In the given $P-V$ diagram,a monoatomic gas $\left(\gamma = \frac{5}{3}\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$. [Given: $\left(\frac{1}{3}\right)^{0.6} \simeq 0.5, \ln 2 \simeq 0.7$].
Which of the following statement$(s)$ is(are) correct?
$(A)$ The magnitude of the total work done in the process $A \rightarrow B \rightarrow C$ is $144 \text{ kJ}$.
$(B)$ The magnitude of the work done in the process $B \rightarrow C$ is $84 \text{ kJ}$.
$(C)$ The magnitude of the work done in the process $A \rightarrow B$ is $60 \text{ kJ}$.
$(D)$ The magnitude of the work done in the process $C \rightarrow A$ is zero.

$A$ cyclic process $ABCD$ is shown in the given $P-V$ diagram. Which of the following diagrams represents the same process in a $P-T$ graph?

Which of the following statement$(s)$ is/are true? "Internal energy of an ideal gas ..........."

When a monatomic gas expands at constant pressure,the percentages of heat supplied that is used to do external work and to increase its internal energy are respectively:

$A$ 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 $v \frac{dL}{L}$

Two gases $A$ and $B$ have the same initial state $(P, V, n, T)$. Gas $A$ is compressed to $V/8$ by an isothermal process,and gas $B$ is compressed to $V/8$ by an adiabatic process. The ratio of the final pressure of gas $A$ to that of gas $B$ is (Both gases are monoatomic,$\gamma = 5/3$).