Three different processes that can occur in an ideal monoatomic gas are shown in the \(P\) vs \(V\) diagram. The paths are labelled as \(A \rightarrow B, A \rightarrow C\) and \(A\rightarrow D\). The change in internal energies during these process are taken as \(E_{AB}\), \(E_{AC}\)  and \(E_{AD}\) and the work done as \(W_{A B}, W_{A C}\) and \( W_{A D}\). The correct relation between these parameters are:

 

1.	\(E_{A B}=E_{A C}=E_{A D}, W_{A B}>0, W_{A C}=0, W_{A D}<0\)
2.	\(E_{A B}<E_{A C}<E_{A D}, W_{A B}>0, W_{A C}>W_{A D}\)
3.	\(E_{A B}=E_{A C}<E_{A D},W_{A B}>0,W_{A C}=0, W_{A D}<0\)
4.	\(E_{A B}>E_{A C}>E_{A D}, \quad W_{A B}< W_{A C}< W_{A D}\)

In an adiabatic process, the density of a diatomic gas becomes \(32\) times its initial value. The final pressure of the gas is found to be \(n\) times the initial pressure. The value of \(n\) is:
1. \(326\)
2. \(\dfrac{1}{32}\)
3. \(32\)
4. \(128\)

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. \(504\)

\(n\) moles of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes.

Total work done in the complete cycle \(ABCA\) is: 

1. \(0\)
2. \(nRT(\ln 2-\frac{1}{2})\)
3. \(nRT\ln 2\)
4. \(nRT(\ln 2+\frac{1}{2})\)

Match List-I with List-II and choose the correct option:

The thermodynamic process is shown below on a \(P-V\) diagram for one mole of an ideal gas. If \(V_2=2V_1\) then the ratio of temperature \(T_2/T_1\) is:

  
1. \(\frac{1}{2}\)
2. \(2\)
3. \(\sqrt{2}\)
4. \(\frac{1}{\sqrt{2}}\)

If one mole of an ideal gas at \((p_1,V_1)\) is allowed to expand reversibly and isothermally \((A\rightarrow B),\) its pressure is reduced to one-half of the original pressure (see figure). This is followed by a constant-volume cooling till its pressure is reduced to one-fourth of the initial value (\((B\rightarrow C).\) Then it is restored to its initial state by a reversible adiabatic compression\((C\rightarrow A).\) The net work done by the gas is:

               
1. \(RT\left(\ln 2-\dfrac{1}{2(\gamma-1)}\right) \)
2. \(-\dfrac{RT}{2(\gamma-1)} \)
3. \(0\)
4. \(RT\ln2\)

A diatomic gas, having \(C_P=\dfrac{7}{2}R\) and \(C_V=\dfrac{5}{2}R\) is heated at constant pressure. The ratio of \(dU:dQ:dW\) is:
1. \(5:7:3\)
2. \(5:7:2\)
3. \(3:7:2\)
4. \(3:5:2\)

In a certain thermodynamical process, the pressure of a gas depends on its volume as \(kV^3.\) The work done when the temperature changes from \(100^\circ \text{C}\) to \(300^\circ \text{C}\) will be:
(where \(n\) denotes number of moles of a gas)

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.