If you pull apart an atom, the individual parts will weigh more than the atom you started with. The extra mass is the binding energy, which is released when the nucleus of atoms break apart (nuclear fission).
If you pull apart the nucleus of an atom, you’ll find that the mass of its parts is greater than the mass of the original nucleus. That extra mass is where nuclear energy comes from; it’s called the binding energy.
How so?
An alpha particle is the nucleus of a helium atom.
Take a helium atom for example. Its nucleus typically has two protons and two neutrons*, which in nuclear physics is usually called an alpha particle (α).
While we usually say that the mass of a proton is 1 atomic mass unit (u), its actually a little heavier. The mass of a proton is 1.00728 atomic mass units (u), while neutrons weigh 1.00866 u.
The alpha particle (helium nucleus) has less mass than sum of the masses of the individual particles that make it up.
The combined mass of the two protons and two neutrons in the helium nucleus is 0.03035 atomic mass units more than the mass of a helium nucleus made up of the very same particles.
Why?
The one equation that everyone remembers from Einstein (perhaps from all the t-shirts) is:
Energy (E) is equal to mass (m) times some constant (c is the speed of light) squared. What it means is that mass is energy, and vice-versa.
When the four nucleons combine, the extra mass is transformed into the energy that holds them together in the nucleus of the atom. The mass can be directly converted to energy, the binding energy of the atom.
How much energy is released?
Somewhere around 10,000 times more energy is released from a single nuclear reaction compared to a single chemical reaction (like the combustion of TNT).
Binding energy per neuleon for the naturally occuring elements. (image from Science in School).
Footnotes
* Helium with two neutrons would be written , where the bottom number is the number of protons and the upper number is the atomic mass, which is the sum of the number of protons and the number of neutrons.
Marketplace’s Jeff Horwich has an excellent article on the uses of the element phosphorus, where it comes from, why it’s getting scarce, and where we might get more.
The answers to these questions are:
It’s a key element in DNA, so the major use is fertilizer,
most of it comes from Morocco these days,
since Morocco supplies about 85% of the world supply, they’re developing a bit of a monopoly and the price is going up,
the main alternative sources are manure and urine that have lots of phosphorous. In fact, burning sewage leaves behind a phosphorous rich ash.
Take what you find interesting and turn it into something challenging, something provocative for someone else.
–Dan Meyer (2011): [anyqs] Hurricane Irene Edition
I’m looking for a good reference for project-based math. Where students face the real-life problems, and learn math as they try to solve them, yet covers the entire curriculum in a complete way.
What I’m considering right now is to swap in some of the real-life questions for some of the sections in the text that consist of rather pedantic word problems, things like: the sum of two numbers is three times less than the square root of the second plus the reciprocal of the first.
Instead, I’d rather do problems like determining the height of a tsunami, which can be treated in different ways depending on which math class you’re teaching, and tie into the science classes (like Physics) as well.
Dan Meyer is a proponent of the project based approach, and he has a lot of interesting problems on his blog.
[A]nger … triggers a less systematic and structured approach to the creativity task, and leads to initially higher levels of creativity. … [However] creative performance should decline over time more for angry than for sad people.
Here are a couple of studies on the interaction between negative emotions and creativity whose implications require some very careful consideration. We want to encourage creativity, but how and at what cost to the student?
Social rejection was associated with greater artistic creativity
Anger, it appears, leads to more unstructured thinking, thinking that is more flexible and able to make new connections among different categories of information. However, anger’s creativity boost does not last that long – strong emotions take a toll – and people soon revert back to a more normal baseline.
These results come from an initial study, and there are a lot of unanswered questions. In particular, I wonder just how much anger is useful for this beneficial outcome. I find it hard to believe that too much anger is terrible useful. And, I’m also curious about the negative consequences in terms of group interactions. Brett Ford points out that some studies have found that anger is useful in negotiation, but only when that negotiation is confrontational. Another study found that angry leaders were better at motivating groups of less agreeable people. Conversely, more agreeable people responded better to less angry leaders.
In a scenario study, participants with lower levels of agreeableness responded more favorably to an angry leader, whereas participants with higher levels of agreeableness responded more favorably to a neutral leader.
It seems that the ability to project anger may be a useful skill to have in one’s toolbox, given the variety of people we will have to deal with in life.
