Thermodynamics

$V=k{T}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$T{V}^{-\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=K$

$\Rightarrow \gamma -1=-3/2$

$\Rightarrow \gamma =-1/2$

Work done = $\frac{nR\Delta T}{-y+1}=\frac{1*R*90}{3/2}=6R$

n = 60

U = 3PV + 4

$\Rightarrow n{C}_{v}T=3PV+4$

$\Rightarrow n*\frac{fRT}{2}=3PV+4$

$\Rightarrow f=6+\frac{8}{PV}$

$\Rightarrow f>6\Rightarrow polyatomic$

n = 1 mole

$W_{AB}=nRTln\frac{v_2}{v_1}=1*RTln\frac{2v_1}{v_1}=RTln2$

$W_{BC}=0$

$W_{CA}=\frac{P_1{V}_1-P_2{V}_2}{\gamma -1}=\frac{\frac{P_1}{4}*2V_1-P_1*V_1}{\gamma }$

$\Rightarrow W_{CA}=\frac{-RT}{2\left(\gamma -1\right)}$

$\therefore W_{total}=RT\left[\mathcal{l}n2-\frac{1}{2\left(\gamma -1\right)}\right]$

P = kV³ Þ PV^-3 = k Þ Polytropic Process with index m = -3

W = $\frac{nR\left(T_2-T_1\right)}{1-m}=\frac{nR\left(300-100\right)}{1-\left(-3\right)}=50nR$ .

At constant pressure,

$dU=n{C}_{V}dT=n.\frac{5}{2}RdT$

$dQ=n{C}_{P}dT=n.\frac{7}{2}RdT$

dW = nRdT

dU : dQ : dW = 5 : 7 : 2

Since process is isochoric

So    $\Delta U=n{C}_v\Delta T$

$\Delta U=n\left(\frac{5}{2}R\right)\Delta T--(i)\left[C_V=\frac{5}{2}R\right]$      

And external work

$\Delta W=nR\Delta T--\left(ii\right)$    

$\frac{5}{2}=\frac{x}{10}\Rightarrow x=25.00$                

from newtan's law of cooling :

$\frac{dT}{dt}=-k\left(T-T_0\right)$

Using average value :

$\frac{75-65}{5}=-k\left[\frac{\left(75+65\right)2}{2}-25\right]$

$\frac{\left(65-T\right)}{5}=-k\left[\frac{\left(65+T\right)}{2}-25\right]$

$\Rightarrow \frac{10}{65-T}=\frac{45}{\frac{65+T}{2}-25}$

$\Rightarrow T=57°C$

Total energy of monoatomic gas at equilibrium

$=\frac{3}{2}{K}_{B}T$

From   $\Delta H=\Delta G+T\Delta S\therefore \Delta S=\frac{\Delta H-\Delta G}{T}$

$\therefore$ $\Delta S=\left[\frac{51.4-\left(-49.4\right)}{300}\right]*1000J{K}^{-1}mo{l}^{-1}=336J{K}^{-1}mo{l}^{-1}$

Since process is isothermal, so

$W=nRT\mathcal{l}n\left(\frac{V_f}{V_i}\right)=1*8.3*300*\mathcal{l}n\left(\frac{4}{2}\right)=1*8.3*300*\mathcal{l}n\left(2\right)$       

$\Rightarrow W=1*8.3*300*0.6931\approx 1725.82J\approx 17258*10^{-1}J$