum. A pellet with a radionuclide inside platinum can be surgically inserted inside of a tumor. In some cases the radionuclide can be ingested; for example, thyroid cancer can be treated by ingesting large quantities of iodine-131. Radiation therapy in deep organs where an implant is impractical often uses a cobalt-60 gun to shoot gamma rays at the organ. Particle accelerators are also used. The problem with it is that the radiation also damages healthy cells causing side effects, sometimes dangerous ones. Research right now is focused on neutron capture therapy. In this treatment a non-radioactive isotope is concentrated in the tumor, such as boron-10. Then it is irradiated with neutrons producing alpha particles. The tumor is damaged by the alpha particles but tissue further away from the tumor is unaffected because of the weak penetrating power of alpha particles. This treatment promises to selectively target cancer cells without irradiating to many healthy cells.

Gamma (γ) Emission

Charge: 0 Mass: 0 Relative Penetrating Power: 10,000 Nature: High energy photons

Beta (β) Emission

Charge: 1- Mass: 9.11 * 10^(-28) Relative Penetrating Power: 100 Nature: Electrons Represented in nuclear equations by e(0, -1) or β⁻

Alpha (α) Emission

Charge: 2+ Mass: 6.64 * 10^(-24) g Relative Penetrating Power: 1 Nature of Radiation: He(4, 2) particle

Positron Emission Tomography (PET)

Compounds containing radionuclides that decay by emitting positrons are injected into the patient. Allows researchers to monitor blood flow, oxygen and glucose metabolic rates, and other biological functions. As an example, PET can be used to monitor glucose metabolism in the brain. Carbon-11 (half life 20.4 min), fluorine-18 (half life 118 min), oxygen-15 (half life 2 min), and nitrogen-13 (half life 10 min). Glucose, for example, is labeled with carbon-11. Because the half lives of these isotopes are so short, the radioisotopes must be generated using a cyclotron onsite, incorporated into the appropriate molecule (e.g., sugar), quickly, and injected immediately into the patient. When the isotope decays, it emits a positron which collides with an electron in the PET device. The positron and electron are annihilated producing gamma rays that are detected using a scintillation counter.

Rad

Corresponds to a radiation absorption of 1 * 10^(-2) J/kG. 1 gray = 100 rads

Scintillation Counter

Detects and counts light flashes when radiation passes through a phosphor.

Einstein's mass/energy equation

E = mc² Mass and energy are equivalent and can be converted into one another. When a system gains mass, it gains energy and when a system loses mass it loses energy

Leukemia

Excessive growth of white blood cells. Major type of radiation caused cancer.

Nuclear Waste

Fission products accumulate as a reactor operates, decreasing the efficiency of the reactor. Periodically, reactors must be stopped to replace or reprocess the nuclear fuel. It was intended that the spent fuel be kept on site for a few months to let short lived radioactive nuclei decay. Then be moved to a reprocessing plant in a shielded container where the unspent fuel would be removed from the products. This does not occur in the United States, where the fuel stays on the reactor site in containers because of public opposition to nuclear waste on roads and because of operational difficulties with reprocessing plants. Reprocessing does occur, however, in France , Russia, The United Kingdom, India, and Japan. Because of the half lives of longer lived radioactive products like Strontium-90 with a half life of 28.8 years. It is estimated that it will take 300 years for the reprocessed waste to decay to acceptable levels of radioactivity. If it is not reprocessed, because of Plutonium-239, which can be used as reactor fuel, it will take 24,000 years.

Nuclear Reactor Components

Fuel elements (typically Uranium-235), control rods, a moderator, and a primary coolant. Because the isotopic abundance of Uranium-235 is very low (about 0.7%) the Uranium must be enriched to reach 3-5% for use in a nuclear reactor. The fuel elements contain enriched Uranium in the form of UO₂ pellets encased in zirconium or stainless steel tubes. The control rods are composed of materials that absorb neutrons such as boron-10 or an alloy of silver, indium and cadmium. These rods regulate the flux of neutrons to keep the reaction chain self sustaining and prevent overheating. The moderator slows down the neutrons, produced by the fission reaction. These typically are ejected during fission at about 10,000 km/s. The moderator slows the neutrons down to a couple km/s. The moderator is typically water or graphite. The primary coolant is a substance that transfers the heat generated by nuclear chain reaction away from the reactor core. In a pressurized water reactor, which is the most common design, water acts both as the primary coolant and moderator. The heat from the primary coolant is transferred to a secondary coolant which turns into high pressure steam that is used to drive a turbine. Additionally a containment shell made of concrete shields personnel from radiation produced in the reactor.