Depression and Creativity
Modupe Akinola and Wendy Berry Mendes point out that highly creative people tend to introversion, emotional sensitivity and, at the extreme, depression and other mood disorders. Unfortunately:
[M]ood disorders are 8 to 10 times more prevalent in writers and artists than in the general population (Jamison, 1993).
On top of the general mood, strong, more transient, activating moods, like anger and happiness, also affect a person’s ability to be creative. Both positive and negative activating moods (the hedonic tone) enhance creativity, but in different ways:
negative activating moods, like anger and fear, increase perseverance;
positive activating moods, like happiness and elatedness, increase mental flexibility.
Curiously enough, although creativity is associated with a baseline of sadness and depression, these two are not among the activating moods that can spur the creativity of the moment.
A Matter of Control
The implications of these studies are complex. I certainly need to think about them a lot more, but it would seem reasonable, or perhaps responsible, to encourage students to carefully monitor their moods and to help them better understand themselves and their behavior. Ultimately, it is probably better if we are able to control how we use our emotions, rather than the other way around.
The pre-frontal lobe, which is responsible for formal thinking, is the part of the brain that can put the brakes on impulsive emotional behavior. It can also, to a degree, modulate how emotions are expressed. As adolescents’ pre-frontal cortex develop, they should be better able to control and use their emotions to their benefit. But to do so, they need to be aware of their emotions and the power of their emotions, which would suggest training in emotional awareness and control.
I’m not aware of any programs or curricula that delve all the way into how to use your emotions proactively, but I’d like to see something that particularly discusses how to use the different activating moods.
Collapsed milk jug.
Place a little hot water (400 ml at 94-100°C) into a plastic, gallon-sized, milk jug. Give it a moment to warm the air in the jug, then put the cap on and seal tightly (hopefully airtightly).
As the air in the jug cools the gas inside with shrink, reducing its pressure, and causing the atmospheric pressure to push in the sides of the jug.
Admittedly, this experiment is a little more dramatic if you use a metal tin, but it works well enough with the milk jug to surprise and impress.
In what one can only hope is an extremely tongue-in-cheek article, Marketplace discusses how we might geoengineer a solution to stop hurricanes forming.
Hurricanes get their energy from the warm surface waters in the tropics. The warm water evaporates, transferring heat from the oceans to the atmosphere as latent heat in the form of water vapor. As the air rises, the water vapor condenses to form water droplets (clouds) releasing the stored heat into the air, causing the air to rise faster, sucking up more moisture, and setting up a positive feedback loop that turns storms and hurricanes.
But they need a constant supply of warm water.
Between 100 and 200 m below the ocean's surface, the temperature drops rapidly. This is called the thermocline. You will notice that the deepest, coldest water is at 4 °C, the temperature at which water is most dense. (Image adapted from the Woods Hole Oceanographic Institute, via Wikipedia).
Unfortunately for the storms, the warm water in the tropics is only a thin layer, a couple of hundred meters deep, that sits above about 3,000 meters of colder deep-water. As the storms suck up the heat and moisture, they stir the oceans, cooling down the surface water, and leaving cooler water in their wakes. The cooler water means that subsequent storms have access to less energy.
The energy in the atmosphere and oceans “wants” to distribute itself evenly over the surface of the Earth. Hurricanes are just one violent means of moving heat from the tropics to the poles, and from the surface to the depths of the oceans.
Cool water from the deep Atlantic is stirred up to the surface by hurricanes. This image shows the cool wake of Hurricane Igor. (Image from
NOAA’s National Hurricane Center monitors sea-surface temperatures closely: it’s one of the key factors that go into their predictions of how bad the hurricane season is going to be, and what path a storm might take.
A Salter Sink for mixing the warm shallow water with the deeper cold water. Image from Intellectual Venture via NewScientist.
The suggestion in the Marketplace article is that we could build about 10,000 long tubes (called Salter Sinks) to connect the warm shallow surface water to the colder water below the thermocline. Wave energy at the surface would drive the warm water downward, causing mixing that would reduce the temperature of the surface water the storms feed off.
The devices might cost tens of millions of dollars per year, but that would be a lot less than the cost in property damage alone of a large storm like Irene, not to mention the loss of life it would prevent.
