Problem Statement of Series 4, Year 36

Robert found out that he had to put 3 batteries with capacity $1~000\,\mathrm{mAh}$ and voltage $U=1.5\,\mathrm{V}$ into his new headlamp. In the headlamp, the batteries are connected in series. How long does it take for the batteries to discharge if they power a headlamp of output power $P=5\,\mathrm{W}$ and efficiency $\eta=90\,\mathrm{\%}$?

A balloon of mass $m_{\mathrm{b}}=2.7\,\mathrm{g}$ and volume $V_0=4\,\mathrm{l}$ was filled with helium of the same temperature as the surrounding air, i.e., $T_0=20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. Inside the balloon, the pressure is $\Delta p=2\,\mathrm{kPa}$ higher than in the surrounding area. To what temperature do we need to cool the balloon and the gas in it so it stops floating? Assume that there will be atmospheric pressure in the balloon after cooling down.

We all know it – road closures and endless standing at traffic lights. The light is green for $60\,\mathrm{s}$, but by the time everyone gets going, it is red again. Consider the $0.5\,\mathrm{s}$ reaction time for a driver to get moving after the car in front of him has done so. By what percentage would the number of cars that pass through the closure increase if everyone in line started moving simultaneously? The first car stands at the traffic light level, the distance of the front bumpers of all cars is estimated to be $5\,\mathrm{m}$, and they all accelerate uniformly for $5\,\mathrm{s}$ to a speed of $30\,\mathrm{km\cdot h^{-1}}$, with which they proceed further into the closure.

We have an astronomical (Keplerian) telescope that we want to launch into space. First, however, we will try it on Earth, where we will measure the magnification $Z$. How does the distance between the lenses have to change for it to have the same magnification in space? Lenses have a refractive index of $n$.

Two aliens each live on their own space station. The stations are in free space and the distance between them is $L$. When one alien wants to visit the other, he has to board his non-relativistic rocket and fly to his neighbor. What is the shortest time an alien can spend on its way there and back? The mass of the rocket with fuel is $m$, without fuel $m_0$. The exhaust velocity is $u$. The fuel flow is arbitrary, and his neighbor won't let him load any fuel (he has little himself).

Discuss what physical phenomena affect the cruising speed of a ship and submarine. What resistive forces act on them? What is the highest cruising speed that a ship or submarine can sail?

We have a rope wrapped around a bar with a weight of mass $m$ at one end. Measure the dependence of the mass of the weight $M$ at the other end, needed to set the rope in motion, on the number of times the rope wraps around the bar.

1. At the beginning of the series, we mentioned a couple of approximations we made – fixing the nuclei and neglecting relativistic effects. For which chemical elements would you expect the motion of electrons to have the strongest influence on the motion of the nuclei, and why? In which part of the periodic table do you think relativistic effects will be most apparent? What is the reason for that? $\left(2\,\mathrm{pts}\right)$
2. The total energy of a water molecule, obtained from a quantum chemical calculation, is approximately $-75\,\mathrm{Ha}$. The energy released by the fusion of hydrogen and oxygen into water is $242\,\mathrm{kJ\cdot mol^{-1}}$. If we calculate the energy of the reactants and products with an error of $1\,\mathrm{\%}$, how big will the error be in determining the reaction energy? Furthermore, try to find an analogy for this based on real-life measurements. (For example: “Weighing myself with a five-crown coin and without it to determine its weight.”) $\left(3\,\mathrm{pts}\right)$
3. Install the program Psi4 and try to calculate the difference of energies of the chair and (twist-)boat conformations of cyclohexane. You can use the attached input files, where the geometry is already optimised. How much does the result differ from the experimental value $21\,\mathrm{kJ\cdot mol^{-1}}$? $\left(2\,\mathrm{pts}\right)$

   Note: If you encounter a problem with Psi4, please feel free to contact me at mikulas@fykos.cz

4. Try calculating the reaction energy for the chlorination of benzene $\ce{C_6H_6} + \ce{Cl_2} \Rightarrow \ce{C_6H_5Cl} + \ce{HCl}$. Compare it with the experimental value of $-134\,\mathrm{kJ\cdot mol^{-1}}$. You can use the included geometry of the benzene molecule. $\left(3\,\mathrm{pts}\right)$

   Bonus: Choose your favorite (or any other) chemical reaction and calculate its energy. (up to $+3\,\mathrm{pts}$)