Homework Assignment Scientific Notation, Unit Conversions, and Radiation Units IEER Workshop 2007

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1 Homework Assignment Scientific Notation, Unit Conversions, and Radiation Units IEER Workshop 2007 This is an optional exercise that will get you on your technical toes for the IEER workshop. IEER board member and teacher extraordinaire Dr. Dave Close developed this homework assignment. He ll review the answers on Day 1 of the workshop. Try your hand at the 6 problems below, perhaps during the plane or train ride for you travelers. No penalty for incorrect answers this is for learning! Scientific Notation Scientific Notation was developed in order to easily represent numbers that are either very large or very small. Here are two examples of large and small numbers. They are expressed in decimal form instead of scientific notation to help illustrate the problem: The Andromeda Galaxy contains at least 200,000,000,000 stars. The weight of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms. As you can see, it could get tedious writing out those numbers repeatedly. So, a system was developed to help represent these numbers in a way that was easy to read and understand: Scientific Notation. Using one of the above examples, the number of stars in the Andromeda Galaxy can be written 2.0 x 100,000,000,000 It is that large number, 100,000,000,000 which causes the problem. But that is just a multiple of ten. In fact it is ten times itself eleven times: 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000,000 A more convenient way of writing 100,000,000,000 is The small number to the right of the ten is called the "exponent," or the "power of ten." It represents the number of zeros that follow the 1. So we would write 200,000,000,000 in scientific notation as: 2.0 x (This number is read: "two point zero times ten to the eleventh.") As we said above, the exponent refers to the number of zeros that follow the 1. So: 10 1 = 10; 10 2 = 100; 10 3 = 1,000, and so on. Similarly, 10 0 = 1, since the zero exponent means that no zeros follow the 1. Negative exponents indicate negative powers of 10, which are expressed as fractions with 1 in the numerator (on top) and the power of 10 in the denominator (on the bottom). So: 10-1 = 1/10 = 0.1; 10-2 = 1/100 = 0.01; 10-3 = 1/1,000 = 0.001, and so on. This allows us to express other small numbers this way. For example: 2.5 x 10-3 = 2.5 x 1/1,000 =

2 Every number can be expressed in Scientific Notation. In our first example, 200,000,000,000 should be written as 2.0 x In theory, it can be written as 20 x 10 10, but by convention the number is usually written as 2.0 x so that the lead number is less than 10, followed by as many decimal places as necessary. It s easy to see that the variations above are just different ways to represent the same number: 200,000,000,000 = 20 x (20 x 10,000,000,000) 2.0 x (2.0 x 100,000,000,000) 0.2 x (.2 x 1,000,000,000,000) This illustrates another way to think about Scientific Notation: the exponent will tell you how the decimal point moves; a positive exponent moves the decimal point to the right, and a negative one moves it to the left. So for example: 4.0 x 10 2 = 400 (2 places to the right of 4); while 4.0 x 10-2 = 0.04 (2 places to the left of 4). Note: Scientific Notation is also sometimes expressed (on some calculators, for instance) as E (for exponent), as in 4 E 2 (meaning 4.0 x 10 raised to 2). Similarly 4 E -2 means 4 times 10 raised to -2, or = 4 x 10-2 = Problem 1: Calculate the number of seconds in a century. Express your answer in scientific Scientific Notation: Addition and Subtraction The key to adding or subtracting numbers in Scientific Notation is to make sure the exponents are the same. For example, (2.0 x 10 2 ) + (3.0 x 10 3 ) can be rewritten as: (0.2 x 10 3 ) + (3.0 x 10 3 ) Now you just add and keep the 10 3 intact. Your answer is 3.2 x 10 3, or 3,200. We can check this by converting the numbers first to the more familiar form. So: 2 x x 10 3 = ,000 = 3,200 = 3.2 x 10 3 Let's try a subtraction example: (2.0 x 10 7 ) - (6.3 x 10 5 ) The problem needs to be rewritten so that the exponents are the same. So we can write (200 x 10 5 ) - (6.3 x 10 5 ) = x 10 5, which in Scientific Notation would be written x Let's check by working it another way: 2 x x 10 5 = 20,000, ,000 = 19,370,000 = x

