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## molecular physics

### (6 points)2. Series 29. Year - S. serial

• Which types of processes (isobaric, isochoric, isothermal and adiabatic) can be reversible?
• Take the relation

$T=\frac{pV}{nR}\$,,

where $n=1mol$, $p=100kPa$ and $V=22l$. How will $T$ change, if we change both $p$ and $V$ by 10$%$, by 1$%$ or by 0$,1%?$ Calculate it in two ways: precisely and by using the relation: $$\;\mathrm{d} T=T_{,p} \mathrm{d} p T_{,V} \mathrm{d} V .$$

What is the difference between the results?

• d gymnastics:
• Show that

$$\;\mathrm{d} (C f(x)) = C \mathrm{d} f(x)\,,$$

where $C$ is constant.

• Calculate

$$\;\mathrm{d} (x^2) \ \quad \mathrm{a} \quad \mathrm{d} (x^3).$$

• Show that

$$\;\mathrm{d}\left( \frac 1x \right)= -\frac {\mathrm{d} x}{x^2}$$

from the definition, that is $$\;\mathrm{d} \left(\frac 1x \right)= \frac {1}{x \mathrm{d} x} - \frac 1x$$

This might be handy: $(x \;\mathrm{d} x)(x-\mathrm{d}$ x) = x^2 - (\mathrm{d} x)^2 = x^2$\$,.

• *Bonus: $This$ holds $$\sin \;\mathrm{d} \vartheta = \mathrm{d} \vartheta \quad a \quad \cos \mathrm{d} \vartheta = 1.$$ And you have the addition formula as well $$\sin (\alpha \beta ) = \sin \alpha \cos \beta \cos \alpha \sin \beta,$$ Prove $$\;\mathrm{d}\left( \sin \vartheta \right)=\, \mathrm{d} \vartheta \cos \vartheta .$$ * Bonus:** Similarly show

$$\;\mathrm{d} \left(\ln x \right)= \frac{\mathrm{d}x}{x}$$

using $$\ln (1 \;\mathrm{d} x) = \mathrm{d} x$$

• Explain, why isobaric temperature is lower than isochoric.

### (8 points)4. Series 28. Year - E. bottled potential

We have a cylindrical container, in which we make a circular hole fromthe side. We shall pour water into. Water shall slowly flow out but at some height above the hole the outpouring of water shall stop. Determine the surface tension of water depending on the height at which it stopped. Repeat the experiment multiple times with three different openings. A plastic bottle would do as a cylinder.

Karel was inspired by what Vojta Zak said he was doing at the Physics seminar.

### (2 points)3. Series 28. Year - 2. bubbles

Determine the difference in surface energy of a spherical bubble and a bubble n the shape of a regular tetrahedron. Both shapes have the same inner volume $V$.

Karel remembered tetrahedric bubbles from Eureka!

### (2 points)2. Series 28. Year - 1. Saint Anne cold from the morning (this is unpunifiable)

In the cold morning mist you are leaving the house and the garden gate works in such a way that it will open after the handle is pushed down and after closing it and letting go of the handle it stays closed shut. You return in the afternoon and ask yourself who didn't close the goddamn gate? Then you try to close the gate but can't. The steel latch won't recede far enough to pass around the aluminum frame, even after pushing down the handle. The gate is also made out of aluminum. What seems to be the problem here? What did the manufacturer forget to think about? Determine the gate's parameters (width, length, breadth) when it is 20 °C, if we know that temperatures don't fall under −30 °C and doesn't rise over 50 °C.

Teresa was happy one morning when observing the evil work of physics.

### (8 points)2. Series 28. Year - E. waterfall(apart)

What is the depth under the tap where the stream of water divides into droplets? How does it depend on the the flow of water?

Lukas went bonkers (again).

### (4 points)6. Series 27. Year - 3. Sphere and shell

Consider a copper sphere and a copper hollow shell (so thin that one can neglect its thickness). Both have the same radius at room temperature. How shall the radius change if we begin warming them up? (Find the relation between the radius and the temperature and comment on it) With the copper shell think that it has small openings which ensure that the inside and outside pressure are both the same.

Karel was inspired by the book Physics for Scientists and Engineers by Serwaye & Jewetta.

### (2 points)4. Series 27. Year - 2. test tubes

Test tubes of volumes 3 ml and 5 ml are connected by a short thin tube in which we can find a porous thermally non-conductive barrier that allows an equilbirum in pressures to be achieved within the system. Both test tubes in the beginning are filled with oxygen at a pressure of 101,25 kPa and a temperature of 20 ° C. We submerge the first test tube (3 ml) into a container which has a system of water and ice in equilbrium inside it and the other one (5 ml) into a container with steam. What wil the pressure be in the system of the teo test tubes be after achieving mechanical equilibrium? What would the pressure be if it would have been nitrogen and not oxygen that was in the test tubes?(while keeping other conditions the same)/p>

Kiki dug up something from the archives of physical chemistry.

### (8 points)4. Series 27. Year - E. some like it lukewarm

Measure the relation between the temperature and time in a freshly made cup of tea. Conduct the measurements for a undisturbed cup of tea and a cup of tea stirred by a teaspoon. Finally determine if the time the tea takes to reach a drinkable temperature depends on the stirring.

Michal altered xkcd.

### (2 points)6. Series 26. Year - 1. disgusting water

Many years ago you drank 2 dcl of water. Imagine that since then all the water on the Earth has mixed. If you drink 2 dcl of water today, how many molecules from the original water you drank does it contain?

Karel is afraid of cholera.

### (6 points)6. Series 26. Year - S. series

• Calculate the time a tokamak COMPASS can store an energy for. The energy of its plasma is 5 kJ, and its ohmic heating is 300 kW.
• Calculate the alpha heating in tokamak COMPASS if it used a DT mixture. Typical plasma temperature is 1 keV, hustota 10^{20} m^{ − 3}, and the volume of the plasma 1 m. Assuming the ohmic heating from the preceeding question, calculate $Q$.
• Using the picture from the main text and knowledge of the DD reaction ^{2}_{1}D + ^{2}_{1}D → ^{3}_{2}He + n + 3,27 MeV (50 %),
{2}_{1}D + {2}_{1}D →

&frac34; energie v of the energy in the first reaction are carried off by a neutron, calculate the total plasma heating that will occure during one DD reaction (assume that it is followed by a DT fusion with the product of the second reaction). Also estimate the requirements on the confinment time assuming density odhadněte nároky na dobu udržení při hustotě 10^{20} m^{ − 3} a teplotě 10 keV.

Robin 