Gamma Emission Example

Gamma rays are typically not written into nuclear equations. They are given off when the nucleons in a radionuclide arrange themselves into more stable arrangements. They do not change the mass number of atomic number.

Mass Energy Equivalence Example 1

Given U(238, 92) → Th(234, 90) + He(4, 2) The masses for the nuclei are U(238, 92) = 238.0003 amu, Th(234, 90) = 233.9942 amu, He(4,2) = 4.0015. We can now calculate the mass and energy change: 233.9942 + 4.0015 - 238.0003 = -0.0046 amu For 1 mole of Uranium-238, we get -0.0046 grams of lost mass. This mass is lost as energy. We can calculate how much energy: E = mc² E = -0.0046g * (1 kg/1000 g) * (2.9979 * 10^(8) m/s) * (2.9979 * 10^(8) m/s) = -413420602860 kg * m^2/s^2 = -4.1 * 10^(11) J This is a massive amount of energy released.

Mass Energy Equivalence Example 2

How much energy is lost or gained when 1 mol of cobalt-60 undergoes beta decay, Co(60, 27) → Ni(60, 28) + e(0, -1). The mass of Co(60, 27) is 59.933819 amu, the mass of Ni(60, 28) is 59.930788 amu. The mass of an electron is 5.4858 * 10^(-4) amu. Since Cobalt-60 has an atomic number of 27 it has 27 protons and 27 electrons. Since Ni(60, 28) has an atomic number of 28 it has 28 protons and 28 electrons. We can calculate the mass of the nuclei of both elements by taking the atomic mass of both and subtracting the mass of the electrons out. Nucleus of Co(60,27) = 59.933819 amu - 27*5.4858 * 10^(-4) amu = 59.91900734 amu Nucleus of Ni(60, 28) = 59.930788 amu - 28*5.4858 * 10^(-4) amu = 59.91542776 amu Now we can calculate the mass change: Mass of an Electron + Mass of Ni(60,28) Nucleus - Mass of Co(60,27) Nucleus 5.4858 * 10^(-4) amu + 59.91542776 amu - 59.91900734 amu = -0.003031 amu If we have 1 mole of Cobalt-60 that has decayed then we have -0.003031g of mass lost in the form of energy. We first convert to kilograms. -0.000003031 kg. E = mc² So E = ((2.9979 * 10^(8) m/s)^2) * -0.000003031 kg = -272408227667 J = -2.724 * 10^(11) J

Mass Energy Equivalence Steps

Identify the masses of the nuclei. If the masses of the nuclei are given we add the masses of the nuclei of the products and subtract by the masses of the nuclei of the reactants. If the masses of the atoms are given (not the nuclei) we must subtract by the mass of the number of electrons each atom has. There will be the same number of electrons as the atomic number (number of protons). Then we use the calculated mass of nuclei. For Alpha decay the products are the new element and a He(4,2) nuclei. For Beta decay the products are the new element and an electron. Given the number of moles of reactants (the radioactive substance that decays), we take the lost mass in amu and convert it to grams. Then we convert it to kilograms. Finally we use E = mc² to find the energy released.

Supercritical Mass

If more than a critical mass is present, very few neutrons escape and this multiplies the chain reaction, which can cause a nuclear explosion. This is called a supercritical mass.

Ernest Rutherford First Nuclear Transmutation

In 1919 used alpha particles emitted by radium to convert nitrogen-14 into oxygen-17. N(14, 7) + He(4,2) → O(17, 8) + H(1,1) Can also be written as N(14, 7) + α → O(17, 8) + p Shorthand for nuclear transmutations can be seen in the image. On the left is the nucleus being bombarded and the particle it is being bombarded with. On the right is the nucleus that forms from the bombardment and the emission from the nucleus that was initially bombarded.