3 Scientific Notation: Multiplication When multiplying numbers expressed in scientific notation, the exponents are simply be added together. This is because the exponent represents the number of zeros following the one. So: 10 1 x 10 2 = 10 x 100 = 1,000 = 10 3 Checking that we see: 10 1 x 10 2 = = 10 3 Similarly 10 1 x 10-3 = = 10-2 =.01 Again when we check we see that: 10 x 1/1000 = 1/100 =.01 Look at another example: (4.0 x 10 5 ) x (3.0 x 10-1 ). The 4 and the 3 are multiplied, giving 12, but the exponents 5 and -1 are added, so the answer is: 12 x 10 4, or 1.2 x Let's check: (4 x 10 5 ) x (3 x 10-1 ) = 400,00 x 0.3 = 123,000 = 1.2 x Interesting note: another way to see that 10 0 = 1 is as follows x 10-1 = = It is also: 10 x 1/10 = 1. So 10 0 = 1 Scientific Notation: Division Let's look at a simple example: (6.0 x 10 8 ) (3.0 x 10 5 ) To solve this problem, first divide the 6 by the 3, to get 2. The exponent in the denominator is then moved to the numerator, reversing its sign. (Remember that little trick from your old math classes?) So we move the 10 5 to the numerator with a negative exponent, which then looks like this: 2 x 10 8 x 10-5 All that's left now is to solve this as a multiplication problem, remembering that all you need to do for the "10 8 x 10-5 " part is to add the exponents. So the answer is: 2.0 x 10 3 or 2,000 Problem 2: The U.S. national debt is about $8.8 trillion. The cumulative amount of high-level waste at the Savannah River Site, Idaho Chemical Processing Plant, Hanford Nuclear Reservation, and the West Valley Demonstration Project is about 25 billion curies. If the entire amount of money associated with the national debt was applied to cleanup of those curies, how many dollars per curie would be spent? 3

4 Fundamental Units of Length, Mass and Time Before we can look at units of radiation, we also have to define some quantities of energy. There are two systems of units that are in common use. First is the SI system of measurement. SI means Système International and is also called the MKS system. MKS stands for Meters, Kilograms and Seconds. A meter is just a little more than a yard (actually 39.4 inches, a kilogram is about 2.2 pounds. The second system is more popular in chemistry. Since a kilogram of some material is a fairly large quantity, chemists prefer to use smaller units. This system is called the CGS system and is bases on Centimeters (1/100 of a meter), Grams (1/1000 of a kilogram) and Seconds. Other units, for example volume, are also conveniently smaller. The SI unit for volume is a cubic meter (a box 1 meter wide, one meter long, and one meter tall). This is the size of a small desk. But in the CGS, volume is measured cubic centimeters (abbreviated cc). This is the size of an eraser on a pencil. Force and Work and Energy To understand energy units, we have to first look at the concept of work. By definition, work involves pushing or pulling some object a certain distance (we say "force times distance"). Newton's famous Second Law tells us that if a steady force in applied to a mass, the mass speeds up (it accelerates). The equation for this phenomenon is as follows: F = m x a (read as mass times acceleration) Suppose we wish to see this expressed in the MKS system discussed above. Mass is in kilograms. Acceleration is defined as the time rate of change in the velocity. Since velocity is in meters/second, acceleration is in units of meters/(second per second). This is usually expressed as meters/second 2 and is pronounced "meters per second squared". So in the MKS system we have Force = mass times acceleration. This would be in units of kilograms times meters/second 2. This is used so often that it has been given the name of a Newton. One Newton (abbreviated N) is by definition one kilogram being accelerated at one meter per second 2. Now that we have the definition of Force we return to the concept of Force times distance which is Work. In the MKS system, work is Force (in Newtons) times distance (in meters). If a one Newton force moves an object one meter, the work done is called one Joule (which is one Newton-meter). We can do the same thing in the CGS system. The unit of force (mass times acceleration) is called a dyne. The work of a one dyne force applied to an object that causes it to move one centimeter (cm) is called an erg (after the Greek word for work which is ergon). One likes to get paid for doing work, but in physics one has to be very precise in order to actually define "work". If I give you a ten pound weight and ask you to hold this out at arms length for some period of time, this may sound like a lot of work to you. But by definition this is no work at all. Work is defined as force times distance. While it takes a considerable force to hold up the ten pounds, the weight is not moving. No motion, no work. There is a second problem that must be considered. I can tell you to push the same ten pound weight across the floor a distance of 20 feet. I didn't say how much time this should take. Someone might do this in ten seconds, and someone else might take 1/2 an hour to do the same amount of work. How then do we assign a dollar amount to a unit of work? Electron Volt Before leaving this subject, we have to mention one more unit of energy, the electron-volt. This is a unit of energy associated with moving electrons around. Suppose we have an electron tightly bound in a hydrogen atom (one proton and one electron). It takes energy to move this electron away from the proton. It takes 13.6 electron-volts of energy to move this electron completely away from the proton. We say then that the atom is "ionized". In the jargon, the ionization energy of the tightly bound 4

