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## chemistry

### (10 points)6. Series 36. Year - S. exciting quanta

The lowest-lying excited singlet state of beta-carotene has an energy $1{,}8 \mathrm{eV}$, which is higher than the ground state energy. However, the transition between this state and the ground state is prohibited, so the molecule does not absorb photons at this energy. On the other hand, the transition to the second lowest-lying singlet state with energy $2{,}4 \mathrm{eV}$ is allowed and responsible for the bright orange color of the molecule. The lowest-lying triplet level is at $0{,}9 \mathrm{eV}$ energy. Draw a Jablonski diagram and use it to explain why beta-carotene does not fluoresce even though it significantly absorbs visible light. $\(3 \mathrm{pts}\)$

**Bonus::** Why is it so important for life on earth that oxygen is a triplet in the ground state? $\(+1 \mathrm{pts}\)$

Try to calculate the approximate limit on the number of orbitals in the active space with the CASSCF method. Consider that you have as many electrons as orbitals in the active space (which corresponds to the fact that half of them in $\ce {HF}$ will be occupied) and that the most of today's supercomputers have at most $1 \mathrm{TB}$ of RAM for computing, in which you need to fit a Hamiltonian. $\(3 \mathrm{pts}\)$

For lithographic manufactured modern semiconductor chips, so-called excimer lasers are used to glow with the spectrum far into UV region. They are based on so-called excimers, which are molecules that are stable only in the excited state, while in the ground state, they decay. As a result, the molecule decays after the photon is emitted, ensuring that a larger fraction of the molecules are in the higher state than in the lower state. That is the necessary condition for the laser to work. Try using Psi4 for the helium dimer ($\ce{He}_{2}^*$) to calculate and plot the dissociation curves of the ground and lowest-laying excited states. ($\ce{He}_{2}^*$) is not yet used for lasers, but for example $\ce{Ar}_{2}^*$ or $\ce{Kr}_{2}^*$ are.) At what wavelength would the laser work? Compare it with the experimental wavelength $66 \mathrm{nm}$. $\(4 \mathrm{pts}\)$

**Note::** In the problem statement on the website, you will find a prepared input file for one geometry. Do not be surprised that it has a total of three states set up. It needs to have those because we have two excited states close to each other. If we were to include only one of them in the calculations for some internuclear distances, this would lead to problems with convergence.

### (3 points)3. Series 34. Year - 1. baking

While baking a gingerbread, baking soda, or more rigidly sodium bicarbonate ($\ce {NaHCO3}$), has to be added into the batter. Let's assume, that at high temperatures sodium bicarbonate decomposes as follows \[\begin{equation*} \ce {2 NaHCO3 \rightarrow Na2CO3 + H2O + CO2} , \end {equation*}\] that is, into sodium carbonate, carbon dioxide and water. How much will the volume of the gingerbread increase as a consequence of creation of water steam and carbon dioxide bubbles in the batter after adding $10 \mathrm{g}$ of sodium bicarbonate? Assume that the water steam and carbon dioxide behave as ideal gases and that the batter solidifies around the bubbles at temperature $200 \mathrm{\C }$ and pressure $1~013 hPa$.

Káťa wanted to bake a cake.

### (10 points)2. Series 33. Year - P. Earth fired up

Estimate about how much would a content of $\ce {CO2}$ in the atmosphere rise, if all flora on Earth was burnt down?

Karel is pyromaniac.

### (10 points)2. Series 31. Year - P. ooh Oganesson

What properties does the $118^{\rm th}$ element in the Periodic table have? Alternatively, what sort of properties would it have, had it been stable? Discuss at least three physical qualities.

Karel wanted to have something on extrapolation.

### (6 points)6. Series 29. Year - S. A closing one

- Find, in literature or online, the change of enthalpy and Gibbs free energy in the following reaction

$$2\,\;\mathrm{H}_2 \mathrm{O}_2\longrightarrow2\,\mathrm{H}_2\mathrm{O},$$

where both the reactants and the product are gases at standard conditions. Find the change of entropy in this reaction. Give results per mole.

- Power flux in a photon gas is given by

$j=\frac{3}{4}\frac{k_\;\mathrm{B}^4\pi^2}{45\hbar^3c^3}cT^4$.

Substitute the values of the constants and compare the result with the Stefan-Boltzmann law.

