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Units and notation

Units and notation

January 29, 2014 00:001 comment

Welcome to my primer on radiation and nuclear topics. I have made an effort to explain things in a way that I believe will be understandable to most people, but in case there is something that you do not know, I suggest searching Wikipedia as a good starting point to find out more. Feel free to ask (relevant!) questions in the comments section and I will do my best to answer them.


Up until this point, I have mostly avoided using numbers, units, and equations but in future articles this will not be possible. So now would be a good time to ensure that you are acquainted with these concepts before things get too complex.

Scientific notation

Scientists often work with very large and very small numbers and also need to keep track of the precision of measurements. For this purpose, scientific notation is used. Numbers in scientific notation take the form:

a \times 10^{b}

where a is a number (often a decimal) between 1 and 10 (negatives are ok too), and b is an integer (no decimals allowed). It is also acceptable to abbreviate  \times 10^{b} by replacing it with: E b.

While this may seem like a funny way to write a number, it is very convenient for very large or very small numbers because it saves you from having to count up the zeros or decimal places. This also simplifies a lot of arithmetic and allows you to quickly make rough estimates.

To convert a number into scientific notation, place a decimal point after the most significant non-zero digit (that is, the first non-zero digit when counting from the left) to form a. Then, the number of times that you just moved the decimal becomes b. If you moved the decimal to the left, b is a positive number, and vice versa. For example:

8675309

would be written as

8.675309\times10^{6}

or

8.675309E6

in scientific notation. Notice how a (8.675309) is a number between 1 and 10, and b (6) is the number of places that you had to move the decimal from the original number (8675309) to change it into a (8.675309).

Element notation

Elements are referred to in chemical equations by their symbols, as listed in the periodic table. Isotopes of elements are written by writing the symbol for the element (for example, gold’s symbol is Au), followed by a dash (-) and then the atomic mass of the isotope. So, gold 197 would be written as: Au-197. In equations, this is often stylized as:

\mathrm{~^{197}_{79}Au}

with the atomic mass written in superscript in the upper-left of the element’s symbol. Sometimes (like in this example) you will see the atomic number written in subscript in the lower-right of the element’s symbol. The atomic number isn’t strictly necessary because the element’s symbol already specifies the atomic number, but when working with equations it can save you the trouble of looking up an element’s atomic number if you don’t have it memorized.

Units of measurement

A unit is a fixed amount that describes a physical quantity, such as length or time, that is used for measurement. You are already familiar with many of these, such as kilograms, meters, liters, and so on.

In science, the SI unit system is used as a worldwide standard of measurement. This means that the average American is handicapped when it comes to science since the USA uses its own system of units. In this guide, I will be using the SI system for all measurements and calculations, so I will leave it as an exercise for my American readers to either convert all quantities into American units or to simply learn the metric system like the rest of the world!

Because of the very small size of nuclei, nuclear scientists use units which may be unfamiliar to the average person, but are convenient when working with quantities seen in nuclear science. I will introduce a few of them to you now and more of them as they become relevant to the topics of discussion.

Energy

Although the SI fundamental unit of energy is the joule (J), a joule is too large when speaking about the energy of individual atoms. Instead, we use a unit called the electronvolt (eV).

1~\mathrm{eV} = 1.602\times10^{-19}~\mathrm{J}

As with all metric units, prefixes (k for 1000, M for 1000000, etc.) are used to easily indicate magnitudes. For example, the energy of the gamma ray released by the decay of a Ba-137m atom has an energy of 662 keV.

Mass

Because atoms are so small, it would be silly to measure their masses in kilograms! Instead, we use a unit called the atomic mass unit (u).

1~\mathrm{u} = 1.661\times10^{-27}~\mathrm{kg}

1 u is approximately equal to the mass of a proton or a neutron, and for ordinary chemistry, this is good enough for most calculations. However, in the nuclear sciences, we must use the exact masses for particles for calculations:

mass of proton = 1.007276466812 u
mass of neutron = 1.00866491600 u

Because of Einstein’s famous E=mc^2 equation, mass can also be expressed using units of energy.

1~\mathrm{u} = 931.494061~\mathrm{\frac{MeV}{c^2}}

The c^2 in \mathrm{\frac{MeV}{c^2}} is often removed, leaving \mathrm{MeV} for convenience. Since this is the same unit to denote energy, context determines whether mass or energy is meant.

Amount of matter

Because atoms are so small, when we want to scale up our calculations to human-sized units, we need a way to easily count how many atoms are in a quantity of matter. Fortunately, the mole1 (mol) allows us to do just that.

1~\mathrm{mol} = 6.022\times10^{23}

You may notice that the mol itself does not have any physical units attached to it. This is not a mistake. A mole is simply a shorthand for that very, very large number, much like how “one dozen” in English means “12 of something”. This may seem to be a very strange choice for a number, but its utility is that it is a conversion factor between atomic mass and grams! For example, the molecular mass (sum of the atomic masses of the atoms in a molecule) of water is 18.015, so one mole of water has a mass of 18.015 grams, but contains 6.022\times10^{23} atoms. In mathematical equations, the number of atoms is often indicated by the letter N.


Summary

  • You can easily write very big or very small numbers using scientific notation. It looks like a \times 10^{b} or ab.
  • Elements are referred to by their symbols from the periodic table. When referring to a specific isotope, write the isotope’s mass after the symbol: Au-197.
  • Energy is often measured in electronvolts (eV). 1~\mathrm{eV} = 1.602\times10^{-19}~\mathrm{J}
  • Mass is measured in atomic mass units (u). 1~\mathrm{u} = 1.661\times10^{-27}~\mathrm{kg} Thanks to it E=mc^2, can also be measured using eV, just like energy! 1~\mathrm{u} = 931.494061~\mathrm{MeV} 
  • The unit for amount of matter is the mole (mol). 1~\mathrm{mol} = 6.022\times10^{23} Think of it as a scientist’s version of a dozen.

Further resources

Crash Course Chemistry: Unit Conversion & Significant Figures

The Hyperphysics Page on Nuclear Units

Chemwiki: SI Units


  1. Yes, it’s got a funny name!

1 Comment

  • Okay, will try to keep up despite my science/math challenges. But please glance at my latest blog for a tiny graphic I made (elements) that will make you smile.

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