Cyclotron

In a cyclotron charged particles move in a spiral path with two D-shaped electrodes. Alternating charges on the electrodes accelerates the particles. While magnets above and below constrain the path of the particles to a spiral path of increasing radius.

Linear Accelerator

In a linear accelerator charged particles are accelerated through a series of tubes. The electrical charge on the tube is changed from positive to negative so that the particle is always attracted to the tube it is approaching and repelled by the tube it is leaving.

Synchrotron

In a synchrotron the same methods as in a cyclotron are used except the magnetic fields are synchronized such that the particles are moved in a circular path instead of a spiral path.

Radiocarbon Dating

In radiocarbon dating, the half life of carbon-14 is used for radiometric dating. Carbon-14 is formed by neutrons created by cosmic rays (mostly protons) hitting the upper atmosphere. These neutrons bombard nitrogen-14 producing carbon-14 in the nuclear reaction: N(14, 7) + n→ C(14, 6) + p The carbon-14 atoms then react with oxygen in the upper atmosphere to form CO₂ which is then taken up by plants and introduced into the food chain via photosynthesis. Because carbon-14 is radioactive it decays through beta emissions with a half life of 5700 years given by the nuclear equation: C(14, 6) → N(14, 7) + β⁻ Because a living plant or animal has a constant intake of carbon compounds it has a carbon-14/carbon-12 ratio almost identical to that of the atmosphere while its alive. When it dies, however, it stops taking in new carbon. This means that the carbon-14 which is radioactive begins to decay into nitrogen. Carbon-12, however, which is stable does not decay. This means that we can look at its carbon-14/carbon-12 ratio and see how much the carbon-14 has decayed and thus how old it is by comparing the ratio to the current ratio in the atmosphere. If the ratio has diminished by half, the organism is as old as the half life of carbon-14 about 5700 years. This method cannot be used to date objects older than 50,000 years old because the radioactivity becomes too old to be measured.

Nuclear Synthesis of the Elements

In the early universe the elements hydrogen and helium and very small amounts of Lithium and beryllium and were formed. All the heavier elements were formed as a result of nuclear fusion in stars. Starts go through three phases: hydrogen burning, Helium burning, and advanced burning. This explains the relative abundance of different elements in the universe.

Binding Energies Per Nucleon and Elemental Stability

Increasing the binding energy per nucleon increases the stability of an element. Intermediate mass numbers are the most stable, having the largest binding energy per nucleon. Low and high mass numbers have the least binding energy per nucleon. The graph attached shows the magnitude of these trends. Heavy nuclei gain stability and therefore give off energy when split into two midsized nuclei. This is fission. Second because of the sharp increase in the graph for light nuclei, if light nuclei are combined they release even more energy. This is fusion.

Beta (β) Emission Example

Iodine-131 emits beta particles as it decays. I(131, 53) → Xe(131,54) + β⁻ Beta decay causes the atomic number of the nucleus to increase by 1. The mass number stays the same Beta emission is equivalent to a neutron being converted into a proton.

Advanced Burning

Most stars cool and dim after converting their Helium to carbon and oxygen and become white dwarfs, a phase in which stars become incredibly dense and hot fusing the elements from neon to sulfur. Eventually progressively heavier elements form at the core until it becomes predominantly iron-56. Because this is such a stable nucleus fusing further requires more energy than it consumes causing the fusion reactions that power the star to dissipate. Intense gravitational forces then collapse the star leading to a dramatic explosion called a supernova. Neutron capture coupled with subsequent radioactive decays in the dying moments of the star are responsible for all elements heavier than Iron and Nickel.

