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## thermodynamics

### (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.

### (7 points)2. Series 34. Year - 4. lifting ice using heat

A man stores small ice blocks in a well $h = 4,2 \mathrm{m}$ deep. To lift the ice up, he uses a heat engine between ice and the surrounding air with efficiency $\eta =12\%$ of the respective Carnot engine. The temperature of available air is $T\_{air}=24 \mathrm{\C }$. How cold must the ice be at the beggining in order to retrieve it with a final temperature $T\_{max}=-9 \mathrm{\C }$? How is it possible even when we heat the ice up in the process?

Karel likes bizzare engines.

### (9 points)2. Series 34. Year - P. costly ice hockey

Estimate how much the complete glaciation of an ice hockey rink costs.

Danka doesn't like ice hockey, but she likes figure skating.

### (10 points)1. Series 34. Year - P. Will we survive in vacuum?

Different movies create different conceptions of what and how fast happens when an astronaut's space suit suddenly gets torn. Some of them are even contradictory. Explain what is most likely to happen, if a healthy person finds himself unprotected in a vacuum. What phenomenon is most likely to cause death first?

Kuba planned to travel the world.

### (10 points)6. Series 33. Year - S.

*We are sorry. This type of task is not translated to English.*

### (12 points)5. Series 33. Year - E. if Jáchym don't oil, Matěj will oil

Measure the time dependence of the temperature of a liquid in an open mug. Use water first, than oil and finally water with a thin layer of oil. The layer should be as thin as possible but still should cover the whole surface. Measure between $90 \mathrm{\C }$ and $50 \mathrm{\C }$. Be careful to keep all conditions same for all experiments (the same mug, the same initial temperature, keep the thermometer on the same place in the liquid etc.). Describe your experimental equipment, compare cooling in individual cases and discuss the result.

Karel ate a bowl of steamy soup in tropically hot weather.

### (10 points)5. Series 33. Year - S. min and max

*We are sorry. This type of task is not translated to English.*

They had to wait a lot for Karel.

### (3 points)4. Series 33. Year - 2. Mach number

Planes at high flight levels are controlled using the Mach number. This unit describes velocity as a multiple of the speed of sound in the given environment. However, the speed of sound changes with height. What is the difference in the speed of a plane, flying at Mach number $0{,}85$, at two different flight levels FL 250 ($7\;600 \mathrm{m}$) and FL 430 ($13\;100 \mathrm{m}$)? At which flight level is the speed higher and by how much (in $\jd {kph}$)? The speed of sound is given by $c =\(331{,}57+0{,}607\left \lbrace t \right \rbrace \) \jd {m.s^{-1}}$, where $t$ is temperature in degrees Celsius. Assume a standard atmosphere, where temperature decreases with height from $15 \mathrm{\C }$ by $0,65 \mathrm{\C }$ per $100 \mathrm{m}$ (for heights between $0$ and $11 \mathrm{km}$) till $-56{,}5 \mathrm{\C }$, and then remains constant till $20 \mathrm{km}$ above mean sea level.

Karel was learning Air Traffic Control.