Suppose $0.5$ mole of an ideal gas undergoes an isothermal expansion as energy is added to it as heat $Q$. Graph shows the final volume $V_f$ versus $Q$. The temperature of the gas is :- (use $ln9=2$ and $R=\frac{25}{3} \mathrm{J}/\mathrm{mol}-\mathrm{K}$ )

(i) The internal energy of gas at $A$ is $10 \mathrm{J}$ and amount of heat supplied to change its state to $C$ through the path $AC$ is $200 \mathrm{J}$. Calculate the internal energy at $C$.

(ii) The internal energy of gas at $B$ is $20 \mathrm{J}$. Find the amount of heat supplied to the gas from $A$ to $B$.

(i) If work done by the gas is $1.8\times {10}^{\mathrm{y}} \mathrm{J}$ then find the value of $y$.

(ii) If increase in internal energy is $1.6z\times {10}^5 \mathrm{J}$ then find the value of $z$.

(iii) If amount of heat supplied is $6.6\times {10}^{\mathrm{x}} \mathrm{J}$ then find the value of $x$.

(iv) If molar specific heat of the gas is $1.71\times {10}^{\mathrm{\alpha }} \mathrm{J} \mathrm{mole}-{\mathrm{K}}^{-1}$ then find the value of $\alpha$.

$A\to B$ : Adiabatic expansion            $B\to C$ : Cooling at constant volume 

$C\to D$ : Adiabatic compression

$D\to A$ : Heating at constant volume

The pressure and temperature at $A, B$, etc., are denoted by $P_A,T_A,P_B,T_B$ etc., respectively. Given that $T_A=1000 \mathrm{K}$, $P_B=\left(\frac{2}{3}\right)P_A$ and $P_C=\left(\frac{1}{3}\right)P_A$.

Calculate the following quantities : (i) The work done by the gas in the process $A\to B$

(ii) The heat lost by the gas in the process $B\to C$

(iii) The temperature $T_D$. (Given : ${\left(\frac{2}{3}\right)}^{2/5}=0.85$)

At $27º \mathrm{C}$ two moles of an ideal monoatomic gas occupy a volume $\mathrm{V}$. The gas expands adiabatically to a volume $2\mathrm{V}$. Calculate:

(i) the final temperature of the gas,

(ii) change in its internal energy and 

(iii) the work done by the gas during the process.