Problem Statement of Series 1, Year 39

A water goblin lives at the bottom of the Dead Sea. He, as is common for such a folklore creature, traps the souls of people who have drowned in the sea, keeping them in small mugs.

However, the jars are quite small and therefore cannot contain the souls, he has to reduce the volume five-fold by an isothermic process.

One day, the goblin got bored of the Dead Sea and decided to move to the bottom of the Orlík reservoir in southern Bohemia. What is the minimum depth he has to live in at his new place such that the souls are not able to escape by lifting the physical lids?

Before moving, the water goblin used to live at the depth of $33\,\mathrm{m}$ such that ascending by any amount would allow the jars to open. Consider the souls to be an ideal gas with the same temperature as that of the surrounding water. At the moment of drowning, just before their compression, the souls are under atmospheric pressure.

The water temperature of the Dead Sea equals $23\,\mathrm{^\circ\mskip-2mu\mathup{C}}$, water temperature of the Orlík reservoir equals $6\,\mathrm{^\circ\mskip-2mu\mathup{C}}$, density of the Dead sea equals $1.24\,\mathrm{g\cdot cm^{-3}}$, while the density of the Orlík reservoir equals $0.99\,\mathrm{g\cdot cm^{-3}}$. Consider water properties to be constant in both bodies of water. The jars were properly sealed for the duration of moving from the Dead Sea to the Orlík reservoir.

Consider a spider web formed by a circle with radius $r$ with two perpendicular crossbars, which go through the middle of the web. We put $N-1$ cocentric circles inside the original circle in a certain way, so that the crossbars always intersect at the distance $r/N$. How much resistance is between the ends of one crossbar? Material of the web has linear resistivity $\lambda$ and knots are conductively connected.

Bonus: Determine resistance for three crossbars, which are rotated by $120^\circ$.

A cake on Jarda's plate sometimes topples over to its other side. What is the work needed for such misfortune? Consider polar symetric cake of weight $M$ and of radius $R$, its center of mass is located in height $h$; solve universally for its cutting to $n$ same pieces.

Bonus: Bake your own cake and send us a photo and a recipe. The best will be shared in the problem solution!

This problem has an open solution, so be sure to cite all sources used.

Imagine that you are riding a bike during a windy day in wavy terrain. Discuss the characteristics of the movement in different directions and different velocities qualitatively. Also, discuss how the description of the movement would change if we neglected the following parameters:

• air resistance,
• rolling resistance of the wheels,
• slope of the terrain,
• other resisting forces.

Sometimes, maybe even some of you are wondering how far a small item can get right after it slips from your hands and falls to the ground. Choose a small item (screw, nail, and so on) and measure the distance between the place where you dropped the item and the place where you have found it, in relation to the height (from the ground) from which you dropped it. For each initial height, repeat the measurement enough times to decrease the statistical error of your measurement. Think about suitable surfaces and items.

1. A layer of iridium $\ce{Ir}$ was sputtered onto a ruthenium $\ce{Ru}$ monocrystal, the process took $240\,\mathrm{min}$. The attenuation of the sublayer signal was measured after the deposition using XPS (Al K$\alpha$, $hf=1486.71\,\mathrm{eV}$), see the final spectrum in the figure below.

   

   Determine the thickness of the deposited layer of $\ce{Ir}$. Assume that the emission was perpendicular to the surface and the effective photoelectron mean free path in $\ce{Ir}$ is $\lambda_{\ce{Ir}}=1.6\,\mathrm{nm}$. – 4 points

2. Consider an iridium sample covered by $2\,\mathrm{nm}$ thick layer of oxide $\ce{IrO_2}$ on its surface. Our goal is to study only the active $\ce{IrO_2}$. This could be achieved by measuring on a synchrotron with the primary photon energy set to $\approx 150\,\mathrm{eV}$, such that we would reach the minimum of photoelectron energy as shown on Figure.

   

   How could we solve this without a synchrotron? Try to find the specific conditions under which this the result can be reached with X-ray tube capadle of energy $1486.71\,\mathrm{eV}$, resulting in the photoelectron mean free path of $\lambda_{\ce{IrO_2}} = 3.2\,\mathrm{nm}$. Does this approach have any disadvantages? – 2 points

   Hint: Use the formula \begin{equation*} I(d) = I_0 \exp\!\left(-\frac{d}{\lambda \cos\theta}\right) \,. \end{equation*}

3. Figure  shows a diffractogram of polycrystalline cerium dioxide measured at photon energy of $77\,\mathrm{keV}$. To which interplanar spacing do the first 3 maxima correspond to? How do these planes look like in the crystal lattice? And at what angle $2\theta$ would we measure them if we used X-ray with energy of $90\,\mathrm{keV}$? – 4 points