Ionizing Radiation and Living Tissue

Most tissue is mostly (70%) water by mass. Thus radiation that ionizes water is ionizing radiation. This process requires a minimum energy of -1216 kJ/mol. Alpha, beta particles, gamma rays, x-rays, and higher energy ultraviolet light possess these energies. When ionizing radiation passes through living tissue, electrons are removed from water forming highly reactive H₂O⁺ molecules. The water molecule then reacts with another water molecule to form H₃O⁺ and a neutral OH. The neutral OH is highly unstable and reactive. Substances with one or more unpaired electrons are called free radicals. This specific free radical is called a hydroxyl radical. These hydroxyl radicals can then attack biological tissues and form more free radicals damaging a large number of cells. Most alpha rays are stopped by skin. Beta rays only go 1 cm deep. X-rays and gamma rays, on the other hand, penetrate the whole body. This makes X-rays and gamma rays more damaging than alpha and beta rays unless the alpha and beta rays are somehow transferred inside the body. Alpha rays inside the body (like after ingesting a radioactive source) are the most damaging. The tissues damaged most are ones that reproduce most rapidly such as bone marrow, blood forming tissues, and lymph nodes. The principle risk of ionizing radiation is cancer.

Belt of Stability

Narrow range of where all stable nuclei exist (N/Z ratio). If outside of this range, the element is unstable and undergoes radioactive decay.

Neutron to Proton Ratio

Neutrons/Protons ratio or n/p ratio. There are no neutrons in a hydrogen atom. For the second (Helium) rest of the first 20 elements, there are about equal numbers of neutrons and protons. After 20, however, ever increasing numbers of neutrons are needed to counteract the repulsive force between protons. All nuclei with 84 or more protons are radioactive

Tokamak

No known structural material is able to withstand the high temperatures involved in fusion. To contain the heat of the reaction, a tokamak is used which uses strong magnetic fields to contain the heat. Temperatures of up to 100,000,000 K have been achieved in a tokamak; however, so far scientists have been unable to generate more power than is consumed over a sustained period of time.

Nuclear Transmutation using Neutrons

No need for acceleration because neutrons are not charged particles and will not be repelled by atomic nuclei. Neutrons are produced in nuclear reactors for use in medical and scientific experiments. For example, Cobalt-60 which is used in cancer radiation therapy is produced by neutron capture. Iron-58 is placed in a nuclear reactor and bombarded with neutrons to trigger the following reactions: Fe(58, 26) + n(1,0) → Fe(59, 26) Fe(59, 26) → Co(59, 27) + e(0, -1) Co(59, 27) + n(1,0) → Co(60, 27)

Helium Burning

Once a stars supply of hydrogen is exhausted it begins to fuse helium. During this phase it becomes a red giant. The decrease in nuclear fusion causes the core to contract, triggering an increase in core temperature and pressure. At the same time the outer regions cool and begin to expand. As the star uses its Helium fuel, a common reaction occurs in which two He(4,2) nuclei fuse to form a Be(8,4) nucleus. This is a slightly endothermic reaction, consuming more energy than it gives off (which eventually causes the star to cool). The Be(8,4) nucleus is very unstable and quickly decays. In a small percentage of cases, however, a third He(4,2) nucleus collides with it to form carbon-12. Some of the C(12,6) particles collide with He(4,2) again to form O(16,8) oxygen-16. This is why there is so much carbon and oxygen in the Universe compared to Lithium and Boron even though those are lighter elements. Nitrogen is relatively abundant because it can emerge from a series of nuclear reactions involving proton capture and positron emission.

Other types of Radiometric Dating

Other isotopes can be similarly used to date other types of objects. For example, it takes 4.5 * 10^(9) years for half of a sample of uranium-238 to decay to lead-206. The age of rocks containing uranium can therefore be determined by measuring the ratio of lead-206 to uranium-238. If the lead-206 had somehow become incorporated into the rock by normal chemical processes instead of by radioactive decay, the rock would also contain large amounts of the more abundant isotope lead-208. In the absence of large amounts of this "geonormal" isotope of lead, it is assumed that all of the lead-206 was at one time uranium-238. This is how the age of the earth is estimated. The oldest rocks on earth are about 3 * 10^(9) years old. This indicates that the earths crust has been solid for at least this length of time. Incorporating estimates of how long it took the earth to cool, the earth is like between 4.0 * 10^(9) and 4.5 * 10^9 years old (4 and 4.5 billion years old)

Radioactive Decay Rules Examples

Predict the mode of decay of a) carbon-14 and b) xenon-118 a) C(14, 6) has a neutron to proton ratio of (14-6/6) = (8/6) = 4/3. With the first 20 elements, there is typically an n/p ratio of 1. Thus carbon-14 lies above the belt of stability with a greater neutron to proton ratio than is stable. Additionally, its atomic number is less than 84. So it will emit a beta particle b) xenon-118. Xe(118, 54). Must use graph. Undergoes both electron capture and positron emission.