5 electron is 13.6 electron volts. Electrons are very light objects, so we don't expect an electron-volt to represent very much energy. In fact 1 electron-volt (abbreviated 1 ev) is only 1.6 x joules of energy. The ev has been around so long that it will not likely be replaced by an SI unit. Power Unlike work, Power has a dollar sign associated with it. Power is work per unit time. In the MKS system, work is in Joules (a Newton-meter). If one Joule of work is done is one second, this is called a Watt (named after the Scottish scientist who made significant improvements to the steam engine). While the unit of power is named after James Watt, these units weren't around in his day. Actually Watt thought of power in terms of how much work a horse could do. He defined power in terms of a big draft horse pulling 550 pounds one foot in one second. Read this as one horsepower equals 550 foot-pounds/sec. This is a great deal of power. A very strong person could develop 1 horsepower for a few seconds by lifting a heavy object off the floor, or by running upstairs. But in a few seconds he would be tired out and would have to rest. Think of a one horsepower motor working all day, or a large draft horse pulling a cart all day, and you can see that this is a great deal of power. That is why we can assign a dollar amount to energy (it is worth something to have a "labor savings device" like a horse or a motor). We have been using English units here because of their historic importance in the definition of energy. To convert to the MKS system, you need to know that one horsepower is equal to 746 watts. Also to pay your electric bill, you have to figure out energy per hour. Somewhere on you electric bill it says kilowatt/hours. That's just one thousand watts (about what a hair dryer, a toaster or any other heavy duty appliance consumes) in one hour. The power company wants about 10 cents for this amount of power. OK, now for your first problem: Problem 3: If one kilowatt/hour costs 10 cents, how much does it cost to leave on a one hundred watt bulb all night (say for 10 hours)? Radiation Units It has taken a while to reach our goal of defining radiation units. This is because these units are defined in terms of energy deposited in a given mass of material. Before this could be done, we needed to have energy properly defined. Now however the rest is easy. Absorbed dose (Rad) The amount of energy absorbed per unit mass of irradiated material is called a Rad. One rad is defined as 100 ergs/gram. There is a movement in science to have all units expressed in the MKS system. We see here that ergs and grams are CGS units. So to convert rads to MKS units we must express energy in Joules and mass in kilograms. One Gray is one Joule/kilogram. This equals 100 rads. Many people are quite comfortable with the rad unit, and don't like the Gray. So one can divide a Gray by 100 (call it a Centi-Gray) and be back to thinking about rads. Suppose time is involved? Then we are talking about dose rate (or dose per unit time). Then the units for dose rate might be rads/minute. Exposure rate, Roentgen The exposure of x-radiation or gamma radiation at a certain place is a measure of the radiation based on its ability to produce ionization in air. Exposure of air to one roentgen produces ions 5

6 carrying a total electrical charge of 2.58 x 10-4 Coulombs/kilogram when all the secondary electrons are stopped in the air. One can show that 1 roentgen is equal to 0.87 rads in air. This is true for both x- and gamma radiation at most energies in air. For other materials the value of the constant varies with atomic composition of the material and the energy of the radiation. Actually computing this dose in other materials is not easy to do. REM The dose discussed above, in rads or Grays measures the amount of energy deposited in a sample. If the sample is tissue, one is interested in biological damage. The amount of biological damage is measured in rems, which is an abbreviation for Roentgen Equivalent Man. The amount of ionizing radiation required to produce the same biological effect as one Roentgen of x-ray is called a rem. Different forms of radiation produce different effects in living tissue. Therefore dose in rems must be calculated as the dose in rads times the Relative Biological Effectiveness (RBE). Rads and rems are almost the same for beta or gamma radiation. Actually 1 rem = 0.93 rad for living tissue. For alpha radiation, things are very different. Alpha radiation is far more damaging per unit of energy deposited in living tissue. Currently the conversion factor from rads to rems for alpha radiation is 20 (multiply rads of alpha radiation by 20 to get rems). It is interesting to note that we say here "current" conversion factor. This factor has changed in the past as the assessment of radiation damage to living tissue has changed. Once more, we note that rems are not MKS units. To bring things up to date, a Sievert (abbreviated Sv) has been defined as 100 rems. So think of rems when you are dealing with rads, and Sieverts when you are dealing with Grays. Problem 4: 5 Grays is how many rads? Problem 5: 0.01 Sieverts is how many rems? Problem 6: 1000 rads is how many Grays? Questions? Dr. Dave Close at CLOSED@mail.etsu.edu 6