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.

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. \(200\)

\(C_P\) 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. \(a=28b\)

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)

'\(n\)' 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. \(10R\)