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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

More or Less?

Are these estimates of physical quantities accurate?

Big and Small Numbers in Chemistry

Age 14 to 16
Challenge Level

1) According to the Royal Mint, a copper coin weighs 3.56g. The relative atomic mass (RAM) is 63.5. The coin therefore contains approximately $\frac{3.56}{63.5}\times6.02\times10^{23} = 3.28\times10^{22}$ copper atoms. Assuming each atom is neutral, each atom contains 29 electrons, so there are $9.52\times10^{23}$ electrons in one coin. 

2) I estimated that a lightbulb contains 100ml of air. The mole fraction of argon in dry air is 0.00934. We therefore need $\frac{100}{0.00934}ml = 1.07\times10^4ml = 10.7l$ of air. 

3) Assuming the density of air is around 1.2kg/m$^3$, the mass of the air from q2 is $1.2\times10.7\times10^{-3}$kg$ = 1.3\times10^{-2}$kg. If this is cooled to a temperature where it's all liquid and it's kept at standard pressure, we can assume it has the standard density of liquid air: 870 kg/m3. It would approximately occupy $\frac{1.3\times10^{-2}}{870}m^3 = 1.5\times10^{-5}m^3 = 0.015l$. 

4) The percentage mass of carbon in teflon is $100\times\frac{2\times12}{2\times12+4\times19} = 24\%$. There's an image of the polymer here

5) The number of molecules of the monomer in a chain of length 1cm is $\frac{0.01m}{140pm\times2} = 3.5\times10^7.$ There are therefore $1.4\times10^8$ atoms of fluorine in such a chain. 

One mole of $F_2$ gas contains 2 moles of F atoms. The number of moles of fluorine gas is therefore $\frac{ 2.8\times10^8}{2\times(6\times10^{23})} = 2\times10^{-16}$ moles. 

6) The reaction is 2M (s) + 2H2O (l)$\rightarrow$ 2MOH (aq) + H2 (g), producing hydrogen gas. To produce 1.5 moles of gas, we'd need 3 moles of water, which weighs 54g and hence has volume 54ml. 

7) The nucleus takes up a percentage volume $100\times\frac{\pi\times(1.75fm)^3}{\pi\times(120pm)^3} = 3\times10^{-13}\%$. The rest of the atom is a cloud of negatively charged electrons, which have much, much smaller mass than the nucleus. 

8) Given the percentage mass of oxygen of your body is 65%, the percentage mass of carbon of your body is $65\times\frac{9.5}{25.5}\times\frac{12}{16} = 18$%. (This accounts for carbon being less abundant, and also a carbon atom is lighter than an oxygen atom.)

Suppose the average atomic mass is 10gmol$^{-1}$. Then the number of atoms in a human weighing 80kg is $\frac{8\times10^4}{10}\times6\times10^{23} \approx 5\times10^{27}$. 

9)A lower bound for this distance is $5\times10^{27}\times120pm  = 6\times10^{17}m$ (as hydrogen is the smallest atom). This is about 63 lightyears (the distance light travels in 63 years). There is apparently a masssive planet this distance away.