Apart from the “benign” environmental impact (according to Stephen Dubner) the only real question left is:
One of the first things you learn in algebra is to use variables to represent numbers. Variables are at the heart of computer programming, and in Python you can use them for more than just numbers. So open the IDLE text editor and get to work.
But first we’ll start with numbers. To assign a number to a variable you just write an equation. Here are a couple (you can copy and paste the code into the IDLE window, but usually typing it in yourself tends to be more meaningful and help you remember):
a = 2
b = 3
Now we can add these two variables together and create a new variable c.
a = 2
b = 3
c = a + b
Which is all very nice, but now we want our program to actually tell us what the result is so we print it to the screen.
a = 2
b = 3
c = a + b
print c
Run this program (F5 or select “Run Module” in the “Run” menu) to get:
Output from the first program: 2+3=5.
Basic Operations and Types of Numbers
Now try some other basic operations:
+ and – are easy as you’ve seen above.
× : for multiplication use *, as in:
a = 2
b = 3
c = a * b
print c
÷ : to divide use a /, as in:
a = 2
b = 3
c = a / b
print c
Now as you know, 2 divided by 3 should give you two thirds, but the running this program outputs 0:
2 divided by 3 to gives 0 because Python thinks we're working with integers.
This is because the Python thinks you’re using integers (a whole number), so it gives you an integer result. If you want a fraction in your result, you need to indicate that you’re using real numbers, or more specifically, rational numbers, which can be integers or fractions, but usually show up as a decimal (these are usually referred to as floating point numbers in programming).
The easiest way to indicate that you don’t just want integers is to make one of your original numbers a decimal:
a = 2.0
b = 3
c = a / b
print c
which produces: Use a = 2.0 to indicate that we're using rational numbers.
Other Things Can Be Variables
In object oriented programming languages like Python you can assign all sorts of things to variables, not just numbers.
To create a box and give it a variable name you can use the program:
from visual import *
c = box()
which produces: A box created with VPython. It looks much more interesting if you rotate it to see it in perspective.
A rotated, zoomed-out view of a box.
To rotate the view, hold down and drag the right mouse button. To zoom in or out, hold down the right and left buttons together and drag in and out. Mac users will probably have to use the “option” button to zoom, and the “command” button to rotate.
The line from visual import * tells the computer that it needs to use all the stuff in the module called “visual”, which has all the commands to make 3d objects (the “*” indicates all).
The c = box() line creates the box and assigns it a variable name of c. You don’t just have to use letters as variable names. In programming you want to use variable names that will remind you of what it’s supposed to represent. So you could just as well have named your variable “mybox” and gotten the same result:
from visual import *
mybox = box()
Now objects like this box have properties, like color. To make the box red you set the color property in one of two ways. The first method is to set the color as you create the object:
from visual import *
mybox = box(color = color.red)
The second is to set the property using the variable you’ve created and “dot” (.) notation.
from visual import *
mybox = box()
mybox.color = color.red
In both of these color.red is a variable name that the computer already knows because it was imported when you imported the “visual” module. There are a few other named colors like color.green and color.blue that you can find out more about in the VPython documentation (specifically here).
You can also find out about the other properties boxes have, like length, width and position (pos), as well as all the other objects you can create, such as springs, arrows and spheres.
At this point, its probably worth spending a little time creating new objects, and varying their properties.
In an interesting application of thermodynamics, BAE Systems has developed panels that can be placed on a tank to mask what it looks like to infra-red goggles, or help it fade into the background.
The panels measure the temperature around them and then warm up or cool so they’re the same temperature and therefore emitting the same wavelength of infrared light. So someone looking at the tank with infra-red goggles would have a harder time distinguishing the tank from the background.
The panels are thermoelectric, which means they use electricity to raise or lower their temperatures, probably using a Peltier device.
Peltier devices, also known as thermoelectric (TE) modules, are small solid-state devices that function as heat pumps. A “typical” unit is a few millimeters thick by a few millimeters to a few centimeters square. It is a sandwich formed by two ceramic plates with an array of small Bismuth Telluride cubes (“couples”) in between. When a DC current is applied heat is moved from one side of the device to the other – where it must be removed with a heatsink. The “cold” side is commonly used to cool an electronic device such as a microprocessor or a photodetector. If the current is reversed the device makes an excellent heater.
, where the bottom number is the number of protons and the upper number is the atomic mass, which is the sum of the number of protons and the number of neutrons. 
