– Heat engine efficiency:
\[
\eta=\frac{A}{Q_{1}}=1-\frac{Q_{i}}{Q_{1}} \text {. }
\]
shery $Q_{1}$ is the heat obtained by the working subancs, $Q_{1}$ is the beat released by the working substance.
– Eliciescy of a Carnot eycle:
\[
\eta=\frac{T_{1}-T_{1}}{T_{1}} \text {, }
\]
where $T_{1}$ and $T_{1}$ are the temperatures of the bet and cold bedies respectively.
– Clausius inequality:
\[
\oint \frac{\infty}{7}<0,
\]
stere $8 Q$ is the elementary amount of beat tranaferred to the system ( $8 Q$ is an alghiraic quantity).
– Entropy increment of a oystem:
\[
\Delta s>\int \frac{Q Q}{T} .
\]
– Foadamental relation of thermedynamic:
\[
T d S>d U+p d V .
\]
– Relation between the entropy asd the statistical weight of (the thermedynamic probability):
\[
s=\boldsymbol{k} \ln \mathrm{Q} .
\]
where $t$ is the Boltumans coestant.
2.113. In which case will the efficiency of a Carnot cycle be higher: when the hot body temperature is increased by $\Delta T$, or when the cold body temperature is decreased by the same magnitude?
2.114. Hydrogen is used in a Carnet cycle as a working substance. Find the efficiency of the cyele, if as a result of an adiabatic expansion
(a) the gas volume increases $n=2.0$ times;
(b) the pressure decreases $n=2.0$ times.
2.115. A heat engine employing a Carnot cycle with an efficiency of $\eta=10 \%$ is used as a refrigerating machine, the thermal reservoirs being the same. Find its refrigerating efficiency s.
2.116. An ideal gas goes through a cycle consisting of alternate isothermal and adiabatic curves (Fig. 2.2). The isothermal processes proceed at the temperatures $T_{1}, T_{8}$, and $T_{3}$. Find the efficiency of such a cycle, if in each isothermal expansion the gas volume increases in the same proportion.
2.117. Find the effieiency of a cyele consisting of two isocheric and two adiabatic lines, if the volume of the ideal gas changes $n=10$ times within the cycle. The working substance is nitrogen.
88
2.118. Find the efficiency of a cycle consisting of twe isobarie and two adiabatic lines, if the pressure changes $n$ times within the cycle. The working substance is an ideal gas whose adiabatic exponent is equal to $\%$.
2.119. An ideal gas whose adiabatic exponent equals $\gamma$ goes through a cycle consisting of two isochoric and two isobaric lines. Find the efficiency of such a cycle, if the absolute temperature of the gas rises $n$ times both in the isochoric heating and in the isobaric expansion.
2.120. An ideal gas goes through a cycle consisting of
(a) isochoric, adiabatic, and isothermal lines;
(b) isobaric, adiabatic, and isothermal lines.
with the isothermal process proceeding at the minimum temperature of the whole cycle. Find the efficiency of each cycle if the absolute temperature varies $n$-fold within the cycle.
Fig. 2.2 .
2.121. The conditions are the same as in the foregoing problem with the exception that the isothermal process proceeds at themaximum temperature of the whole cyele.
2.122. An ideal gas goes through a cycle consisting of isothermal. polytropic, and adiabatic lines, with the isothermal process proceeding at the maximum temperature of the whole cycle. Find the efficiency of such a cycle if the absolute temperature varies $n$-fold within the cycle.
2.123. An ideal gas with the adiabatic exponent $\gamma$ goes through a direct (clockwise) cycle consisting of adiabatic, isobaric, and isochorie lines. Find the efficiency of the cyele if in the adiabatic process the volume of the ideal gas
(a) increases $n$-fold; (b) decreases $n$-fold.
2.124. Calculate the efficiency of a cycle consisting of isothermal, isobaric, and isochoric lines, if in the isothermal process the volume of the ideal gas with the adiabatic exponent $\gamma$
(a) increases $n$-fold; (b) decreases $n$-fold.
2.125. Find the efficiency of a cyele consisting of two isochoric and twe isothermal lines if the volume varies v-fold and the absolute temperature $\tau$-fold within the cycle. The working substance is an ideal gas with the adiabatic exponent $\%$.
2.126. Find the efficiency of a cycle consisting of two isobaric and two isothermal lines if the pressure varies $n$-fold and the absolute temperature $\mathrm{r}$-fold within the cycle. The working substance is an ideal gas with the adiabatic exponent $\gamma$.
2.127. An ideal gas with the adiabatic exponent $\gamma$ goes through – cyele (Fig. 2.3) within which the absolute temperature varies t-fold. Find the efficiency of this cycle.
