Question

One mole of a monatomic ideal gas is taken through the cycle shown in the figure. A $\to$ B adiabatic expansionB $\to$ C cooling at constant volumeC $\to$ D adiabatic compressionD $\to$ A heating at constant volumeThe pressures and temperature at A, B etc., are denoted by $P_A$, $T_A$, $P_B$, $T_B$, etc., respectively. Given $T_A$ = 1000 K, $P_B$ = $\left(\frac{2}{3}\right)P_A$ and $P_B$ = $\left(\frac{1}{3}\right)P_A$ and $P_C$ = ($\frac{1}{3})P_A$. Then choose the incorrect option.

• A. The work done by the gas in the process A $\to$ B is 1869.75 J.
• B. The heat lost by the gas in the process B $\to$ C is -5297.625 J.
• C. Temperature $T_D$ is 500 K.
• D. Work done from B $\to$ C is 40 J.

Answer 1

The correct option is D Work done from B $\to$ C is 40 J.

Given that

$T_A=1000Kn=1$

$P_B=\left(\frac{2}{3}\right)P_A,\lambda =\frac{5}{3}$

$P_C=\left(\frac{1}{3}\right)P_A,{\left(\frac{2}{3}\right)}^{2/5}=0.85$

For Process $A\to B$

$P_A^{1-y}{T}_A^y=P_B^{1-y}{T}_B^y$

$T_B=T_A{\left(\frac{P_A}{P_B}\right)}^{\frac{1-\lambda }{\lambda }}=1000\times {\left(\frac{3}{2}\right)}^{-2/5}$

$=1000\times 0.85=850K$

For process $B\to C$

$\frac{P_B}{T_B}=\frac{P_C}{P_c}\Rightarrow T_c=T_B\left(\frac{P_C}{P_B}\right)=850\times \frac{1}{2}=425K$

(i) $W_{A\to B}=\frac{nR(T_A-T_B)}{\lambda -1}$

$=\frac{1\times 8.134\times (1000-850)}{\frac{5}{3}-1}=1870.2J$

(ii) $\Delta Q_{B\to C}=\Delta U_{B\to C}+\Delta W_{B\to C}=nCv\Delta T+0$

$=\frac{3}{2}R(T_C-T_B)=1\times \frac{3}{2}\times 8.314(425-850)$

$=-5300.175J$

(ii) For process $A\to B{P}_A{V}_A^{\gamma }=P_B{V}_B^{\gamma }\Rightarrow \frac{V_B}{(V_A{)}^{\gamma }}=\frac{P_A}{P_B}=\frac{3}{2}$

For process $C\to D{P}_C{V}_C^{\gamma }=P_D{V}_D^{\gamma }$

${\left(\frac{V_C}{V_D}\right)}^{\gamma }={\left(\frac{V_B}{V_A}\right)}^{\gamma }=\frac{3}{2}=\frac{P_D}{P_C}\Rightarrow P_D=\frac{3}{2}{P}_C$

At end points $A$ and $D$ $\Rightarrow \frac{P_A}{T_A}=\frac{P_D}{T_D}\Rightarrow =\frac{3P_C}{1000}=\frac{\frac{3P_C}{2}}{T_D}\Rightarrow T_D=500K$

We can see, Work done from B $\to$ C is $0$.

Hence, wrong statement given in the $option\left(D\right)$.
So, the correct option is $\left(D\right)$