# Serial of year 29

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## Text of serial

### (6 points)1. Series 29. Year - S. Ideal Gas

• As a little warm-up, to help you understand the numbers we'll be using, try to find height to what should be an average person (70 kg), lifted in order to use up all the energy of a standard Mars bar ( 250 cal for 50 g bar). Determine also what is the energy equivalent to $k_{B}T$ at room temperature and express it in electronvolts (i.e. the unit of energy equivalent to the kinetic energy electron gains when accelerated at potential difference of 1 V. Explicitly 1 eV = 1,602 \cdot 10^{-19} J).
• The Ideal Gas Law can be modified in many ways. If you rewrite it using amount of substance, instead of number of particles, you get $$pV = n N_\;\mathrm{A} k_\mathrm{B} T\,,$$ where $N_{A}k_{B}$ together is labeled as $R$ and is called universal gas constant. Express its value. Then modify the equation once again using mass of the gas and third time into a form containing gas density.
• Evaluate the volume of a single mole of gas at room temperature. It is useful to remember this number.
• And finally, a small consideration. Notice, when we were discussing the work of ideal gas, we automatically reached for the inner gas pressure value. Try to reason this choice of pressure. We might be objecting we should use the surrounding pressure or even the pressure difference between the inner and outer pressure. $Evaluation$ of this section will be moderate, do not be afraid to write whatever you think of yourself..</a>

### (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.

### (6 points)3. Series 29. Year - S. serial

• All states of ideal gas can be shown on various diagrams: $pV$ diagram, $pT$ diagram and so on. The first quantity is shown is on vertical axis, the second on horizontal. Every point therefore determines 2 parameters. Sketch in a $pV$ diagram the 4 processes with ideal gas that you know. Do the same on a $Tp$ diagram. How would $UT$ diagram look like? Explain how would the unsuitability of these two variables appear on the diagram.
• What are the dimensions of entropy? What other quantities with the same dimensions do you know?
• In the text for this series we analysed a case of entropy increasing as heat flows into a gas. Perform a similar analysis for the case of heat flowing out of the gas.
• We know that entropy does not change during an adiabatic process. Therefore, the expression for entropy as a function of volume and pressure $S(p,V)$ can only contain a combination of pressure and volume that does not change during an adiabatic process.

What is this expression? Draw lines of constant entropy on a $pV$ diagram ($p$ on vertical axis, $V$ on horizontal). Does this agree with the expression for entropy we have derived?

• Express the entropy of an ideal gas as functions $S(p,V)$, $S(T,V)$ and $S(U,V)$.

### (6 points)4. Series 29. Year - S. serial

• From the inequality

$$\Delta S_{tot} \ge 0 }$$

and given the equation from the text of the serial

$$\Delta S_{tot} = \frac{-Q}{T_H} \frac{Q-W}{T_C}$$

express $W$ and derive this way the inequality for work

$$W\le Q\left( 1 - \frac {T_C}{T_H} \right).$$

• Calculate the efficiency of the Carnot cycle without the use of entropy.

Hint: Write out 4 equations connecting 4 vertices of the Carnot cycle

$$p_1 V_1 = p_2 V_2$$

$$p_2 V_2^{\kappa} = p_3V_3^{\kappa}$$