Light Water Reactors

Pressurized Water Reactors and Boiling Water Reactors. Water is used as primary coolant and moderator.

Nuclear Fusion

Produces energy via fusion light nuclei. The main obstacle is that very high temperatures and pressures are needed to overcome the electrostatic repulsion between nuclei in order to fuse them. The lowest temperature required for fusion is 40,000,000 K the temperature required to fuse deuterium and tritium in the reaction: H(2,1) + H(3,1) → He(4,2) + n The high temperature used to achieve fusion have been achieved during atomic blasts. This is the principle behind the thermonuclear or hydrogen bomb.

Nucleons

Protons and neutrons, which reside in the nucleus of an atom.

What product is formed when radium(226,88) undergoes alpha emission?

Ra(226,88) → X(Y, Z) + He(4, 2) We get 226 = Y + 4 and 88 = Z + 2 Thus Y = 222 and Z = 86. Looking on on the periodic table we see the element with atomic number = 86 is radon. Thus: Ra(226,88) → Rn(222, 86) + He(4, 2)

How Radiation was Discovered

Radiation was first discovered by Henri Becquerel who noticed that it caused photographic plates to fog up. It also affects film. The greater the extent of radiation the darker the area of the developed negative.

Radioactive Decay and Half Life

Radioactive decay follows first order kinetics. That means half life is given by the formula tsub(1/2) = -ln(1/2)/k = 0.693147/k. Given the half we can calculate k and given k we can calculate the half life. We also have formula: N(t) = N₀e^(-kt) N(t) = N₀/2^(time/halflife) = N₀/2^(t/tsub(1/2)) Time = t is the input where N₀ is the initial quantity of substance, N is the quantity of substance that remains and has not decayed, t is the time, and k is the reaction/decay rate. If t is the half life This means ln(N/N₀) = -kt(sub(1/2)) another useful formula

First Order Rate Law

Rate = k[A]¹ [A] vs time is linear with slope = -k

Electron Capture Example

Rb(81, 37) + e(0, -1) → Kr(81, 36)

Radioactive Emissions

Reduce the mass of the radioactive nucleus

Becquerel (Bq)

SI unit for expressing activity. 1 nuclear disintegration per second

First vs. Second vs. Third vs. etc Half Lives

Second half life is the time it takes for the concentration of the reactant to cut in half a second time. Substitute the concentration of the reactant at the first half life for the initial concentration. Solve. Same with the third half life, etc.

Radioactive Decay Chain or Nuclear Disintegration Series

Some nuclei cannot attain stability through a single emission. They must go through multiple to reach the belt of stability. Uranium-238 is one example of this. It must undergo a lot of emissions to reach lead-206. Similarly, Uranium-235 must undergo several emissions to reach lead-207 and Thorium 232 to lead-208. All of the decay processes in these series are either alpha or beta emissions.

Fission

Splits heavy atoms (typically Uranium-235) into lighter elements by colliding a slow moving neutron with a Uranium-235 atom. The slow moving neutron joins the nucleus of the Uranium-235 atom creating a super heavy atom, which then spontaneously splits to move back to a lower energy level. This spontaneous splitting then releases additional neutrons, on average 2.4 for Uranium-235, which then either hit other Uranium-235 atoms causing a chain reaction or escape from the reaction entirely. If this chain reaction goes unchecked it can cause a violent explosion. For a chain reaction to occur the sample of fissionable material used must have a certain minimum mass.

Hydrogen burning

Stars are born in clouds of gas called nebulas. Gravitational forces collapse the cloud and its density and core temperature rise until fusion begins. Hydrogen nuclei fuse to form deuterium, and eventually helium. Because Helium (He(4,2)) has a larger binding energy than any of its neighbors this releases an enormous amount of energy.

Phosphors

Substances that emit light when radiation passes through them. The radioactivity excites the atoms moving them to a higher energy state. To get back to ground level they release this energy as light.