- Calculate the internal energy and the Gibbs free energy of a photon gas. Use the internal energy to write the temperature of a photon gas as a function of its volume for an adiabatic expansion (a process with $δQ=0)$.

**Hint:** The law for an adiabatic process with an ideal gas was derived in the second part of this series (Czech only).

- Considering a photon gas, show that if $δQ⁄T$ is given by

$$\delta Q / T = f_{,T} \;\mathrm{d} T f_{,V} \mathrm{d} V\,,$$

then functions $f_{,T}$ and $f_{,V}$ obey the necessary condition for the existence of entropy, that is

$$\frac{\partial f_{,T}(T, V)}{\partial V} = \frac{\partial f_{,V}(T, V)}{\partial T} $$

### (6 points)5. Series 29. Year - S. naturally variant

- Use the relation for entropy of ideal gas from the solution of third serial problem

$$S(U, V, N) = \frac{s}{2}n R \ln \left( \frac{U V^{{\kappa} -1}}{\frac{s}{2}R n^{\kappa} } \right) nR s_0$$

and the relation for the change of the entropy

$$\;\mathrm{d} S = \frac{1}{T}\mathrm{d} U \frac{p}{T} \mathrm{d} V - \frac{\mu}{T} \mathrm{d} N$$

to calculate chemical potential as a function of $U$, $VaN$. Modify it further to get the function of $T$, $pandN$.

**Hint:** The coefficients like 1 ⁄ $T$ in front of d$U$ can be calculated as a partial derivative of $S(U,V,N)$ by $U$. Don't forget that ln$(a⁄b)=\lna-\lnb$ and that $n=N⁄N_{A}$.

**Bonus:** Express similarly temperature and pressure as functions of $U$, $VandN$. Eliminate the pressure dependence to get the equation of state.

- Is the chemical potential of an ideal gas positive or negative? (Assume $s_{0}$ is negligible.)?

- What will happen with a gas in a piston if the gas is connected to a reservoir of temperature $T_{r}?$ The piston can move freely and there is nothing acting on it from the other side. Describe what happens if we allow only quasistatic processes. How much work can we extract? Is it true that the free energy is minimized?

**Hint:** To calculate the work, this equation can be useful:

$$\int _{a}^{b} \frac{1}{x} \;\mathrm{d}x = \ln \frac{b}{a}.$$

- We defined the enthalpy as $H=U+pV$ and the Gibb's free energy as $G=U-TS+pV$. What are the natural variables of these two potentials? What other thermodynamic quantities do we obtain by differentiating these potentials by their most natural variables?

- Calculate the change of grandcanonic potential d$Ω$ from its definition $Ω=F-μN$.

### (2 points)3. Series 29. Year - 2. alchemist's apprentice

The young alchemist George has learnt to measure electrochemical equivalents. He measured quite precisely the electrochemical equivalent $A=(6.74±0.01)\cdot 10^{-7}\;\mathrm{kg}\cdot C^{-1}$ of an unknown sample. How can he determine what substance was his sample made of?

### (8 points)3. Series 29. Year - E. hydrogel

Examine the dependence of a weight of a hydrogel ball on a time of submersion in a water and on a concentration of salt dissolved in water.
*Note* We do not send the experimental material abroad, therefore the hydrogel you buy must be described in detail.

### (2 points)2. Series 29. Year - 2. numismatic

Once in a while, a situation may occur, that the nominal value of coins is lower that their manufacturing costs. Assume we have two coins, made of a gold-silver alloy. The first one has diameter $d_{1}=1\;\mathrm{cm}$, second one $d_{2}=2\;\mathrm{cm}$, both have thickness $h=2\;\mathrm{mm}$. If we submerge them in mercury, the smaller one sinks to the bottom, whilst the larger one rises to the surface. If we submerge both coins, smaller one on top of the larger, they neither rise nor sink. Assuming the smaller coin is made of pure gold, determine the fraction of silver in the larger coin (in percent of mass).

**Bonus:** How would the result change if the smaller one could contain silver as well?

Mirek má radši mince než bankovky.

### (8 points)2. Series 29. Year - E. let's do some Fizzics!

Buy any effervescent (i.e. fizzy) tablets and measure the time that takes for the tablet to fully dissolve in water as a function of temperature of this water. Discuss the possible causes and propose why is the relation the way it is.

Aleš Podolník umíral na rýmu.