Serial of year 29

• 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>

• 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.

• 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 $S(U,V)$.

• 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

$$p_1 V_1 = p_2 V_2 $$

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

$$p_3V_3 = p_4V_4$ p_4V_4^{\kappa} = p_1V_1^{\kappa}$

and multiply all of them together. By modifying this equation you should be able to get

$$\frac {V_2}{V_1} = \frac {V_3}{V_4}.$$

Next step is using the equation for the work done in an isothermal process: when going from the volume $V_{A}$ to the volume $V_{B}$, the work done on a gas is

$nRT\,\;\mathrm{ln}\left(\frac{V_A}{V_B}\right)$.

Now the last thing we need to realize is that the work in an isothermal process is equal to the heat (with the correct sign) a calculate the work done by the gas (there is no contribution from the adiabatic processes) and the heat taken away.

$ For the correct solution, you only need to fill in the details.$

• In the last problem you worked with $pV$ and $Tp$ diagram. Do the same with $TS$ diagram, i. e. sketch there the isothermal, isobaric, isochoric and adiabatic process. In addition sketch the path for the Carnot cycle including the direction and labeling of the individual processes.

• Sometimes it is important to check if we give or receive heat. Because sometimes this fact can change during the process. One of the examples is the process

$p=p_0\;\mathrm{e}^{-\frac{V}{V_0}}$,

where $p_{0}$ and $V_{0}$ are constants. Show for which values of $V$ (during the expansion) the heat is going into the gas and for which out of it.

• 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 $Ω=F-μN$.

• 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

$$\frac{\partial f_{,T}(T, V)}{\partial V} = \frac{\partial f_{,V}(T, V)}{\partial T} $$