Gray

The SI unit of absorbed radiation dose. One gray is equal to an absorbed dose of 1 Joule/kilogram. 1 gray = 100 rads

Sievert (Sv)

The Sievert (Sv) is the SI unit of effective dose. Number of Sieverts = number of grays * RBE 1 Sv = 100 rem

Half Life

The amount of time it takes for one-half of a reactant to be used up in a chemical reaction. Concentration of a Reactant at its Half Life = [A]sub(tsub(1/2)) = 1/2[A]₀ To find the half life we substitute this into the integrated rate law equations: For a zero order reaction: [A]sub(tsub(1/2)) = 1/2[A]₀ = -ktsub(1/2) + [A]₀ -1/2[A]₀ = -kt(sub(1/2) Half life = tsub(1/2) = (1/2[A]₀)/k = [A]₀/2k For a first order reaction, ln[A]sub(t) = -kt + ln[A]₀ we substitute 1/2[A]₀ for [A]sub(tsub(1/2)) and tsub(1/2) for t. We get ln(1/2([A]₀) - ln[A]₀ = -ktsub(1/2) Half Life = tsub(1/2) = -(ln(1/2([A]₀)) - ln[A]₀)/k = -(ln(1/2([A]₀)/[A]₀)/k = -ln(1/2)/k For a Second Order Reaction we can make the same substitution. The resulting formula is: Half Life = tsub(1/2) = 1/(k[A]₀)

Mass Defect

The difference between the mass of the nucleus and the mass of its constituents (parts). The mass of a nucleus is less than that of its individual parts summed (mass of protons and neutrons added). This occurs because energy must be added to break the nucleus into separated protons and neutrons.

Fission Products

The elements produced from splitting a material in fission. For Uranium-235 more than 200 isotopes of 35 elements have been found after splitting Uranium-235. Most of them are radioactive.

Decay Constant

The first order rate constant for radioactive decay, k.

Radioactive Decay Example 1

The half life of 218Po is 3.1 minutes. How much of a 170g sample remains after 0.64 hours? Remember we have tsub(1/2) = -ln(1/2)/k and N(t) = N₀e^(-kt) N(t) = N₀/2^(time/halflife) = N₀/2^(t/tsub(1/2)) Method 1: We first calculate k 3.1 minutes = -ln(1/2)/k k = -ln(1/2)/3.1 mins = 0.2236 mins⁻¹ 0.64 hours = 0.64*60 mins = 38.4 mins N(t) = N₀e^(-kt) N(38.4 mins) = 170g*e^(-0.2236 mins⁻¹ * 38.4 mins) = 0.0317g Method 2: N(t) = N₀/2^(time/halflife) = N₀/2^(t/tsub(1/2)) N(38.4 mins) = 170g/2^(38.4 mins/3.1 mins) = 0.0317 g

Radioactive Decay Example

The half life of Cobalt-60 is 5.27 years. How much of a 1.000mg sample of Cobalt-60 is left after 15.81 years. Remember we have two equations. tsub(1/2) = -ln(1/2)/k. We can use this to find k. We also have N(t) = N₀e^(-kt) N(t) = N₀/2^(time/halflife) = N₀/2^(t/tsub(1/2)) tsub(1/2) = halflife = 5.27 years N₀ = 1.000 mg = 1.000 * 10^(-3) g t = 15.81 years We can do this two ways. First find k then use the first N(t) equation. Or we can use the second N(t) equation. Method 1: 5.27 years = -ln(1/2)/k k = -ln(1/2)/5.27 years = 0.1315 yrs⁻¹ Now we plug in to the first N(t) equation N(15.81 years) = (1.000 * 10^(-3))e^(-0.1315 yrs⁻¹ * 15.81 years) = 0.00012505 grams = 1.25 * 10^(-4) grams. Method 2: N₀/2^(time/halflife) = N₀/2^(t/tsub(1/2)) So we have (1.000 * 10^(-3))/(2^(15.81/5.27)) = 0.000125 = 1.25 * 10^(-4) grams

Why Nuclear Reactions are associated with so much more energy than Chemical Reactions

The mass changes in chemical reactions are too small to detect; for example, the mass change in associated with the combustion of 1 mol of CH₄ is -9.9 * 10^(-9) g. The mass change for the decay of 1 mol of Uranium-238 is 50,000 times greater. This means, per Einstein's equation, that nuclear reactions involve much greater energy.

