Deadline for submission: Jan. 12, 2025, 23:59, CET.

Problem Statement of Series 3, Year 38

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Text of Serial Number 3

1... lazy river

3 points

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 0.5ms1 relative to the current. When the distance between Jarda and his friend is only 3m, the cross-section area of the artificial river bed narrows from 4m2 to 3m2. How much time does Jarda have to prepare for the capsize if the initial current speed was 0.8ms1?

We will publish the solution to this problem soon.
~ Target destroyed. Enemy vessel neutralized.

2... Mišo doesn't want to die

3 points

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

We will publish the solution to this problem soon.
~ Mišo found a radioactive source at work.

3... non-coloumbic

6 points

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.

We will publish the solution to this problem soon.
~ Fykosaurus should be awarded a Nobel prize for discovering new fundamental interaction.

4... forever Young

7 points

Marek has a double slit with negligible slit width, immense slit height and a distance between the slits equal to b. Light of wavelength λ is incident on the slit. A nearby screen, where 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?

We will publish the solution to this problem soon.
~ Marek sympathized with the Master.

5... a toast down under

10 points

Consider a wine glass, modeled as a spherical shell with a spherical cap of polar angle θ0,π cut off. What is the angular velocity ω 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 h0,R(1+cosθ).

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~ Radka had to clean a stained carpet.

P... bread

9 points

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.

We will publish the solution to this problem soon.
~ Jarda is studying at a Department of Surface (and Plasma Science).

E... salted

12 points

Determine the specific heat capacity of kitchen salt.

Instructions for Experimental Tasks
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~ Jarda is cooling his tea with salt.

S... electrochemistry 3 – kinetics and efficiency

10 points

  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 sf>0 for the forward reaction and sb<0 for the reverse reaction. Find the relationship between α and sf and sb. – 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 [HA+]=1M. One electrode is connected to a reservoir (i.e., an infinite volume) of gaseous hydrogen with a pressure of 1bar, and the other is also connected to hydrogen of the same pressure, but only a volume of 10l. We apply a voltage of 25mV to the cell, causing gaseous hydrogen to start forming at the second electrode. We reached a pressure of 2bar in the bottle at time t2bar=1.2h. How long did it take for the pressure to increase to 90% of its maximum value? Assume the beakers with HA+ are large enough that their concentrations remain constant during the process, and everything occurs at a temperature of 25C. – 5 points

  3. Consider a Carnot heat engine with the corresponding efficiency, where the cooler temperature is Tc=40C. What is the temperature Th from which would this heat source achieve higher efficiency than the electrochemical reaction of hydrogen forming water, which also occurs at temperature Th? For water vapor, use ΔG100=225kJmol1 and ΔH100=248kJmol1, both valid at a temperature of 100C. – 2 points

Text of Serial Number 3
We will publish the solution to this problem soon.
~ Jarda thinks the 3rd part of the series should be the hardest.