An ideal gas is taken from state $1$ to state $2$ through optional path $A, B, C \& D$ as shown in $P-V$ diagram. Let $Q, W$ and $U$ represent the heat supplied, work done $\&$ internal energy of the gas respectively. Then

One mole of an ideal gas at initial temperature $T$, undergoes a quasi-static process during which the volume $V$ is doubled. During the process, the internal energy $U$ obeys the equation $U=a V^3$, where $a$ is a constant. The work done during this process is

What is the nature of change in internal energy in the following three thermodynamical processes shown in figure

Graphs below show the entropy versus energy $U$ of two systems $1$ and $2$ at constant volume. The initial energies of the systems are indicated by $U_{1, i}$ and $U_{2, i}$, respectively. Graphs are drawn to the same scale. The systems are then brought into thermal contact with each other. Assume that, at all times the combined energy of the two systems remains constant. Choose the most appropriate option indicating the energies of the two systems and the total entropy after they achieve the equilibrium.

One mole of an ideal diatomic gas undergoes a transition from $A$ to $B$ along a path $AB$ as shown in the figure. 

The change in internal energy of the gas during the transition is …………$\;kJ$