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. \(\frac{\gamma-1}{2}\)

'\(n\)' 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 given diagram shows four processes i.e., isochoric, isobaric, isothermal and adiabatic. The correct assignment of the processes, in the same order, is given by:

   
1. \(d,~a,~b,~c\)
2. \(a,~d,~c,~b\)
3. \(a,~d,~b,~c\)
4. \(d,~a,~c,~b\)

An engine takes in \(5\) moles of air at \(20^\circ\) C and \(1\) atm, and compresses it adiabaticaly to \(1/10\) th of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be:
1. \(46~ \text{J}\)
2. \(46~\text{kJ}\)
3. \(23~\text{J}\)
4. \(23~\text{kJ}\)

A balloon filled with helium (\(32^\circ \mathrm{C}\) and \(1.7\) atm) bursts. Immediately afterwards the expansion of helium can be considered as:

A closed vessel contains \(0.1\) mole of a monatomic ideal gas at \(200\) K. If \(0.05\) mole of the same gas at \(400\) K is added to it, the final equilibrium temperature (in K) of the gas in the vessel will be closed to:
1. \(133\)
2. \(57\)
3. \(266\)
4. \(504\)

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.

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. \(200\)