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## chemistry

### (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 \[\begin{equation*} \ce{2 NaHCO3} \rightarrow \ce{Na2CO3} + \ce{H2O} + \ce{CO2} , \end {equation*}\] 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 $1\;013 \mathrm{hPa}$.

Káťa wanted to bake a cake.

### (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.

### (2 points)1. Series 29. Year - 1. densifying Hofmann

During the electrolysis in a Hofmann voltameter the electolyte is a soluton of sulfuric acid in water. The mass of the acid in the solution is practicaly constant but as the name says the water slowly dissolves into hydrogen and oxygen. So the concentration of the acid in the solution rises. How long will take for the mass fraction of the acid in the solution to rise to twice the original amount if there was a current of $I=1A$ passing through the solution, the original mass fraction of the acid in the solution was $w_{0}=5%$ and the volume of the solution in the container was$V_{0}=2l?$

Karel was thinking about electrolysis again.

### (4 points)6. Series 27. Year - 4. insatiable spider

In a dark corner there lurks a spider that has just caught a fly and is slowly devouring it. Assume that the consumption follows such an equation:

$$\;\mathrm{A} + \mathrm B \mathop{\rightleftharpoons}_{k_{-1}}^{k_1} \mathrm{AB} \stackrel{k_2}{\longrightarrow} \mathrm C + \mathrm B\,,$$

where A is *fly substrate*, B are the digestive compounds (there is always enough of themu) and C is the product of digestion. AB denotes the unstable intermediate product. The reaction is of the first order, in other words the speed is directly proportional to the concentration of the said substance. Determine how long will take the spider to digest the fly and begin hunting again, if its receptors will tell it that it is hungry once the substrate reaches 10 % of the original value.

*Tip* Use the approximation of the stationary state of intermediate product.

Mirek reminiscing about Bestvina.