Critical Mass

The minimum mass (of Uranium-235) that must be present so that neutrons can cause a chain reaction. The critical mass of Uranium-235 is about 50kg for a bare sphere of metal. When a critical mass is present about 1 neutron on average from each fission produces another fission.

Atomic Number

The number of protons in an atom

Activity

The rate at which a sample decays, expressed in disintegrations per unit time.

Mass Number

The sum of the number of neutrons and protons in an atom. While all atoms of the same element have the same number of protons, the number of neutrons can differ.

Hydrogen, Deuterium, and Tritium

Three isotopes of hydrogen. H(1, 1) is hydrogen, H(2, 1) is deuterium, and H(3,1) is tritium.

Atomic Bomb

Two subcritical masses of Uranium-235 are slammed together using a chemical explosion to form a supercritical mass. This then leads to a chain reaction and violent explosion. The simplicity of the technique has led to concerns about nuclear proliferation.

Transuranium Elements

Typically nuclear elements with atomic numbers higher than 92 (above uranium) are typically not found in nature. They need to be produced through nuclear transmutation. Elements 93 and 94 were produced through neutron bombardment. Higher atomic numbers must be produced using particle accelerators.

Nuclear Equation Example

U(238, 92) → Th(234,90) + He(4,2) This shows Uranium-238 decaying into Thorium-234 and Helium-4. U(238,92) is the one undergoing radioactive decay in this equation. Because an alpha particle is emitted this is often called alpha decay or an alpha emission. Note the sum of the mass numbers and atomic numbers on both sides are equal. Mass numbers and atomic numbers must be balanced in all nuclear equations. The radioactive properties of a nucleus are independent of the chemical properties of the element (if it is in a compound, for instance).

rem

Unit of effective dose. Number of rem = number of rads * RBE The rem is the most common unit in medicine.

Difference between Uranium-238 and Uranium-235

Uranium-235 has a mass number of 235. Uranium 238 has a mass number of 238. Both have the same number of protons but different numbers of neutrons. A shorthand 238U 92 where 238 is the mass number, and 92 is the atomic number is used. We will write this as U(238, 92)

Geiger Counter

Used to detect radiation in particles like alpha particles, beta particles and gamma rays. The radiation ionizes matter, creating an electrical current that can be measured in the Geiger counter.

Radioactive Decay Example 2

What percentage of a radioactive sample remains after 175.0 yr if it has a half life of 28.8 years. Suppose we have 100g of the substance initially. Using the formula N(t) = N₀/2^(t/tsub(1/2)) 100g/2^(175.0yrs/28.8yrs) = 1.48 g So 1.48g/100g = 1.48%

Positron Reaction with Electron

When a positron runs into an electron it is annihilated producing gamma rays. This is why it has such a short life. e(0, +1) + e(0, -1) → 2γ(0, 0)

Ionizing vs Non-ionizing Radiation

When matter absorbs radiation the radiation energy can cause the atoms to become either excited or ionized. Ionizing radiation is more harmful to living systems than non-ionizing radiation. Non-ionizing radiation is typically slow moving neutrons or lower energy electromagnetic radiation such as radio waves.

Nuclear Transmutation Equation Example

Write the balanced nuclear equation for (27,13)Al(n, α)Na(24, 11) Al(27, 13) + n(1, 0) → Na(24,11) + He(4, 2) or Al(27, 13) + n → Na(24,11) + α

Nuclide

a nucleus containing a specified number of protons and neutrons

Radionuclide

a radioactive nuclide. Spontaneously emit particles and electromagnetic radiation. The emission of radiation is how radionuclides gradually go from unstable to stable.

Nuclear Equation

a type of equation that shows what a radioactive nucleus decays into and the atomic numbers and mass numbers of the particles involved.

First Order Integrated Rate Law

ln[A]sub(t) = -kt + ln[A]₀, where t is time, [A]₀ is the concentration of a reactant A at time 0, and k is the rate constant for the reaction.