(1)

For adiabatic process

$T{V}^{\gamma -1}=\text{constant}$

$T_a·V_a^{\gamma -1}=T_d·V^{\gamma -1}$

${\left(\frac{V_a}{V_d}\right)}^{\gamma -1}=\frac{T_d}{T_a}$                                ...(i)

$T_b·V_b^{\gamma -1}=T_c·V_c^{\gamma -1}$

${\left(\frac{V_b}{V_c}\right)}^{\gamma -1}=\frac{T_c}{T_b}$                           ...(ii)

As, $T_d=T_c$ and $T_a=T_b$$\Rightarrow \frac{V_a}{V_d}=\frac{V_b}{V_c}$

(2)

For A to B:

Given $P{V}^3=RT\Rightarrow P=\frac{RT}{V^3}$

Work done $W_{AB}=\int PdV$

$W_{AB}=\int \frac{RT}{V^3}dV=RT{\left[\frac{{\displaystyle V^{-2}}}{{\displaystyle -2}}\right]}_{V_A=2}^{V_B=4}$

$=-\frac{RT}{2}\left[\frac{{\displaystyle 1}}{{\displaystyle V^2}}\right]=-\frac{RT}{2}[\frac{{\displaystyle 1}}{{\displaystyle 16}}-\frac{{\displaystyle 1}}{{\displaystyle 4}}]$

$=-\frac{RT}{2}\left(\frac{{\displaystyle -3}}{{\displaystyle 16}}\right)=\frac{3RT}{32}$

$\Rightarrow W_{AB}=\frac{3}{32}\times 8\times 300=225\text{ J}$

$W_{BC}$ (Isobaric process)

$W_{BC}=P\Delta V=10(2-4)=-20\text{ J}$

$W_{AC}$ (Isochoric process) = 0

$W_{\text{net}}=W_{AB}+W_{BC}+W_{CA}=225-20+0=205\text{ J}$

(3)

Work done $AB=\frac{1}{2}(8000+4000){\text{ Dyne/cm}}^2\times 4{\text{ m}}^3$

                          $=\left(6000{\text{ Dyne/cm}}^2\right)\times 4{\text{ m}}^3$

Work done $BC=\left(4000{\text{ Dyne/cm}}^2\right)\times 4{\text{ m}}^3$

Total work done $=2000{\text{ Dyne/cm}}^2\times 4{\text{ m}}^3$

         $=2\times {10}^3\times \frac{1}{{10}^5}\frac{N}{{\text{cm}}^2}\times 4{\text{ m}}^3$

         $=2\times {10}^{-2}\times \frac{N}{{10}^{-4}{\text{ m}}^2}\times 4{\text{ m}}^3$

          $=2\times {10}^2\times 4\text{ Nm}=800\text{ J}$

(1)

Slope of isothermal < Slope of adiabatic

${\left(\mathrm{Slope}\right)}_A<{\left(\mathrm{Slope}\right)}_B$

Process B $\to$ adiabatic $\to$ $P{V}^{\gamma }=K$

Process A $\to$isothermal$\to$$PV=K$