Problem Statement of Series 2, Year 38

A certain computer science student is mining bitcoins. His graphics cards yield $0.2\mathrm{BTC}$ per year and have the energy consumption of $3000\mathrm{W}$. However, the student reaches an epiphany thinking about how much carbon dioxide he releases into the Earth's atmosphere. He therefore decides to use the money gained from Bitcoin mining to buy trees which act as natural carbon capture mechanism. What would the price of a bitcoin have to be to make such an activity profitable? Suppose the price of one tree is $1000\mathrm{CZK}$ and each tree can capture $25\mathrm{kg}\cdot {\mathrm{year}}^{-1}$ $C{O}_2$. Consider two energy sources – coal with the price $5.32\mathrm{CZK}\cdot {\mathrm{kWh}}^{-1}$ and emissions of $0.82\mathrm{kg}\cdot {\mathrm{kWh}}^{-1}$ and a hydroelectric power plant with the price of $4.00\mathrm{CZK}\cdot {\mathrm{kWh}}^{-1}$ and emissions of $0.012\mathrm{kg}\cdot {\mathrm{kWh}}^{-1}$.

Monika has attempted to create her own kind of tree. However, as a physicist, she disregarded all biological phenomena and aspects. For her tree, she took a long capillary tube with a diameter of $d=0.1\mathrm{mm}$. How high does the water rise in this physics-based tree? For which fluid are such trees the tallest? Additionally, by neglecting which physical principle does the height of this “tree” differ so significantly from that of real trees?

Consider a homogeneous pyramid with density ${\rho }_{\mathrm{j}}=250\mathrm{kg}\cdot {\mathrm{m}}^{-3}$ floating on water with density $\rho =1000\mathrm{kg}\cdot {\mathrm{m}}^{-3}$. While floating, its axis is vertical. Is the position more stable when the apex of the pyramid is pointing up or down? The height of the pyramid is $h=20\mathrm{cm}$ and the surface area of its base is $S=49{\mathrm{cm}}^2$.

We have an empty container of height $H$. The container has a square base of side length $a$. A faucet is positioned directly above the container, and at time $t=0\mathrm{s}$, water begins to flow out of the faucet with an initial velocity $v_0$. Calculate the dependence of the water level in the tank on time $t$. The volumetric flow rate of water $Q$ is constant. Assume that $Q$ is small enough for the water level in the container to settle instantaneously at a uniform height. However, do not forget about the time it takes the liquid to fall.

Consider a cylindrical capacitor with internal radius $r_1$ and external radius $r_2$. The capacitor is charged so that the voltage between the two electrodes is $V$. Electrons with a small angle distribution $\mathrm{\Delta }\alpha$ are flying out perpendicular to the radius of the cylinder at a distance $r$ ($r_1<r<r_2$) and at such speed that their distance from the center of the cylinder is approximately constant. Determine the location of the first point in which the electrons focus again. The situation is planar and do not consider the space charge of the electrons.

Let the quality of acclimatization to cold temperatures be defined based on the thermal power that a person must exert. Is it better to acclimatize under flowing water or by immersing oneself in a bathtub of cold water? Consider at least the different water temperatures, flow rate, and the temperature of the surroundings.

Measure the deformation curve for an ordinary rubber band.

1. Determine the voltage of the following electrochemical reactions under standard conditions. Does the reaction occur spontaneously? (2 points)
   1. $\mathrm{CaCl}{\phantom{A}}_2(\mathrm{l})⟶\mathrm{Ca}(\mathrm{s})+\mathrm{Cl}{\phantom{A}}_2(\mathrm{g})$,
   2. $\mathrm{Pb}(\mathrm{s})+\mathrm{PbO}{\phantom{A}}_2(\mathrm{s})+2\mathrm{H}{\phantom{A}}_2\mathrm{SO}{\phantom{A}}_4(\mathrm{aq})⟶2\mathrm{PbSO}{\phantom{A}}_4(\mathrm{s})+2\mathrm{H}{\phantom{A}}_2\mathrm{O}(\mathrm{l})$.
2. Determine which of the following electrochemical half-cells will reduce and what will be the voltage after they combine. (2 points)
   1. $\mathrm{Ni}{\phantom{A}}^{2+}(\mathrm{aq})|\mathrm{Ni}(\mathrm{s})$ a $\mathrm{Au}{\phantom{A}}^{3+}(\mathrm{aq})|\mathrm{Au}(\mathrm{s})$,
   2. $(\mathrm{NO}{\phantom{A}}_3){\phantom{A}}^{-}(\mathrm{aq})|\mathrm{NO}(\mathrm{g})|\mathrm{Pt}(\mathrm{s})$ in an acidic solution and $\mathrm{Fe}{\phantom{A}}^{2+},\mathrm{Fe}{\phantom{A}}^{3+}|\mathrm{Pt}$
3. Let us consider a fuel cell which, under standard conditions, produces electric energy during the water-producing reaction $2\mathrm{H}{\phantom{A}}_2(\mathrm{g})+\mathrm{O}{\phantom{A}}_2(\mathrm{g})⟶2\mathrm{H}{\phantom{A}}_2\mathrm{O}(\mathrm{g})$. Determine the energy density in relation to the mass of hydrogen (in $\mathrm{J}\cdot {\mathrm{kg}}_{\mathrm{H}{\phantom{\mathrm{A}}}_2}^{-1}$) that is released in this situation. Determine the constant of equilibrium $K$ for this reaction and discuss its value. (2 points)
4. Now, let us consider a electrochemical cell $\mathrm{Cu}|\mathrm{Cu}{\phantom{A}}^{2+}(\mathrm{aq})||\mathrm{Ag}{\phantom{A}}^{+}(\mathrm{aq})|\mathrm{Ag}$. The initial concentration of copper in the solution is $\left[\mathrm{Cu}{\phantom{A}}^{2+}\right]=0.40\mathrm{M}$ while the concentration of silver is $\left[\mathrm{Ag}{\phantom{A}}^{+}\right]=0.50\mathrm{M}$. What is the concentration of silver when the cell's voltage is equal to $0.40\mathrm{V}$? (4 points)

   Special motivation: You might encounter a problem like this on the electrochemistry exam during your master's studies at MFF. Can you solve it already in high school?