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## thermodynamics

### (10 points)5. Series 35. Year - S. stabilizing

- What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of

- $0,2 \mathrm{\micro m}$,
- $5 \mathrm{\micro m}$.

- How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
- What else can help us minimize the RT instability?

### (10 points)3. Series 35. Year - 5. blacksmith's

Gnomes decided to forge another magic sword. They make it from a thin metal rod with radius $R=1 \mathrm{cm}$, one end of which they maintain at the temperature $T_1 = 400 \mathrm{\C }$. The rod is surrounded by a huge amount of air with the temperature $T_0 = 20 \mathrm{\C }$. The heat transfer coefficient of that mythical metal is $\alpha = 12 \mathrm{W\cdot m^{-2}\cdot K^{-1}}$ and the thermal conductivity coefficient is $\lambda = 50 \mathrm{W\cdot m^{-1}\cdot K^{-1}}$. The metal rod is very long. Where closest to the heated end can gnomes grab the rod with their bare hands if the temperature on the spot they touch is not to exceed $T_2 = 40 \mathrm{\C }$? Neglect the flow of air and heat radiation.

Matěj Rzehulka burnt his fingers on metal.

### (10 points)3. Series 35. Year - S. igniting

- Determine the reach of helium nuclei in central hot spot (using the figure ).
- What energy must be released in the fusion reactions in order for the fusion to spread to the closest layer of the pellet? How thick is the layer?
- Estimate the most probable amount of energy transferred from helium nucleus to deuterium. How many collisions on average does the helium nucleus undergo in the central hot spot before it stops?

### (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.

### (14 points)1. Series 35. Year - E. Is the pasta ready?

Measure the dependency of the time it takes for water to start boiling on its volume. Repeat the measurement several times for at least five different volumes. Pay attention to the consistency of the external conditions, especially the criterion you use for assessing when the water starts boiling and the initial temperature of the water, vessel and stove. Try to explain the resulting relation.

Dodo's fight with the stove at the dormitory.

### (10 points)1. Series 35. Year - S. commencing fusion

- Determine the energy gain of the following reactions and the kinetic energy of their products

\[\begin{align*} {}^{2}\mathrm {D} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {T} + \mathrm {p} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {He} + \mathrm {n} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {n} ,\\ {}^{3}\mathrm {He} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} + \mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + {}^{2}\mathrm {D} ,\\ \mathrm {p} + {}^{11}\mathrm {B} &\rightarrow 3\;{}^{4}\mathrm {He} ,\\ {}^{2}\mathrm {D} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {p} . \end {align*}\]

- 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?

### (6 points)5. Series 34. Year - 3. involuntary breatharianism

Lukáš wanted to cook himself a dinner. He put a pot onto a stove, but forgot to fill it with water (or anything else). The teperature of the pot and the air inside stabilized at $100 \mathrm{\C }$ (do not ask, how he managed that without water). Lukáš realized his mistake and removed the pot from the stove. When the pot had cooled down to the room temperature, however, he was unable to remove its lid with the area $S$ and mass $m$. Calculate the force with which the lid resisted being removed if Lukáš put the lit on the pot

- just before removing it from the stove and,
- before the start of dinner preparation.

Assume the air to be an ideal gas.

Lukáš and his culinary art.

### (8 points)4. Series 34. Year - 4. ants

The ants have a peculiar way of keeping the anthill warm – they crawl out, let the sunlight heat them up, and then crawl back in, where the heat is transferred to the anthill. The anthill can be approximated as a cone of height $H=0{,}8 \mathrm{m}$ with base radius of $R_0=1,5 \mathrm{m}$. The walls are made of cellulose with heat conductivity $\lambda = 0{,}039 \mathrm{W\cdot m^{-1}\cdot K^{-1}}$ and are $2 \mathrm{cm}$ thick.

Assume that the entire heat exchange between the anthill and its surroundings (which have temperature $T\_o = 10 \mathrm{\C }$) is only mediated by the ants and by the conduction of heat through the walls, i.e. neglect the heat exchange with the ground. An ant weighs $m =5 \mathrm{mg}$ and has a specific heat capacity of approximately $4~000 J.kg^{-1}.K^{-1}$. How many ants, heated up to $T\_m = 37 \mathrm{\C }$, have to enter the anthill every second in order to keep the inner volume of the anthill at constant temperature of $T\_M = 20 \mathrm{\C }$?

Káťa missed biology classes.

### (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.

### (12 points)3. Series 34. Year - E. diffusion

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?

Káťa enjoys labs even during the quarantine.