(10 points)3. Series 36. Year - S. quantum of orbital

Similarly to the series, use the Hückel method to create the Hamiltonian matrix for the cyclobutadiene molecule and verify that its eigenvalues are $\alpha +\beta $, $\alpha $, $\alpha $, $\alpha -\beta $. Sketch the diagram of the final energies in the resulting orbitals. And show how the electrons will occupy them. $(4~b)$ Bonus: What is the main difference in the characterics of these orbitals and their occupancy compared to a benzene molecule we showed in the series? What are the consequences for the cyclobutadiene molecule? $(2~b)$

Try going back to the beta-carotene molecule and calculate again at what wavelength it should absorb using the Hückel method. What should the value of the parameter $\beta $ be equal to in order to be consistent with the experimental results Alternative: If you encounter a problem with the diagonalisation of the hamiltonian, solve the problem statement with the hexa-1,3,5-triene molecule. The experimentally determined absorption value in this case is at a wavelength of $250 \mathrm{nm}$. $(4~b)$

What happens to a molecule (a molecule with only simple bonds is sufficient) if we use UV light to excite an electron from the $\sigma $ to the $\sigma ^\ast $ orbital? $(2~b)$

(10 points)2. Series 36. Year - P. planetary atmosphere

What parameters does a planet need to have to keep its atmosphere comparable to the Earth? What conditions are essential for the planet to gain such an atmosphere?

Imagine we create a small soap bubble with a bubble blower. How fast does it fall to the ground? The bubble has an outer radius $R$ and an areal density $s$.

How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma^{1)}

^{1)}

The density of plasma $n_e$ is typically expressed as a funciton $n_e = f\(\frac {x}{x_c}\)$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume

that the density of the plasma decreases exponentially with distance from the target,

that the density of the plasma decreases linearly with distance from the target.

What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.

What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?

Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?

(13 points)2. Series 35. Year - E. light or dense ethanol

Measure the dependance of the density of an alcohol solution in water on its volume concentration in water. Include also the measurement of pure alcohol and pure water for comparison.

Be careful when mixing alcohol and water – remember that the volume of the mixture is not exactly the sum of their original volumes.

Karel was thinking that the participants might have a little sniff.

(10 points)2. Series 35. Year - S. compressing

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? Note: Assume that the initial plasma is a monatomic ideal gas.

By using the graph of fusion reaction rate (sometimes called volume rate) as a function of temperature in the Serial study text, derive the Lawson criterion for the inertial-confinement-fusion time for a temperature of your choosing, while considering the following reactions:

deuterium - deuterium,

proton - boron,

deuterium - helium-3.

Determine the product of the size of a fuel pellet, and the density of a compressed fuel for each case. Are there any advantages of these reactions compared to the traditional DT fusion?

What form would the Lawson criterion take for the non-Maxwellian velocity distribution, considering the case with the following kinetic energy of a particle

$E\_k = k\_B T^\alpha $,

$E\_k = a T^3 + b T^2 + c T$.

Could such a fusion be even possible? If so, what (the fuel) should drive the fusion reaction, what is the ideal size of the fuel pellet and what density should it be compressed to?

(10 points)4. Series 34. Year - P. Fykos bird on vacation

How would aviation work on other planets (with atmosphere)? Consider mainly jet aircraft. Which planetary parameters would influence the aviation positively and which negatively, compared to Earth's?

You have probably heard at school about the thermal motion of molecules such as diffusion or Brownian motion. Measure the time dependance of the size of a color spot in water and calculate the diffusion constant. Make measurements for several different temperatures and plot the temperature dependance of the diffusion constant in a graph. How could you arrange the experiment so that the temperature would stay constant during the measurement?