$A$ gas may expand either adiabatically or isothermally. $A$ number of $P-V$ curves are drawn for the two processes over different ranges of pressure and volume. It will be found that:

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.

Process	Condition
$(I)$ Adiabatic	$(A) \Delta W = 0$
$(II)$ Isothermal	$(B) \Delta Q = 0$
$(III)$ Isochoric	$(C) \Delta U \neq 0, \Delta W \neq 0, \Delta Q \neq 0$
$(IV)$ Isobaric	$(D) \Delta U = 0$

An ideal gas is subjected to a cyclic process $ABCD$ as depicted in the $p-V$ diagram given below. Which of the following curves represents the equivalent cyclic process?

$1\,g$ of a liquid is converted to vapour at $3 \times 10^5\,Pa$ pressure. If $10\%$ of the heat supplied is used for increasing the volume by $1600\,cm^3$ during this phase change,then the increase in internal energy in the process will be $............\,J$.

$A$ small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in the figure. When the distance of the piston from the closed end is $L = L_0$, the particle speed is $v = v_0$. The piston is moved inward at a very low speed $V$ such that $V \ll \frac{dL}{L} v_0$, where $dL$ is the infinitely small displacement of the piston. Which of the following statement(s) is/are correct?
$(1)$ The rate at which the particle strikes the piston is $v / (2L)$
$(2)$ After each collision with the piston, the particle speed increases by $2V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_0$ to $L_0 / 2$
$(4)$ If the piston moves inward by $dL$, the particle speed increases by $v \frac{dL}{L}$

One mole of an ideal gas undergoes a cyclic process,consisting of two isochores and two isobars. Temperatures at points $1$ and $3$ are $T_1$ and $T_3$ respectively. Find the work done by the gas over the cycle,if points $2$ and $4$ lie on the same isotherm.