'\(n\)' 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. \(\frac{9 P_0 V_0}{n R}\)

The volume \(V\) of a given mass of monoatomic gas changes with temperature \(T\) according to the relation \(\mathrm{V}=\mathrm{K} \mathrm{T}^{2 / 3}\). The work done when temperature changes by \(90\) K will be:
1. \(60R\)
2. \(30R\)
3. \(20R\)
4. \(10R\)

One mole of a rigid diatomic gas performs a work of \(\dfrac{Q}{5}\) when heat \(Q\) is supplied to it. The molar heat capacity of the gas during this transformation is:
1. \(\dfrac{15}{8} R\)

2. \(\dfrac{5}{8} R\)

3. \(\dfrac{25}{8} R\)

4. \(\dfrac{5}{7} R\)

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 \({u}=\dfrac{U}{V}\propto T^4\) and \(P=\dfrac{1}{3}\left(\dfrac{U}{V}\right ) \). 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. \({T} \propto \dfrac{1}{{R}} \)
4. \({T} \propto \dfrac{1}{{R}^3}\)

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. \(\frac{\gamma-1}{2}\)

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:

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)

\(C_P\) 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. \(a=28b\)

The change in the magnitude of the volume of an ideal gas when a small additional pressure \(\Delta P\) is applied at constant temperature, is the same as the change when the temperature is reduced by a small quantity \(\Delta T\) at constant pressure. The initial temperature and pressure of the gas were \(300~\text{K}\) and \(2~\text{atm}\). respectively. If \(|\Delta T|=C|\Delta P|\) then the value of \(C\) in ( K/atm) is:
1. \(50\)
2. \(100\)
3. \(150\)
4. \(200\)

Match the thermodynamic processes taking place in a system with the correct conditions. In the table: \(\Delta Q\) is the heat supplied, \(\Delta W\) is the work done and \(\Delta U\) is change in internal energy of the system.