2.128. Making use of the Clausius inequality, demonstrate that all cycles having the same maximum temperature $T_{\max }$ and the same minimum temperature $T_{\text {min }}$ are less efficient compared to the Carnot eycle with the same $T_{\operatorname{mox}}$ and $T_{\text {min }}$.
2.129. Making use of the Carnot theorem, show that in the case of a physically uniform substance whose state is defined by the parameters $T$ and $V$
\[
(\partial U / \partial)_{r}=T(\partial p / \partial T)_{r}-p .
\]
where $U(T, V)$ is the internal energy of the substance.
Instruction. Consider the infinitesimal Carnot cycle in the variables p. $\boldsymbol{V}$.
2.130. Find the entropy increment of one mole of carbon dioxide when its absolute temperature increases $n=2.0$ times if the process of heating is
(a) isochorie; (b) isobaric.
The gas is to be regarded as ideal.
2.131. The entropy of $v=4.0$ moles of an ideal gas increases by $\Delta S=23 \mathrm{~J} / \mathrm{K}$ due to the isothermal expansion. How many times should the volume $v=4.0$ moles of the gas be increased?
2.132. Two moles of an ideal gas are cooled isochorically and then expanded isobarically to lower the gas temperature back to the initial value. Find the entropy increment of the gas if in this process the gas pressure changed $n=3.3$ times.
2.133. Helium of mass $m=1.7 \mathrm{~g}$ is expanded adiabatieally $n=3.0$ times and then compressed isobarically down to the initial volume. Find the entropy increment of the gas in this process.
2.134. Find the entropy increment of $v=2.0$ moles of an ideal gas whose adiabatic exponent $\gamma=1.30 \mathrm{if}$, as a result of a certain process, the gas volume increased $\alpha=2.0$ times while the pressure dropped $\beta=3.0$ times.
2.135. Vesels $I$ and 2 contain $y=1.2$ moles of gaseous helium. The ratio of the vessels’ volumes $V_{1} / V_{1}-a=2.0$, and the ratio of the absolute temperatures of helium in them $T_{1} / T_{2}=\beta=1.5$. Assuming the gas to be ideal, find the difference of gas entropies in these vessels, $s_{2}-s_{1}$.
2.136. One mole of an ideal gas with the adiabatic exponent $\gamma$ goes through a polytropic process as a result of which the absolute temperature of the gas increases v-fold. The polytropic constant equals $n$. Find the entropy increment of the gas in this process.
2.137. The expansion process of $v=2.0$ meles of argon proceeds $m$ that the gas pressure increases in direct proportion to its volume.
90
ne. 2.3 .
Find the entropy increment of the gas in this process provided its volume increases $a=2.0$ times.
2.138. An ideal gas with the adiabatic exponent $y$ goes through – process $p=p_{0}-a \boldsymbol{V}$, where $p_{0}$ and $a$ are positive constants, and $\boldsymbol{V}$ is the volume. At what volume will the gas entropy have the maximum value?
2.139. One mole of an ideal gas goes through a process in which the entropy of the gas changes with temperature $T$ as $S=a T+$ $+C_{V}$ In $T$, where $a$ is a positive constant, $C_{V}$ is the molar heat capacity of this gas at constant volume. Find the volume dependence of the gas temperature in this process if $T=T_{0}$, at $V=V_{0}$.
2.140. Find the entropy increment of one mole of a $V_{a n}$ der $W_{\text {aals }}$ gas due to the isothermal variation of volume from $V_{1}$ to $V_{\text {, }}$. The Van der Wals corrections are assumed to be known.
2.141. One mole of a Van der Waals gas which had initially the volume $V_{1}$ and the temperature $T_{1}$ was transferred to the state with the volume $\boldsymbol{V}_{1}$ and the temperature $\boldsymbol{r}_{2}$. Find the corresponding entropy increment of the gas, assuming its molar heat capacity $c_{v}$ to be known.
2.142. At very low temperatures the heat capacity of crystals is equal to $C=a T^{\circ}$, where $a$ is a constant. Find the entropy of a crystal as a function of temperature in this temperature interval.
2.143. Find the entropy increment of an aluminum bar of mass $m=3.0 \mathrm{~kg}$ on its heating from the temperature $T_{1}=300 \mathrm{~K}$ up to $T_{1}=600 \mathrm{~K}$ if in this temperature interval the specific heat capacity of aluminum varies as $c=a+b T$, where $a=0.77 \mathrm{~J} /(\mathrm{g} \cdot \mathrm{K})$, $b^{2}=0.46 \mathrm{~mJ} /\left(\mathrm{g} \cdot \mathrm{K}^{0}\right)$.
2.144. In some precess the temperature of a substance depends on its entropy $S$ as $F=a S^{*}$, where $a$ and $n$ are constants. Find the correspending heat capacity $C$ of the substance as a function of $S$. At what condition is $C<0$ ?
