One mole of a diatomic ideal gas undergoes a cyclic process \(ABC\) as shown in figure. The process \(BC\) is adiabatic. The temperatures at \(A,B\) and \(C\) are \(400~\text{K},800~\text{K}\) and \(600~\text{K}\) respectively. Choose the correct statement:

Consider a spherical shell of radius \(R\) at temperature \(T\). The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume \({u}=\frac{U}{V}\propto T^4\) and \(P=\frac{1}{3}\left(\frac{U}{V}\right )\). If the shell now undergoes an adiabatic expansion the relation between \(T\) and \(R\) is:
1. \({T} \propto {e}^{-{R}} \)
2. \({T} \propto {e}^{-3 {R}} \)
3. \({T} \propto \frac{1}{{R}} \)
4. \({T} \propto \frac{1}{{R}^3}\)

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as \(V^q\), where \(V\) is the volume of the gas. The value of \(q\) is: \((\gamma =\frac{C_P}{C_V})\)
1. \( \frac{3 \gamma+5}{6} \)
2. \(\frac{3 \gamma-5}{6} \)
3. \(\frac{\gamma+1}{2} \)
4. \(\frac{\gamma-1}{2}\)

An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity \(C\) remains constant. If during this process the relation of pressure \(P\) and volume \(V\) is given by \(PV^n\) = constant, then \(n\) is given by: (here \(C_P\) and \(C_V\) are molar specific heat at constant pressure and constant volume, respectively) 
1. \( n =\dfrac{C_P}{C_V} \)
2. \(n =\dfrac{C-C_P}{C-C_V} \)
3. \(n =\dfrac{C_P-C}{C-C_V} \)
4. \(n =\dfrac{C-C_V}{C-C_P}\)

'\(n\)' moles of an ideal gas undergo a process \(A\rightarrow B\) as shown in the figure. The maximum temperature of the gas during the process will be:
                       
1. \( \frac{9 P_0 V_0}{4 n R} \)
2. \(\frac{3 P_0 V_0}{2 n R} \)
3. \(\frac{9 P_0 V_0}{2 n R} \)
4. \(\frac{9 P_0 V_0}{n R}\)

\(C_P\) and \(C_V\) are specific heats at constant pressure and constant volume respectively. It is observed that
\(C_P-C_V=a\) for hydrogen gas
\(C_P-C_V=b\) for nitrogen gas
The correct relation between \(a\) and \(b\) is:
1. \(a=\frac{1}{14} b\)
2. \(a= b\)
3. \(a=14b\)
4. \(a=28b\)