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STUFF THAT INTERESTS MR REID, A PHYSICIST AND TEACHER


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WHY IS THE EARTH HOT?

10th Aprilearth, geology

Not on the surface, I mean, but underneath? As you dig down into the Earth’s
crust you will find yourself getting warmer, at a rate of about 0.025°C per
metre of depth (25°C per kilometre). But why?

There are two sources of internal heat that make up the Earth’s internal heat
budget: radiogenic heat and primordial heat. This heat flows to the surface at a
rate of around 47 terawatts or 92 milliwatts per square metre of surface. This
flow is uneven, with more heat flowing at mid-ocean ridges where two tectonic
plates move away from each other, and more heat flowing through oceanic crust
(which underlies the oceans) than through continental crust (which underlies
continental land). The exact contribution of radiogenic v primordial heat to the
total is unknown, but is thought to be approximately 50:50.[1]



Radiogenic heat is the heat that comes from the decay of radioactive isotopes in
the Earth’s crust and mantle; most of it comes from the decay of four isotopes:
potassium-40, thorium-232, and uranium-235 and -238.



Concentration is on the x-axis, with the more common isotopes further to the
right, and the y-axis measures the amount of heat produced by each isotope, with
“hotter” isotopes towards the top. The area of the bubble is proportional to the
amount of heat that isotope contributes per kilogram of mantle, with thorium-232
contributing the most at 3.3 picowatts per kilogram.

Primordial heat is the heat left over from the formation of the Earth. As gas
and dust left over from the formation of the Sun coalesced, they lost
gravitational potential energy and gained kinetic/thermal energy. The huge
amount of initial heat means that Earth is still cooling even now, 4.5 billion
years later.



[1] Gando A, Gando Y, Ichimura K, Ikeda H, Inoue K, Kibe Y et al. Partial
radiogenic heat model for Earth revealed by geoneutrino measurements. Nature
Geoscience. 2011;4(9):647-651.

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PHASE DIAGRAMS

21st Decemberpressure, statesofmatter, temperature

Everybody knows that water freezes at 0°C and melts at 100°C, right?

Except that’s not always true. The melting point and boiling point of water
depends on the pressure of the water: water only freezes at 0°C and melts at
100°C when it’s at standard atmospheric pressure: 101325 pascals. For example,
you cannot make a good cup of tea at the peak of Mount Everest because the
pressure is lower there and therefore water boils at a lower temperature (around
71°C), lower than the temperature required to properly release the flavour from
the tea.



Information about a substance’s melting and boiling points at different
pressures can be represented on a phase diagram. The phase diagram for water is
shown below:



From the diagram, we can see that at pressures below around 600 Pa, water
transforms from a solid to a gas without passing through a liquid phase. This is
a process known as sublimation, and is most well-known from the carbon dioxide
“fog” created when dry ice is placed into hot water. We can also see the triple
point, a combination of temperature and pressure (0.01°C and 611.73 Pa) at which
ice can exist in all three states simultaneously.

Cyclohexane at its triple point boiling and freezing simultaneously.

At higher pressures, the melting point of water changes. Water can remain as
solid ice up to temperatures of more than 300°C when the pressure is above ten
gigapascals. What is also interesting is the different types of solid water that
are formed at different pressures and temperatures.

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ONE MILLION HATS

21st Octobermathematics, probability

From the always-excellent Futility Closet, a problem by Paul J. Nahin:

> Each of a million people puts his or her hat into a very large box. Each hat
> has its owner’s name on it. The box is given a good shaking, and then each
> person, one after another, randomly draws a hat out of the box. What is the
> probability that at least one person gets their own hat back?



Most people might think that the chance is very, very small, but it’s not. It’s
actually more than 60%. How can this be true?

We can view this problem as having only two possible solutions (events):
either nobody gets their hat back, or at least one person gets their hat back.
The sum of the probabilities of these events must be equal to one and therefore
if we can work out the probability that nobody gets their own hat back, then the
probability that at least one person does is one minus that.

For each person, there is only a one-in-a-million chance that they have picked
their own hat, and thus a 999999 in 1000000 chance that they have not got their
own hat.







If the probability of all one million people picking the incorrect hat in 0.368,
then via our previous reasoning, the probability of at least one person picking
the correct hat is 1-0.368, or 63.2%.

This is a very counterintuitive result, but the wording of the question is key.
If we changed the wording to exactly one person getting their hat back then our
answer changes dramatically. Starting with our 63.2% chance, we would have to
subtract the chance of two people getting their hats back, and of three people
getting their hats back, and so on … until we reached the very small chance of
one person, and one person only, getting their hat back.

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FIXED AREA, INFINITE PERIMETER

5th Octobergeometry, mathematics

The Koch Snowflake (named after its inventor, the Swedish mathematician Helge
von Koch) is a fractal with a number of interesting properties.



The first four generations of the Koch Snowflake

As the number of generations increases, the area of the snowflake increases, but
it increases towards a limit: eight-fifths of the size of the first (triangular)
generation. This is because each additional generation adds three triangles
which are one-ninth the size of the triangle added in the previous generation
(for a total increase of one-third), and the additional of increasingly small
triangles has a lesser and lesser effect on the overall area as the number of
generations increases.

But as the number of generations increases, the perimeter of the shape continues
to grow, without approaching a limit: each generation of the Snowflake has a
perimeter which is four-thirds of the previous generation’s perimeter. This is
because the additional perimeter added each time is four times one-third of the
length of each side, for a total increase of four-thirds, as opposed to the
one-third increase in area. Because four-thirds is greater than one, the
perimeter tends to infinity, whereas the area (which at one-third is less than
one) does not.



As you can see from the graph above, the area approaches its limit very quickly,
whereas the perimeter grows very quickly (which is why it has to be shown on a
different axis).

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VEBLEN AND GIFFEN GOODS

25th Septembereconomics

There is a fundamental law of economics that says that as the price of a good or
service increases, the demand for that product decreases. This is the Law of
Demand: if prices are high, people cannot buy as much. But there are some
products for which this is not the case.

Veblen Goods do not obey the Law of Demand: as their price increases, so does
demand for them. This is a case of conspicuous consumption, people show off and
demonstrate their status by buying increasingly expensive products. A £10000
gold Rolex watch does not really tell time any better than a £10 Casio, but
owning a £10000 gold Rolex demonstrates that you are so wealthy that you can
afford to spend £10000 on a watch. People do not want to buy a £10 Rolex, but
they do want to buy a £10000 Rolex, to show off how much money they have.



There are some goods to which the Law of Demand does not apply, and which are
not Veblen Goods. These are called Giffen Goods, and on the face of it, they
seem to disobey all rational economic rules: demand for them increases even when
their price increases, despite the fact that they cannot be used to demonstrate
status via conspicuous consumption.

But imagine this situation: You need to feed your family. You normally buy a
mixture of expensive (tasty) and inexpensive (staple) products to provide enough
nutrition, but also some variety. But one day the price of rice quadruples: now
you can’t afford the more expensive products and the rice, and you still need
the staple rice to provide basic nutrition. Thus you buy more rice, even though
its price has increased, and the rice is a Giffen Good.

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RECENT POSTS

 * Why is the Earth Hot? 10th April
 * Phase diagrams 21st December
 * One Million Hats 21st October
 * Fixed Area, Infinite Perimeter 5th October
 * Veblen and Giffen Goods 25th September
 * Types of Perpetual Motion Machines 15th September
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