2.145. Find the temperature $T$ as a function of the entropy $S$ of a substance for a polytropic process in which the heat capacity of the substance equals $C$. The entropy of the substance is known to be egual to $S_{\text {g at }}$ at the temperature $T_{4}$. Draw the approximate plots $r(S)$ for $C>0$ and $C<0$.
2.146. One mole of an ideal gas with heat capacity $C_{V}$ goes through a process in which its entropy $S$ depends on $T$ as $S=a / T$, where $a$ is a constant. The gas temperature varies from $T_{1}$ to $T_{1}$. Find:
(a) the melar heat capacity of the gas as a function of its temperature;
(b) the amount of heat transferred to the gas;
(c) the work perlormed by the gas.
2.147. A working substance goes through a cycle within which the absolute temperature varies $n$-fold, and the shape of the cyele is shown in (a) Fig. 2.4a; (b) Fig. 2.4b, where $T$ is the absolute temperature, and $s$ the entropy. Find the efficiency of each eycle.
2.148. One of the two thermally insulated vessels interconnected by a tube with a valve contains $v=2.2$ moles of an ideal gas. The other vesel is evacuated. The valve having been opened, the gas increased its volume $n=3.0$ times. Find the entropy increment of the gas.
2.149. A weightless piston divides a thermally insulated cylinder inte two equal parts. One part contains one mele of an ideal gas with adiabatic exponent $\gamma$, the other is evacuated. The initial gas temperature is $T_{6}$. The piston is released and the gas fills the whole
Fig. 2.4.
volume of the cylinder. Then the piston is slowly displaced back to the initial position. Find the increment of the internal energy and the entropy of the gas resulting from these twe processes.
2.150. An ideal gas was expanded from the initial state to the volume $V$ without any heat exchange with the surrounding bodies. Will the final gas pressure be the same in the case of (a) a fast and in the ease of (b) a very slow expansion proces?
2.151. A thermally insulated vessel is partitioned into two parts so that the volume of one part is $n=2.0$ times greater than that of the other. The smaller part contains $v_{1}=0.30$ mole of nitrogen, and the zreater one $\mathrm{v}_{2}=0.70$ mole of exygen. The temperature of the gases is the same. A hole is punctured in the partition and the gases are mixed. Find the corresponding increment of the system’s entropy, assuming the gases to be ideal.
2.152. A piece of cepper of mass $m_{1}=300 \mathrm{~g}$ with initial temperature $t_{1}=97^{\circ} \mathrm{C}$ is placed into a calorimeter in which the water of mass $m_{1}=100 \mathrm{~g}$ is at a temperature $t_{1}=7{ }^{\circ} \mathrm{C}$. Find the entropy increment of the system by the moment the temperatures equalize. The heat capacity of the calorimeter itself is negligibly small.
2.153. Two identical thermally insulated vesels interconnected by a tube with a valve contain one mole of the same ideal gas each. The gas temperature in one vessel is equal to $T_{1}$ and in the other, $T_{2}$. The molar heat capacity of the gas of constant volume equals $c_{v}$. The valve having been opened, the gas comes to a new equilibrium state. Find the entropy increment $\Delta S$ of the gas. Demonstrate that $\Delta s>0$.
2.154. $N$ atoms of gaseous helium are enclosed in a cubic vessel of volume $1.0 \mathrm{~cm}^{3}$ at room temperature. Find:
92
(a) the probability of atems gathering in one half of the vessel;
(b) the approximate numerical value of $N$ ensuring the occurrence of this event within the time interval $t \approx 10^{10}$ years (the age of the Universe).
2.155. Find the statistical weight of the most probable distribution of $\boldsymbol{N}=10$ identical molecules over two halves of the cylinder’s voleme. Find alse the probability of such a distribution.
2.156. A vessel contains $N$ melecules of an ideal gas. Dividing mentally the vessel inte two halves $A$ and $B$, find the probability that the half $A$ contains $n$ molecules. Consider the cases when $N=5$ and $n=0,1,2,3,4,5$.
2.157. A vessel of volume $V_{0}$ contains $N$ melecules of an ideal gas. Find the probability of $n$ molecules getting inte a certain separated part of the vessel of volume V. Examine, in particular, the case $\boldsymbol{V}=\mathrm{V}_{4} / 2$.
2.158. An ideal gas is under standard cenditions. Find the diameter of the sphere within whose volume the relative fuctuation of the number of molecules inside such a sphere?
2.159. One mole of an ideal gas consisting of monatomic molecules is enclosed in a vessel at a temperature $T_{0}=300 \mathrm{~K}$. How many times and in what way will the statistical weight of this system (gas) vary if it is heated isochorically by $\Delta T-1.0 \mathrm{~K}$ ?