A monoatomic ideal gas, initially at temperature \(T_1\), is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \(T_2\) by releasing the piston suddenly. If \(L_1\) and \(L_2\) are the lengths of the gas column before and after expansion, respectively, then \(\frac{T_1}{T_2}\) is given by:
1. \(\left(\frac{L_1}{L_2}\right)^{\frac{2}{3}}\)
2. \(\frac{L_1}{L_2}\)
3. \(\frac{L_2}{L_1}\)
4. \(\left(\frac{L_2}{L_1}\right)^{\frac{2}{3}}\)

The initial pressure and volume of a gas are \(P\) and \(V\), respectively. First, it is expanded isothermally to volume \(4V\) and then compressed adiabatically to volume \(V\). The final pressure of the gas will be: [Given: \(\gamma = 1.5\)]

If \(n\) moles of an ideal gas is heated at a constant pressure from \(50^\circ\text C\) to \(100^\circ\text C,\) the increase in the internal energy of the gas will be:
\(\left(\frac{C_{p}}{C_{v}} = \gamma\   ~\text{and}~\   R = \text{gas constant}\right)\)

The latent heat of vaporisation of water is \(2240~\text{J/gm}\). If the work done in the process of expansion of \(1~\text{g}\) is \(168~\text{J}\), then the increase in internal energy is:
1. \(2408~\text{J}\)
2. \(2240~\text{J}\)
3. \(2072~\text{J}\)
4. \(1904~\text{J}\)

The figure below shows two paths that may be taken by a gas to go from state A to state C. In process AB, \(400~\text{J}\) of heat is added to the system and in process BC, \(100~\text{J}\) of heat is added to the system. The heat absorbed by the system in the process AC will be:

The first law of thermodynamics is based on:

The efficiency of an ideal heat engine is less than \(100\%\) because of:

Work done during the given cycle is:
                               
1. 4$P_0{V}_0$

2. 2$P_0{V}_0$

3. $\frac{1}{2}$$P_0{V}_0$

4. $P_0{V}_0$

An ideal gas goes from \(A\) to \(B\) via two processes, \(\mathrm{I}\) and \(\mathrm{II},\) as shown. If \(\Delta U_1\) and \(\Delta U_2\) are the changes in internal energies in processes \(\mathrm{I}\) and \(\mathrm{II},\) respectively,  (\(P:\) pressure, \(V:\) volume) then: