<h4>Statement</h4>
<p>
$5.9.10^*$
Find the efficience of cycles shown in following figures, if as power agent it is taken a monoatomic ideal gas.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" alt="5.9.10" width="350" />
<figcaption>
For problem 5.9.10
</figcaption>
</figure>
</center>
<h4>Solution</h4>
<p class="TxtSolutions">
a) Let enumerate the states from 1 to 4 in clockwise sense starting in the coordinate $(P_1,V_1)$. The efficience is defined as
</p>
<center>
$\eta = 1-\frac{|Q_c|}{Q_h}=1-\frac{|Q_{34}|}{Q_{12}} \\;(1)$
</center>
<p class="TxtSolutions">
where $Q_{34}$ is the heat released by gas (cold reservoir) and $Q_{12}$ is the heat absorbed by gas (hot reservoir), while $Q_{23}$ and $Q_{41}$ are zero (adiabatic processes). Since $3-4$ is an isochoric process,
</p>
<center>
$Q_{34}=\Delta U_{34} = nC_v\left(T_4-T_3\right) \\;(2)$
</center>
<p class="TxtSolutions">
From Poisson Law
</p>
<center>
$P_1V_1^\gamma = P_4V_2^\gamma$
</center>
<p class="TxtSolutions">
and the state 4 is characterized by
</p>
<center>
$P_4V_2=nRT_4$
</center>
<p class="TxtSolutions">
So, separating $P_4$ and putting it into above Poisson law
</p>
<center>
$T_4 = \frac{P_1V_1^\gamma}{nRV_2^{\gamma-1}} \\;(3)$
</center>
<p class="TxtSolutions">
Proceeding similarly, $T_3$ can be found. Applying Poisson Law for process $2-3$,
</p>
<center>
$P_2V_1^\gamma=P_3V_2^\gamma$
</center>
<p class="TxtSolutions">
State equation for state 3 is
</p>
<center>
$P_3V_2=nRT_3$
</center>
<p class="TxtSolutions">
Separating $P_3$ and substituting into respective Poisson Law
</p>
<center>
$T_3 = \frac{P_2V_1^\gamma}{nRV_2^{\gamma-1}} \\;(4)$
</center>
<p class="TxtSolutions">
Substituting $(3)$ and $(4)$ into $(2)$, and as $C_v=iR/2$
</p>
<center>
$Q_{34} = \frac{i}{2}\frac{V_1^\gamma}{V_2^{\gamma-1}}\left(P_1-P_2\right) \\;(5)$
</center>
<p class="TxtSolutions">
For the process $1-2$
</p>
<center>
$Q_{12}=\Delta U_{12} = nC_v\left(T_2-T_1\right)$
</center>
<p class="TxtSolutions">
Taking account the state equations for states 1 and 2, temperatures can be found,
</p>
<center>
$Q_{12}=nC_v\left(\frac{P_2V_1}{nR}-\frac{P_1V_1}{nR}\right)$
</center>
<center>
$Q_{12}=\frac{i}{2}V_1\left(P_2-P_1\right) \\;(6)$
</center>
<p class="TxtSolutions">
Putting $(5)$ and $(6)$ into $(1)$
</p>
<center>
$\eta = 1-\left(\frac{V_1}{V_2}\right)^{\gamma-1}$
</center>
<p class="TxtSolutions">
Since $\gamma=C_p/C_v=1+2/i$ and $i=3$ (monoatomic gas),
</p>
<center>
$\boxed{\eta = 1-\left(\frac{V_1}{V_2}\right)^{\frac{2}{3}}}$
</center>
<p class="TxtSolutions">
b) Let enumerate again the states following a clockwise way, this time starting in coordinate $(P_1,T_1)$. The efficience is
</p>
<center>
$\eta = 1-\frac{|Q_c|}{Q_h} \\;(I)$
</center>
<p class="TxtSolutions">
For identifying correctly the reservoirs, let's see the following graph corresponding to the cycle in a $P$ vs $V$ plot.
</p>
<center>
<figure>
<img src="PVb.png"
loading="lazy" alt="5.9.10" width="300" />
<figcaption>
$P$ vs $V$ plot
</figcaption>
</figure>
</center>
<p class="TxtSolutions">
The hot reservoir is defined as
</p>
<center>
$Q_h = Q_{23} + Q_{34} \\;(II)$
</center>
<p class="TxtSolutions">
and cold one
</p>
<center>
$Q_c = Q_{12} + Q_{41} \\;(III)$
</center>
<p class="TxtSolutions">
Process $2-3$ is isobaric, then
</p>
<center>
$Q_{23}=nC_p\left(T_2-T_1\right) \\;(IV)$
</center>
<p class="TxtSolutions">
Process $3-4$ is isothermal, so
</p>
<center>
$Q_{34} = A_{34} = nRT_2\ln{\frac{V_4}{V_3}}$
</center>
<p class="TxtSolutions">
Since $V=nRT/P$ according state equation
</p>
<center>
$Q_{34} = nRT_2\ln{\frac{P_2}{P_1}} \\;(V)$
</center>
<p class="TxtSolutions">
Substituting $(IV)$ and $(V)$ into $(II)$and taking account that $C_p=(i+2)R/2$
</p>
<center>
$Q_h = \frac{i+2}{2}nR\left(T_2-T_1\right)+nRT_2\ln{\frac{P_2}{P_1}} \\;(VI)$
</center>
<p class="TxtSolutions">
Process $1-2$ is isothermal, so
</p>
<center>
$Q_{12}=A_{12}=nRT_1\ln{\frac{V_2}{V_1}}$
</center>
<p class="TxtSolutions">
and newly, taking account state equation
</p>
<center>
$Q_{12}=A_{12}=nRT_1\ln{\frac{P_1}{P_2} \\;(VII)}$
</center>
<p class="TxtSolutions">
Finally, process $4-1$ is isobaric
</p>
<center>
$Q_{14} = nC_p\left(T_1-T_2\right) \\;(VIII)$
</center>
<p class="TxtSolutions">
Substituting $(VII)$ and $(VIII)$ into $(III)$
</p>
<center>
$Q_c = nRT_1\ln{\frac{P_1}{P_2}}+\frac{i+2}{2}nR\left(T_1-T_2\right) \\;(IX)$
</center>
<p class="TxtSolutions">
Substituting $(VI)$ and $(IX)$ into $(I)$ and developing algebraically
</p>
<center>
$\eta = \frac{2\left(T_2-T_1\right)\ln{\frac{P_2}{P_1}}}{(i+2)\left(T_2-T_1\right)+2T_2\ln{\frac{P_2}{P_1}}}$
</center>
<p class="TxtSolutions">
Since gas is monoatomic, $i = 3$, hence
</p>
<center>
$\boxed{\eta = \frac{2\left(T_2-T_1\right)\ln{\frac{P_2}{P_1}}}{5\left(T_2-T_1\right)+2T_2\ln{\frac{P_2}{P_1}}}}$
</center>
<p class="TxtSolutions" style="text-align: right; font-style: italic; font-size: 14;">
BSc. Luis Daniel Fernández Quintana<br>
Physics Department (FCNE)<br>
Universidad de Oriente, Cuba<br>
</p>
</article>