Problem Statement of Series 3, Year 38

Jarda is lying on his pool float in a water park and he is being carried away by an artificial river current, when he notices his friend swimming toward him, intending to capsize Jarda. His friend is swimming at speed $0.5\,\mathrm{m\!\cdot\! s^{-1}}$ relative to the current. When the distance between Jarda and his friend is only $3\,\mathrm{m}$, the cross-section area of the artificial river bed narrows from $4\,\mathrm{m^2}$ to $3\,\mathrm{m^2}$. How much time does Jarda have to prepare for the capsize if the initial current speed was $0.8\,\mathrm{m\!\cdot\! s^{-1}}$?

Mišo found an old radioactive source that contains $\ce{^{90}Sr}$ at work. In 1974, it had an activity of $5\,\mathrm{mCi}$. How long would Mišo have to be irradiated evenly with this radioactive source to reach a lethal dose of $10\,\mathrm{Sv}$? Assume that all subsequent decays are instantaneous and that all decay products are absorbed uniformly by Mišo. Mišo weighs $65\,\mathrm{kg}$ and the source was discovered in 2020.

Fykosaurus discovered a previously unknown type of interaction while being in a lab. He found a small spherical object in a dusted cabinet. When he placed a point mass with a mass $m$ and released it, the point mass always collided with the sphere after a specific time $t$. Determine the force by which the mass point is attracted to the unknown object as a function of their mutual distance. Consider that everything takes place on a horizontal plane without resistive forces in the framework of classical mechanics. In addition, the Fykosaurus attached the unknown object to a mounting pad so it remains at rest relative to the room.

Hint: Try to find an analogy to a force you know.

Marek has a double slit with negligible slit width, immense slit height; the distance between the slits equals $b$. Light of wavelength $\lambda$ is incident on the slit. A nearby screen on which the interference pattern is formed is moving away from the double slit at a small velocity $v$. What is the velocity of the $m$-th order maximum on the screen?

Consider a wine glass, modeled as a spherical shell with a spherical cap of polar angle $\theta\in \langle 0,\pi\rangle$ cut off. What is the angular velocity $\omega$ needed to keep its contents from spilling if the glass is rotating around its axis of symmetry while being upside-down? If the glass is stationary on a table, the level of wine is at height $h\in \langle 0,R(1+\cos\theta)\rangle $.

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

We can squeeze bread quite well, as there are a lot of cavities filled with gas. Determine the inner surface of all such cavities in a sourdough loaf.

Determine the specific heat capacity of kitchen salt.

1. In previous parts, we used a model where Gibbs free energy increases linearly and then decreases depending on the reaction coordinate to calculate the change in activation barriers for both the forward and the reverse reaction. Consider the slopes of these straight lines in figure 4 of this part of the series to be $s_{\mathrm{f}} > 0$ for the forward reaction and $s_{\mathrm{b}} < 0$ for the reverse reaction. Find the relationship between $\alpha$ and $s_{\mathrm{f}}$ and $s_{\mathrm{b}}$. – 3 points

2. It is possible to use an electrochemical cell to compress gases instead of mechanical pistons. Consider a simplified model of such a cell for hydrogen compression. Assume we have two standard hydrogen electrodes in beakers, filled with a solution of concentration $\left[\ce{H^+} \right] = 1\,\mathrm{M}$. One electrode is connected to a reservoir (i.e., an infinite volume) of gaseous hydrogen with a pressure of $1\,\mathrm{bar}$, and the other is also connected to hydrogen of the same pressure, but only a volume of $10\,\mathrm{l}$. We apply a voltage of $25\,\mathrm{mV}$ to the cell, causing gaseous hydrogen to start forming at the second electrode. We reached a pressure of $2\,\mathrm{bar}$ in the bottle at time $t_{2\,\mathrm{bar}} = 1.2\,\mathrm{h}$. How long did it take for the pressure to increase to $90\,\mathrm{\%}$ of its maximum value? Assume the beakers with $\ce{H^+}$ are large enough that their concentrations remain constant during the process, and everything occurs at a temperature of $25\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. – 5 points

3. Consider a Carnot heat engine with the corresponding efficiency, where the cooler temperature is $T_{\mathrm{c}} = 40\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. What is the temperature $T_{\mathrm{h}}$ from which would this heat source achieve higher efficiency than the electrochemical reaction of hydrogen forming water, which also occurs at temperature $T_{\mathrm{h}}$? For water vapor, use $\Delta G_{100} = 225\,\mathrm{kJ\!\cdot\! mol^{-1}}$ and $\Delta H_{100} = 248\,\mathrm{kJ\!\cdot\! mol^{-1}}$, both valid at a temperature of $100\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. – 2 points